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Classical and quantum traces coming from SLₙ(ℂ) and U_q(slₙ)

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Classical and quantum traces coming from SLₙ(ℂ) and U_q(slₙ)

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Content CLASSICAL AND QUANTUM TRACES COMING FROM SL n (C) AND U q (sl n ) by Daniel Charles Douglas A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) August 2020 Copyright 2020 Daniel C. Douglas Dedication To my sister, Karen Loranne Douglas, my mother Linda Olson Douglas, and my father Michael Charles Douglas; in memory of my paternal grandfather, Charles Neathery Douglas, and my maternal grandmother, Loretta Ann Olson, both of whom always encouraged me to forge my own path; and, in memory of Cheri Kaiser, who always believed in me. ii Acknowledgements First and foremost, this thesis would not have been possible without the seemingly endless supply of time and energy committed by my advisor Francis Bonahon, whose passion for mathematics, education, pretty pictures, and \applied dierential geometry" is a gift to all. I would like to profoundly thank Charlie Frohman for his guidance and enthusiasm, without whom the second part of this thesis would not have been possible. Survival may not have been assured without the unwavering support of G uher and my family, as well as my old friends Alec, Benji and Kyle, Catherine, Dave, Jesse, Laura and Paul, and Noah. My collaborators Zhe Sun and Dylan Allegretti generously gave me the opportunity to work on interesting and challenging problems, for which I am extremely grateful. Special thanks goes out to my undergraduate mentors Ezra Getzler and Mike Stein, who got me started. I would also like to thank my committee members Aaron Lauda, Rajiv Kalia, Fedor Malikov, and Sheel Ganatra. Many thanks to Tommaso Cremaschi and Mareike Pfeil for reading some sections of this thesis and for oering their very helpful feedback. During my educational years, I have had the great opportunity to interact with and get to know many incredible people, including friends, teachers, mentors, and colleagues. The list is long, and I have inevitably missed some names. To those people, I oer my sincere apologies and also my immense gratitude as well. iii That being said, many thanks go out to: my academic siblings, Guillaume Dreyer, Giuseppe Martone, Jihoon Sohn, and Dong Zhang; USC faculty, Eric Friedlander, Su- san Montgomery, and Neelesh Tiruviluamala; non-USC faculty, Philip Boalch, Fran cois Costantino, Tudor Dimofte, Vladimir Fock, Jim Hoste, Yi Huang, David Jordan, Rick Kenyon, Greg Kuperberg, Thang L^ e, Yi Liu, Greg McShane, Blake Mellor, Adam Sikora, Dylan Thurston, Li-Sheng Tseng, Harold Williams, Helen Wong, and Ying Zhang; USC administrators, Arnold Deal, Michael Shields, Adriana Del Villar, Chaunt e Williams, and Amy Yung; pre-graduate teachers and mentors, Erling Antony, Michael Greenblatt, Bill Halperin, Cheri Kaiser, Jens Koch, Fred Rasio, Heidi Schellman, Steve Urban, Dave Wa- try, and Andrew Zuercher; the \Stein 4-group", Ian Coley, Daniel Kaplan, and Andrew Srisuwananukorn; \Bonahon visitors", Hongtaek Jung, Filippo Mazzoli, Mareike Pfeil, and Hatice Zeybek; participants for \that seminar with the really long name", Ross Akhmechet, Aaron Calderon, Mark Greeneld, Charles Ouyang, Nancy Scherich, K ur sat S ozer, Kostas Tsouvalas, Dai Xian, and Feng Zhu; and, last but not least, many other friends and col- leagues, Bahar Acu, Andrea Appel, Martin Bobb, Nate Bottman, G uher C amlyurt, Qingtao Chen, Tommaso Cremaschi, Brad Drew, Ozlem Ejder, Sabrina Enriquez, Nicolle Gonz alez, Matt Hogancamp, Ezgi Kantarc O guz, Viktor Kleen, Jesse Levitt, Zhen Liang, Eilidh McK- emmie, Shawn Nevalainen, Can Ozan O guz, J. E. Paguyo, Alex Port, John Rahmani, Gus Schrader, Sasha Shapiro, Zhanerke Temirgaliyeva, J er emy Toulisse, Harold Williams, Ka Ho Wong, Haiping Yang, Tian Yang, and Bradley Zykoski. The author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 \RNMS: Geometric structures And Representation varieties" (the GEAR Network). The author was partially supported by the grants DMS-1711297 and DMS-1406559 from the National Science Foundation. iv Contents Dedication ii Acknowledgements iii List of Figures ix Abstract xiii 1 Introduction 1 1.1 Quantum aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Classical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 I Background 9 2 Fock-Goncharov snakes for SL n (C) 10 2.1 Vectors and co-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Linear groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Linear bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Projective bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Change of basis matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6 Dual subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 v 2.7 Complete ags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 Dual ags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.9 Generic pairs of ags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.10 Generic triples and quadruples of ags . . . . . . . . . . . . . . . . . . . . . 17 2.11 Discrete triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.12 Fock-Goncharov triangle and edge invariants . . . . . . . . . . . . . . . . . . 20 2.13 Action of PGL n (C) on generic ag triples . . . . . . . . . . . . . . . . . . . . 21 2.14 Snakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.15 Line decomposition of (C n ) associated to a triple of ags and a snake . . . . 27 2.16 Projective basis of (C n ) associated to a triple of ags and a snake . . . . . . 28 2.17 Elementary snake moves for triangles . . . . . . . . . . . . . . . . . . . . . . 32 2.18 Proof of Proposition 2.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.19 Sketch of proof of Fock and Goncharov's theorem . . . . . . . . . . . . . . . 43 2.20 Right snakes and right matrices . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.21 Triangle invariants as shears . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.22 Elementary snake moves for edges . . . . . . . . . . . . . . . . . . . . . . . . 54 2.23 Edge invariants as shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 II Quantum aspects 62 3 Points of quantum SL n coming from Fock-Goncharov snakes 63 3.1 Quantum tori, matrix algebras, and Weyl ordering . . . . . . . . . . . . . . . 63 3.2 Fock-Goncharov quantum torus for a triangle . . . . . . . . . . . . . . . . . 66 3.3 Shearing and triangle matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 Classical left, right, and edge matrices . . . . . . . . . . . . . . . . . . . . . 69 3.5 Quantum left and right matrices . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.6 Quantum SL n and its points . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.7 First theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 vi 3.8 Concrete formulas and examples . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.9 Snake-move quantum tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.10 Quantum elementary snake-move matrices . . . . . . . . . . . . . . . . . . . 81 3.11 Embedding a Fock-Goncharov sub-algebra into a tensor product of snake-move algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.12 Finishing the proof of the theorem . . . . . . . . . . . . . . . . . . . . . . . 84 3.13 Setup for the quantum right matrix . . . . . . . . . . . . . . . . . . . . . . . 86 4 Quantum traces for SL n (C): the case n = 3 88 4.1 Topological setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2 Two classical trace polynomials . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3 Classical and quantum Fock-Goncharov coordinates on a triangulated surface 91 4.4 Local monodromy matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Computing the classical trace polynomial . . . . . . . . . . . . . . . . . . . . 99 4.6 Framed oriented links in the thickened surface . . . . . . . . . . . . . . . . . 101 4.7 Stated links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.8 Second theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.9 Matrix conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.10 Biangles and the Reshetikhin-Turaev invariant . . . . . . . . . . . . . . . . . 105 4.11 Denition of the SL 3 (C)-quantum trace map . . . . . . . . . . . . . . . . . . 112 4.12 Computer check of local moves for n = 3 . . . . . . . . . . . . . . . . . . . . 115 4.13 HOMFLYPT skein relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.14 Going to SL n (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.15 Proposed denition of the SL n -quantum trace map . . . . . . . . . . . . . . 119 vii III Classical aspects 121 5 Tropical Fock-Goncharov coordinates for Kuperberg webs on surfaces 122 5.1 Webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Essential webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.4 Knutson-Tao cone of an ideal triangulation . . . . . . . . . . . . . . . . . . . 127 5.5 Goal: mapping from webs to the cone . . . . . . . . . . . . . . . . . . . . . . 131 5.6 Webs-with-boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.7 External faces of webs-with-boundary . . . . . . . . . . . . . . . . . . . . . . 132 5.8 Flat Riemannian metrics associated to webs-with-boundary . . . . . . . . . . 134 5.9 Non-elliptic webs-with-boundary . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.10 Essential and rung-less webs-with-boundary . . . . . . . . . . . . . . . . . . 136 5.11 Ladders in ideal biangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.12 Honeycombs in ideal triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.13 Minimal position of a web on the surface . . . . . . . . . . . . . . . . . . . . 147 5.14 Denition of the mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.15 Third theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.16 Integer cones: proof of Lemma 5.28 . . . . . . . . . . . . . . . . . . . . . . . 158 References/Appendices 166 Bibliography 167 Appendix A 172 Appendix B 178 viii List of Figures 2.1 Discrete triangle and triangle invariants for a generic triple of ags . . . . . . 19 2.2 Edge invariants for a generic quadruple of ags . . . . . . . . . . . . . . . . . 22 2.3 Head-, left-, and right-coordinates associated to a snake-head . . . . . . . . . 24 2.4 Snake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Re-naming a ag triple with respect to a snake-head . . . . . . . . . . . . . 29 2.6 Three co-planar lines used to dene a projective basis . . . . . . . . . . . . . 31 2.7 Diamond-move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.8 Tail-move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.9 Proof of Proposition 2.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.10 Shearing snakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.11 Clockwise U-turn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.12 Edge invariants as shears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.1 Quiver dening relations in the Fock-Goncharov algebra . . . . . . . . . . . . 67 3.2 Snake sequences for n = 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3 The case n = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.4 Diamond snake-move algebra (j = 2;:::;n 1) . . . . . . . . . . . . . . . . 80 3.5 Tail snake-move algebra (j = 1) . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6 Quantum snake sweep, shown in the case n = 4 . . . . . . . . . . . . . . . . 82 3.7 Embedding into the tensor product of snake-move algebras . . . . . . . . . . 85 3.8 Right diamond snake-move algebra (j = 2;:::;n 1) . . . . . . . . . . . . . 87 ix 3.9 Right tail snake-move algebra (j = 1) . . . . . . . . . . . . . . . . . . . . . . 87 4.1 Examples of ideal triangulations . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.2 Fock-Goncharov coordinates on a triangulated surface, shown in the case n = 4 91 4.3 Interior edge coordinates as tensor products of generators, shown for n = 3 . 93 4.4 Left matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.5 Right matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.6 Edge matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.7 U-turn matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.8 Decreasing U-turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.9 Increasing U-turns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.10 Same direction crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.11 Opposite direction crossings . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.12 Single strand crossing the biangle . . . . . . . . . . . . . . . . . . . . . . . . 110 4.13 Skein relation for positive kinks . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.14 Skein relation for negative kinks . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.15 Quantum left and right matrices . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.16 One of the oriented versions of Move (II) . . . . . . . . . . . . . . . . . . . . 115 4.17 One of the oriented versions of Move (IV) . . . . . . . . . . . . . . . . . . . 116 4.18 HOMFLYPT skein relation, in the case n = 3 . . . . . . . . . . . . . . . . . 118 5.1 Web on the once punctured torus . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 Global parallel-move on the once punctured torus . . . . . . . . . . . . . . . 124 5.3 Web face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.4 Prohibited face forming a \hat" . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.5 Prohibited 4-face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.6 Seven tropical coordinates for a triangle and nine diamond inequalities, three of which are shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 x 5.7 Eight points in the Knutson-Tao cone for a triangle, four of which are shown 130 5.8 Web-with-boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.9 External face . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.10 Cap, fork, H, and external 4-face of type II . . . . . . . . . . . . . . . . . . . 134 5.11 Tiling the disk with the dual graph of a web . . . . . . . . . . . . . . . . . . 135 5.12 Small webs-with-boundary in the disk . . . . . . . . . . . . . . . . . . . . . . 137 5.13 Denition of an essential web . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.14 Examples and non-examples of essential webs . . . . . . . . . . . . . . . . . 139 5.15 Adding and removing an H from a web . . . . . . . . . . . . . . . . . . . . . 140 5.16 Examples and non-examples of rung-less webs . . . . . . . . . . . . . . . . . 141 5.17 Essential web in a biangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.18 Ladders in biangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.19 Local move dening the Bird's-Eye View (BEV) multicurve . . . . . . . . . . 143 5.20 Interpreting non-ellipticity from the standpoint of the BEV multicurve . . . 144 5.21 Honeycomb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.22 Proof of Proposition 5.22: 1 of 3 . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.23 Proof of Proposition 5.22: 2 of 3 . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.24 Proof of Proposition 5.22: 3 of 3 . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.25 Disconnected rung-less essential web in the ideal triangle . . . . . . . . . . . 150 5.26 Cap- and fork-moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.27 H-move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.28 Split ideal triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.29 Modied H-move across a triangle between two biangles . . . . . . . . . . . . 152 5.30 Local parallel-move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.31 Good position with respect to a split ideal triangulation (compare Figure 5.32) 158 xi 5.32 Dierent corner arc permutations for local triangle pictures yield dierent local ladder pictures (compare Figure 5.31), but not dierent global parallel- equivalence classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.33 (Recall of Figure 5.6) Seven tropical coordinates for a triangle and nine dia- mond inequalities, three of which are shown . . . . . . . . . . . . . . . . . . 159 5.34 (Recall of Figure 5.7) Eight points in the Knutson-Tao cone for a triangle, four of which are shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 xii Abstract The thesis is divided into two parts. The rst part discusses the quantum world. We generalize Bonahon and Wong's SL 2 (C)-quantum trace map for nite-type surfaces to the case of SL 3 (C), and then we propose a denition for an SL n (C)-version of the quantum trace map. Along the way, we establish a \building block" result that we expect to be an important ingredient in the construction of the SL n (C)-quantum trace map. In particular, we relate Fock and Goncharov's geometric theory of SL n (C) to the algebraic theory of the quantum group U q (sl n ). The second part studies the classical setting. We show that the Sikora-Westbury linear basis for the algebra of functions on the SL 3 (C)-character variety, namely those algebraic functions that correspond to planar essential webs on the surface, can be indexed by an inte- ger cone dened by Knutson-Tao diamond inequalities. The theory of Fock and Goncharov furnishes the correspondence between webs and their integer coordinates inside the cone. xiii Chapter 1 Introduction The work presented in this thesis lies at the interface between geometry, topology, and representation theory. The geometry component draws its origins from the study of homo- geneous geometric structures on manifolds, the topology component from the study of curves or graphs on surfaces as well as knots in 3-dimensional manifolds, and the representation theory component from the study of Lie groups and quantum groups. A historically important example with roots in geometry involves the special linear groups SL 2 (R) or SL 2 (C) of 22 matrices with determinant 1. In this case, the underlying geometry is hyperbolic geometry in 2 or 3 dimensions. A primary object is the SL 2 -character variety X SL 2 (M) consisting of representations of the fundamental group 1 (M) of a manifold M into SL 2 (R) or SL 2 (C), considered up to conjugation. Namely, one associates a (conjugacy class of a) 22 matrix( ) to each closed oriented curve inside the manifoldM. Examples of geometric families of monodromy representations come from the Teichm uller theory of surfaces and from the hyperbolic geometry of 3-dimensional manifolds. A popular trend considers more general Lie groups G other than SL 2 . When working with a surface S this is the realm of higher Teichm uller theory, which is a fertile research area that can be explored using a wide variety of mathematical skill sets. Some examples include the Higgs bundle approach of Hitchin [34], the dynamics approach of Labourie [42], 1 and the representation theory approach of Fock and Goncharov [19]. A principal object is theG-character varietyX G (S) consisting of representations of the fundamental group 1 (S) into the Lie group G. Another related trend investigates quantizations of the G-character varietyX G (S), aris- ing from the natural Poisson structure on the character variety. In this thesis, we study a quantum version of the monodromy representations, called the quantum trace map. More specically, we construct a SL 3 -quantum invariant for knotsK in thickened surfacesS[0; 1], coming from an interaction between higher Teichm uller theory and quantum groups. This research explores both the quantum and classical aspects of this large and fascinating story. 1.1 Quantum aspects The discussion in this section primarily concerns the quantum world. We are interested in building a bridge between two competing quantizations of the character variety X SLn(C) (S) of a nite-type surface S for the Lie group G = SL n (C). Both quantizations depend on a complex parameter q =e 2i~ . The rst quantization, developed by several groups of researchers [60, 51, 61, 8, 11, 52, 41, 57, 58], is the SL n (C)-skein algebra S q SLn(C) (S). The skein algebra is motivated by the classical algebraic geometric approach to the character variety. More specically, a closed oriented curve on the surface S determines a trace function Tr :X SLn(C) (S)!C on the character variety by the assignment 7! Tr(( )). A theorem of Procesi [50] implies that the trace functions Tr (up to taking powers of the representation ) generate the algebra of functionsC[X SLn(C) (S)] on the character variety. According to the philosophy of Turaev and Witten, quantizations of the character variety should be of a 3-dimensional nature. Indeed, the elements of the skein algebra S q SLn(C) (S) are represented by (framed) oriented knotsK in the 3-dimensional thickened surface S [0; 1]. The skein algebraS q SLn(C) (S) has the advantage of being natural, but is dicult to work with in practice. 2 The second quantization, due to Chekhov and Fock [17] and independently Kashaev [36] in the case of SL 2 (C), and due to Fock and Goncharov [21] in the case of more general Lie groups G, is what we call the Fock-Goncharov algebra FG q SLn(C) (S). At the classical level, Fock and Goncharov [19] introduced a framed character variety X PSLn(C) (S) FG , whose charts can be parametrized by complex coordinates (X i ) i=1;:::;N for some positive integer N depending only on n and the topology of the surface S. The framed character variety X PSLn(C) (S) FG is very similar to the original character variety X SLn(C) (S), and in fact can be realized as (the projective version of) the character variety X PSLn(C) (S) equipped with additional boundary data. When expressed in the coordinatesX i , the trace functions Tr = Tr (X i ) take the form of Laurent polynomials in the variablesX i . At the quantum level, Fock and Goncharov dened a non-commutative algebra FG q SLn(C) (S) consisting of polynomials in q-deformed versions b X i of their coordinates, which no longer commute but q-commute with each other. Compared to the skein algebra S q SLn(C) (S), the Fock-Goncharov algebra FG q SLn(C) (S) has the advantage of being easy to use in practice, but is less intrinsic than the skein algebra, for example depending on the choice of an ideal triangulation of the surface. It is natural to ask if there is also aq-deformed version Tr q of the trace functions, assigning to a closed curve a Laurent polynomial in theq-deformed Fock-Goncharov coordinates b X i . To construct this, one is forced to leave the 2-dimensional setting of curves on the surface S, and must instead work in the 3-dimensional setting of knots K in the thickened surface S [0; 1]. We recall that a knot K represents an element of the skein algebraS q SLn(C) (S). Conjecture 1 (Quantum trace map). There exists an injective algebra map Tr q :S q SLn(C) (S),−!FG q SLn(C) (S); (1.1) from the skein algebra S q SLn(C) (S) to the Fock-Goncharov algebra FG q SLn(C) (S), such that if K is a framed oriented knot (or, more generally, a closed link) in the thickened surface S [0; 1], then the assignment K7! Tr q K satises the property that the specialization q = 1 3 recovers the classical trace polynomial, up to sign, Tr 1 K =Tr (X i ) 2FG 1 SLn(C) (S) =C[X 1 1 ;X 1 2 ;:::;X 1 N ]; (1.2) where is the closed oriented curve obtained by projecting the knot K to the surface S. Moreover, this map is natural with respect to the action of the Mapping Class Group on the surface S, meaning that it is well-behaved with respect to choosing a dierent ideal triangulation of the surface S. This was proved \by hand" in the case n = 2 by Bonahon and Wong [5]. See also [27], as well as [49]. In Theorem 4.8, we prove a slightly weaker version of Conjecture 1 in the case n = 3 for knots K in the thickened surface S [0; 1]. Theorem 1. For each framed oriented knot (or, more generally, a closed link) K in the thickened surface S [0; 1], there exists a Laurent polynomial Tr q K 2 FG q SL 3 (C) (S) in the q-deformed SL 3 (C)-Fock-Goncharov coordinates, such that the assignment K 7! Tr q K is a deformation (up to sign) of the classical trace polynomial 7! Tr (X i ), where is the closed oriented curve obtained by projecting the knot K to the surface S. In particular, this assignment is an isotopy invariant of the knot K. The proof is also by hand, and some of its computations require computer assistance. Proving Conjecture 1, or a general version of Theorem 1 in the case of arbitraryn, requires a more conceptual approach. We make progress in this direction, embracing the theory of the quantum group U q (sl n ) or, more precisely, of its Hopf dual SL q n . Specically, in Theorem 3.10 we prove a local \building block" result for general n that makes contact with the quantum group SL q n . We brie y describe this now. In the classical case, the trace functions Tr (X i ), which are polynomials in the Fock- Goncharov coordinates X i , may be considered as elements of the algebra of commuting polynomialsFG 1 SLn(C) (S) =C[X 1 1 ;X 1 2 ;:::;X 1 N ]. To compute these polynomials, Fock and Goncharov, roughly speaking, associate to each triangleTS in the ideal triangulation of 4 the surface S a nn matrix in the special linear group SL n (FG 1 SLn(C) (S)) with coecients in the commutative algebra FG 1 SLn(C) (S). Then the trace polynomial Tr (X i ) is obtained simply by multiplying these \building block" matrices and taking the usual matrix trace. For the quantum setup, let SL q n (FG q SLn(C) (S)) denote thosenn matrices with coecients in the Fock-Goncharov algebraFG q SLn(C) (S) whose entries satisfy the dening relations of the dual quantum group SL q n . Recall from above that the Fock-Goncharov algebraFG q SLn(C) (S) consists of Laurent polynomials in the q-deformed Fock-Goncharov coordinates b X i . There is an obvious quantization of Fock and Goncharov's classical \building block" matrices, using a physics symmetrizing trick. In Theorem 3.10, we show that these quantized \building block" matrices satisfy the relations of SL q n , thus connecting the geometry of higher Teichm uller theory with the algebraic properties of quantum groups. Theorem 2. The q-deformed \building block" matrices, which are, roughly speaking, asso- ciated to each triangle TS in the xed ideal triangulation of the surface S, and whose coecients are Laurent polynomials in the q-deformed Fock-Goncharov coordinates b X i , are elements of SL q n (FG q SLn(C) (S)). In other words, they are points of the quantum group SL q n valued in the non-commutative Fock-Goncharov algebra FG q SLn(C) (S). The proof uses a quantum version of the technology invented by Fock and Goncharov called snakes. Other useful references for Fock-Goncharov coordinates and snakes are [35, 29]. For very related work, see also [56, 55, 14]. 1.2 Classical aspects After describing the rst part of this thesis, which explores the quantum world, we now turn to the second part, which focuses on the classical setting of the character varietyX SLn(C) (S) and its algebra of functionsC[X SLn(C) (S)]. The work discussed in this section was carried out in collaboration with Zhe Sun. We have seen that the trace functions Tr of curves on S are important functions on 5 the character variety X SLn(C) (S). Our main goal is to investigate \canonical" linear bases for the algebra of functionsC[X SLn(C) (S)] on the character variety. The casen = 2 is well-understood. Following Turaev and Przytycki, it was demonstrated by Bullock [8, 9] among others [52] that the trace functions 7! Tr provide an explicit iso- morphism from the skein algebra S 1 SL 2 (C) (S) evaluated at q = 1 to the algebra of functions C[X SL 2 (C) (S)] on the SL 2 (C)-character variety. Moreover, a \canonical" basis for the alge- bra of functions C[X SL 2 (C) (S)] on the SL 2 (C)-character variety is identied through trace functions with the family of planar non-self-intersecting multi-curves on the surface S. To study what happens when n > 2, we need to generalize our notion of \curve". We know that knots K represent elements of the SL n (C)-skein algebraS q SLn(C) (S). Going back even to Procesi's work, it turns out that knots (or links) are not enough to fully describe skein algebras in the higher rank setting. More precisely, the skein algebra S q SLn(C) (S) has more relations than can be satised by the usual well-known skein relations for knots. Reshetikhin and Turaev [54] as well as Kuperberg [41], among others, realized that one needs to consider elements of the skein algebra represented by framed oriented graphs, often called webs W , embedded in the thickened surface S [0; 1]. Sikora [57] extended the notion of trace func- tions Tr W to include webs on the surface S, thus providing an explicit natural isomorphism C[X SLn(C) (S)] =S 1 SLn(C) (S) between the algebra of functionsC[X SLn(C) (S)] on the character variety and the skein algebraS q SLn(C) (S) when q is evaluated to 1. In the case n = 3, Sikora and Westbury [59] proved that a certain family of planar trivalent graphs W on the surface S, called non-elliptic webs, form a \canonical" basis for the algebra of functionsC[X SL 3 (C) (S)] on the SL 3 (C)-character variety. Re-visiting the casen = 2, although the natural isomorphismC[X SL 2 (C) (S)] =S 1 SL 2 (C) (S) is conceptually important, it does not provide explicit coordinates, in the spirit of Fock and Goncharov, for the \canonical" basis consisting of non-self-intersecting multicurves on the surface S. We can obtain such an explicit description by studying the asymptotics of the trace functions Tr , borrowing techniques from the tropical geometry of curves on surfaces. 6 The crucial property, of a tropical nature, is that the geometric intersection numbers of a curve with respect to the xed ideal triangulation of the surface S can be read from the trace polynomial Tr (X i ) as the exponents of the highest term in the Fock-Goncharov coordinates X i . Equivalently, it can be shown that this \canonical" basis for the algebra of functionsC[X SL 2 (C) (S)] on the SL 2 (C)-character variety is indexed by a cone C 2 Z N 2 >0 of positive integers dened by explicit rational inequalities. In Theorem 5.27, we generalize this result to the case n = 3, employing the SL 3 (C)- representation theoretic ideas of Kuperberg [41] together with the tropical geometric ideas of Goncharov-Shen [31] in uenced by the work of Knutson-Tao [40]. Our work also builds on ideas of Xie [62]. Theorem 3 (with Zhe Sun). The Sikora-Westbury \canonical" basis for the algebra of func- tionsC[X SL 3 (C) (S)] on the SL 3 (C)-character variety, namely the set of homotopy classes of non-elliptic websW on the surfaceS, can be indexed by a coneC 3 Z N 3 >0 of positive integers, obtained by studying the asymptotics of the trace polynomials Tr W (X i ). The cone C 3 Z N 3 >0 is dened by explicit rational \Knutson-Tao diamond inequalities". A very similar theorem was independently proved in [26]. See also [46]. One big reason to believe there exists a \canonical" basis for the algebra of functions on the SL n (C)-character variety is the Fock-Goncharov Duality Conjecture [19, 21, 33, 32], which has both classical and quantum incarnations. Fock and Goncharov dene two \mirror-dual" spaces, the framed PSL n (C)-character variety X PSLn(C) (S) FG and the decorated SL n (C)- character varietyA SLn(C) (S) FG . We have already encountered the former of these two spaces. There are also tropical integer versionsX PSLn(C);S [Z t ] FG andA SLn(C);S [Z t ] FG of these spaces. Choosing an ideal triangulation of the surface S corresponds to choosing a chart for each of these spaces, which is parametrized by X- or A-coordinates, and their tropical versions, respectively. The Duality Conjecture says that the integerA-coordinates for the tropicalized decorated SL n (C)-character varietyA SLn(C);S [Z t ] FG index a \canonical" basis for the algebra of functionsC[X PSLn(C) (S) FG ] on the framed PSL n (C)-character variety. 7 In this new light, Theorem 3 might be interpreted as a positive skein theoretic version of the classical Duality Conjecture in the case n = 3, where the positive tropical integer A-coordinates are modeled as points in the Knutson-Tao coneC 3 . 1.3 Conclusion The quantum trace map, combined with the classical properties of character varieties, can be applied to answer fundamental questions about skein algebrasS q SLn(C) (S) for an arbitrary deformation parameterq. In this way, Bonahon and Wong [5] showed that the quantum trace is injective, re-constructed the \canonical" basis for the SL 2 (C)-skein algebraS q SL 2 (C) (S) [51], and re-proved thatS q SL 2 (C) (S) has no zero divisors [52, 53]; and Frohman and Abdiel [1] re- proved that the skein algebraS q SL 2 (C) (S) is nitely generated [10]. Moreover, the injectivity of the quantum trace map was used [6, 25] to study the representation theory of skein algebras, which plays an important role in Topological Quantum Field Theories [61, 3]. Moreover, Allegretti and Kim [2] employed Bonahon and Wong's quantum trace map to prove the quantum Fock-Goncharov Duality Conjecture in the case n = 2. Conjecture 2. The SL n (C)-quantum trace map (Conjecture 1) is an essential tool for un- derstanding the quantum Fock-Goncharov Duality Conjecture for SL n (C). Going from SL 2 or SL 3 to SL n can be challenging, as demonstrated in [13] which solved a problem of this fashion that was open for over a decade. This is sometimes called the search for higher laminations. There are many competing approaches, coming from both mathematics and physics. Popular methods use the theory of \buildings" [24, 43] from geometry, and \spectral networks" [28] from physics. Our work continues a complementary skein theoretic approach, exploiting the powerful theory of Fock and Goncharov. 8 Part I Background 9 Chapter 2 Fock-Goncharov snakes for SL n (C) In this chapter, we present the classical geometric part of Fock and Goncharov's theory [19, 20, 12] that underlies our quantum work later on. Other useful references for Fock- Goncharov coordinates and snakes are [35, 29]. I also greatly beneted from un-published notes on Fock-Goncharov snakes that were communicated by Francis Bonahon. Throughout the chapter, let n2Z >2 be a xed integer greater than or equal to 2. The integerN plays a more uid role, and should be interpreted according to the context. 2.1 Vectors and co-vectors Dene the vector spaceC n to be the set of column vectors of length n with coecients in C, or equivalently as the set of n 1 complex matrices. We identify the dual vector space (C n ) with the set of row vectors of lengthn with coecients inC, or equivalently as the set of 1n complex matrices. An element ofC n is called a vector, and an element of (C n ) is called a co-vector. The action of a co-vector u2 (C n ) on a vector v2C n can be expressed as u(v) = uv 2C; (2.1) 10 where the product on the right-hand side of the equation is the matrix product, and where we have identied the set of 1 1 complex matrices withC. 2.2 Linear groups The general linear group GL n (C) is the set of invertiblenn matrices. There is a left action of the matrix group GL n (C) on the set of vectorsC n by left matrix multiplication, and there is a right action of GL n (C) on the set of co-vectors (C n ) by right matrix multiplication. The special linear group SL n (C) is the subgroup of GL n (C) consisting of all matrices with determinant equal to 1. The standard upper triangular Borel subgroup B n is the subgroup of GL n (C) consisting of all upper triangular matrices with nonzero diagonal entries. By identifyingCf0g with the subgroup of GL n (C) consisting of nonzero scalar multiples of the identity matrix, thenCf0g acts on GL n (C) by matrix multiplication. The quotient group GL n (C)=(Cf0g) under this action is called the projective general linear group and is denoted by PGL n (C). By thinking of the n-th roots of unity inC as a subgroupZ=nZ Cf0g, then the inclusionCf0g ,−! GL n (C) induces an inclusionZ=nZ ,−! SL n (C). The resulting quotient group SL n (C)=(Z=nZ) is called the projective special linear group and is denoted by PSL n (C). Because complex numbers always possess n-th roots, an intriguing elementary fact is that the inclusion SL n (C),−! GL n (C) induces a group isomorphism PSL n (C) = PGL n (C): (2.2) 2.3 Linear bases IfV =fv 1 ;:::; v n g is a basis forC n , then we may arrange these basis vectors as the columns of an invertible matrix V = ( v 1 v 2 vn )2 GL n (C). On the other hand, ifU =fu 1 ;:::; u n g is a basis for (C n ) , then we may arrange these basis co-vectors as the rows of an invertible 11 matrix U = u 1 u 2 . . . un ! 2 GL n (C). IfV =fv 1 ;:::; v n g is a basis forC n , then there is a unique basisV =U =fu 1 ;:::; u n g for (C n ) , called the dual basis corresponding to the basisV, dened by the property that u i (v j ) = ij 2C (16i;j6n); (2.3) where ij 2f0; 1g is the Kronecker delta. This same equation is used to dene the notion of the dual basis U =V =fv 1 ;:::; v n g forC n corresponding to a basisU =fu 1 ;:::; u n g for (C n ) . Using equation (2.1) together with our matrix interpretation of bases, we see that the notion of dual bases is equivalent to the matrix property V = v 1 v 2 v n = 0 B B B B B B B @ u 1 u 2 . . . u n 1 C C C C C C C A 1 = U 1 2 GL n (C): (2.4) As a consequence, using the identication (C n ) = C n , the operation of taking the dual basis satises the symmetryV =V andU =U. For example, the standard vector basis forC n isE =fe 1 ; e 2 ;:::; e n g, and the standard co-vector basis for (C n ) isE =fe 1 ; e 2 ;:::; e n g. Then e 1 e 2 e n = 0 B B B B B B B @ e 1 e 2 . . . e n 1 C C C C C C C A = 0 B B B B B B B @ 1 0 0 0 1 0 . . . . . . . . . . . . 0 0 1 1 C C C C C C C A = Id n 2 GL n (C): (2.5) These bases are dual since the inverse of the identity matrix Id n = Id 1 n is itself. 12 2.4 Projective bases Let V =C n or (C n ) . The nonzero complex numbersCf0g act on the set of linear bases for V . A projective basis is an equivalence class for this action. Equivalently, thinking of linear bases as columns (resp. rows) of an invertible matrix in GL n (C) whenV =C n (resp. V = (C n ) ), a projective basis can be thought of as an element of PGL n (C). 2.5 Change of basis matrices Given a basisV =fv 1 ; v 2 ;:::; v n g of vectors inC n , and given a vector v inC n , the coordinate vector [v] V of the vector v with respect to the basisV is the unique vector [v] V = x 1 x 2 . . . xn ! in C n such that v = P n i=1 x i v i . If in addition we are given another basisV 0 =fv 0 1 ; v 0 2 ;:::; v 0 n g forC n , then the change of basis matrix B V 0 V going from the basis V to the basis V 0 is the unique matrix in GL n (C) satisfying the property that [v] V 0 = B V 0 V [v] V 2C n (v2C n ); (2.6) where we have used the left action of GL n (C) onC n . In fact, arranging the basis V as the columns of a matrix V = ( v 1 v 2 vn ) in GL n (C) and similarly arranging the basis V 0 into a matrix V 0 in GL n (C), then the property dening the change of basis matrix B V 0 V is equivalent to the property that V 0 B V 0 V = V 2 GL n (C): (2.7) On the other hand, given a basisU =fu 1 ; u 2 ;:::; u n g of co-vectors in (C n ) , and given a co-vector u in (C n ) , the coordinate co-vector [u] U of the co-vector u with respect to the 13 basisU is the unique co-vector [u] U = ( y 1 y 2 yn ) in (C n ) such that u = P n i=1 y i u i . If in addition we are given another basisU 0 =fu 0 1 ; u 0 2 ;:::; u 0 n g forC n , then the change of basis matrix B U!U 0 going from the basis U to the basisU 0 is the unique matrix in GL n (C) satisfying the property that [u] U B U!U 0 = [u] U 0 2 (C n ) u2 (C n ) ! ; (2.8) where we have used the right action of GL n (C) on (C n ) . In fact, arranging the basisU as the rows of a matrix U = u 1 u 2 . . . un ! in GL n (C) and similarly arranging the basis U 0 into a matrix U 0 in GL n (C), then the property dening the change of basis matrix B U!U 0 is equivalent to the property that U = B U!U 0U 0 2 GL n (C): (2.9) For example, it follows from equations (2.7) and (2.9) that if V;V 0 ;V 00 are bases forC n and ifU;U 0 ;U 00 are bases for (C n ) , then B V 00 V 0B V 0 V = B V 00 V ; B U!U 00 = B U!U 0B U 0 !U 00 2 GL n (C): (2.10) As another example, ifV andU are dual bases, and ifV 0 andU 0 are as well, then B U!U 0 = B V V 0 2 GL n (C): (2.11) 2.6 Dual subspaces Given a linear subspace W of a nite-dimensional vector space V , the dual subspace W ? is a linear subspace of the dual vector space V dened by V ? =f2V ;(v) = 0 for all v2Vg: (2.12) 14 Fact 2.1. The dual subspace construction satises the following elementary properties: (1) (W +W 0 ) ? =W ? \W 0? ; (2) (W\W 0 ) ? =W ? +W 0? ; (3) dim(W ? ) =n dim(W ); (4) W ?? =W under the identication V =V . 2.7 Complete ags A complete ag (or simply ag) E inC n is a collection of linear subspaces E (a) ofC n for each 06 a6 n, satisfying the property that each subspace E (a) is properly contained in the subspace E (a+1) . In particular, E (0) =f0g and E (n) =C n . Denote the space of ags by Flag(C n ). For example, the standard ascending ag E 0 2 Flag(C n ) and the standard descending ag G 0 2 Flag(C n ) are dened by E (a) 0 = Span(e 1 ; e 2 ;:::; e a ); G (c) 0 = Span(e nc+1 ;:::; e n1 ; e n ) C n ; (2.13) for all 06a;c6n. It is worth mentioning an alternative denition for the space of ags Flag(C n ). The left action of the general linear group GL n (C) on the linear spaceC n of vectors induces an action of the projective general linear group PGL n (C) on the space of ags Flag(C n ). By elementary linear algebra, the action of PGL n (C) on Flag(C n ) is transitive, namely it has a single orbit. In addition, the stabilizer of the standard ascending ag E 0 is the subgroup PB n PGL n (C) obtained by projectivizing the standard upper triangular Borel subgroup B n . Consequently, there is a bijection Flag(C n ) ! PGL n (C) PB n : (2.14) 15 2.8 Dual ags Of course, we can study ags E in any n-dimensional vector space V , where the denition is analogous to the case V = C n . Given a complete ag E in V , its dual ag E ? is the complete ag E ? in the dual space V dened by (E ? ) (a) = (E (na) ) ? V (06a6n): (2.15) 2.9 Generic pairs of ags For two ags, the notion of generic is well-understood. Denition 2.2. A pair of ags (E;G)2 Flag(C n ) Flag(C n ) is generic if it satises any of the following four equivalent properties: For all 06a;c6n, (1) the sum E (a) +G (c) =E (a) G (c) =C n is direct for all a +c =n; (2) the intersection E (a) \G (c) =f0g is trivial for all a +c =n; (3) we have the dimension formula dim(E (a) +G (c) ) = min(a +c;n); (4) we have the dimension formula dim(E (a) \G (c) ) = max(a +cn; 0). The equivalence of these properties can be deduced from the classical relation dim(E (a) +G (c) ) + dim(E (a) \G (c) ) =a +c: (2.16) Recall from§2.7 that PGL n (C) acts transitively on the left on the space of ags Flag(C n ). Proposition 2.3. The diagonal left action of PGL n (C) on the space Flag(C n ) Flag(C n ) restricts to a transitive left action of PGL n (C) on the space of generic ag pairs. Proof. Let (E;G)2 Flag(C n ) be a generic ag pair. By genericity, for each 16a6n, the subspaceL a =E (a) \G (na+1) is a line inC n . It follows by genericity that the linesL a form 16 a line decomposition ofC n , namelyC n = n a=1 L a . Let' :C n !C n be a linear isomorphism sending the line L a to the a-th standard basis vector e a . Then ' maps the ag pair (E;G) to the standard ascending-descending ag pair (E 0 ;G 0 ). 2.10 Generic triples and quadruples of ags Notions of genericity for triples of ags (E;F;G)2 Flag(C n ) 3 are more subtle. Indeed, there are at least two competing denitions, which coincide for ag pairs. Denition 2.4. First, a ag triple (E;F;G) satises the Maximum Span Property if either of the following two equivalent conditions hold: for all 06a;b;c;6n, (1) the sum E (a) +F (b) +G (c) =E (a) F (b) G (c) =C n is direct for all a +b +c =n; (2) the dimension formula dim(E (a) +F (b) +G (c) ) = min(a +b +c;n). In this case, the ag triple (E;F;G)2 Flag(C n ) is called a maximum span ag triple. Second, a ag triple (E;F;G) satises the Minimum Intersection Property if either of following two equivalent conditions hold: For all 06a;b;c;6n, (1) the intersection E (a) \F (b) \G (c) =f0g is trivial for all a +b +c = 2n; (2) the dimension formula dim(E (a) \F (b) \G (c) ) = max(a +b +c 2n; 0). In this case, the ag triple (E;F;G) is called a minimum intersection ag triple. Maximum span and minimum intersection ag quadruples (E;F;G;H) are similarly de- ned. Recall from §2.8 that given a ag E in a nite-dimensional vector space V , we may consider the dual agE ? inV . In the context of triples of ags, we have the following fact. Proposition 2.5. Let V be a nite-dimensional vector space. A ag triple (E;F;G) in V satises the Maximum Span Property if and only if the dual ag triple (E ? ;F ? ;G ? ) in V satises the Minimum Intersection Property. 17 Proof. Using the elementary properties of dual subspaces, as described in Section 2.6, we derive the following two relations • dim E ?(a) +F ?(b) +G ?(c) =n dim E (na) \F (nb) \G (nc) ; • dim E ?(a) \F ?(b) \G ?(c) =n dim E (na) +F (nb) +G (nc) . From the rst of these relations, we deduce that the rst (resp. second) condition dening the Minimum Intersection Property for (E;F;G) implies the rst (resp. second) condition dening the Maximum Span Property for (E ? ;F ? ;G ? ). Conversely, from the second of these relations, we deduce that the rst (resp. second) condition dening the Maximum Span Property for the ag triple (E;F;G) implies the rst (resp. second) condition dening the Minimum Intersection Property for the dual ag triple (E ? ;F ? ;G ? ). We have prolonged mentioning why the two conditions dening the Maximum Span Prop- erty (resp. Minimum Intersection Property) are indeed equivalent. This is straightforward for the Maximum Span Property. That this is true for the Minimum Intersection Property follows by the fact that it is true for the Maximum Span Property and then using the proof of the previous proposition. Proposition 2.6. A maximum span ag triple (E;F;G) in a nite-dimensional vector space need not be a minimum intersection ag triple, and vice versa. Proof. By taking double duals and applying Proposition 2.5, it suces to show just one of these directions. We check inC 3 that E (1) = Span(e 1 ); E (2) = Span(e 1 ; e 2 ); G (1) = Span(e 3 ); G (2) = Span(e 3 ; e 2 ); F (1) = Span( 1 1 1 ); F (2) = Span( 1 1 1 ; 0 1 0 ); denes a maximum span ag triple that is not a minimum intersection ag triple. 18 2.11 Discrete triangle The discrete n-triangle n is n = (a;b;c)2Z 3 ; a;b;c> 0; a +b +c =n ; (2.17) as shown in Figure 2.1. The hexagon shown in the gure will be discussed in Section 2.12. Figure 2.1: Discrete triangle and triangle invariants for a generic triple of ags Dene the interior of the n-discrete triangle to be int( n ) = (a;b;c)2Z 3 ; a;b;c> 0; a +b +c =n : (2.18) We will also need the (n 1)-discrete triangle, for which we use a dierent notation n1 = (;; )2Z 3 ; ;; > 0; + + =n 1 : (2.19) 19 For m =n;n 1, an element v2 m is called a vertex of the discrete triangle m . An element v2f(m; 0; 0); (0;m; 0); (0; 0;m)g m is called a corner vertex of m . The set of corner vertices of m is denotedC( m ). For m = n;n 1, we choose an orientation for the m-discrete triangle m so that the sequence ((m; 0; 0); (0;m; 0); (0; 0;m))2 C( m ) 3 of corner vertices of the triangle goes clockwise around the boundary of the triangle. 2.12 Fock-Goncharov triangle and edge invariants We now take the integerm from the previous section to be equal to our xed integer n> 2. Fix a maximum span triple of ags (E;F;G) 2 Flag(C n ) 3 . To each interior point (a;b;c) 2 int( n ) of the discrete triangle n , Fock and Goncharov associate a quantity abc (E;F;G)2Cf0g called a triangle invariant of the maximum span ag triple (E;F;G) which is dened by the formula abc (E;F;G) = e (a1) ^f (b+1) ^g (c) e (a+1) ^f (b1) ^g (c) e (a) ^f (b1) ^g (c+1) e (a) ^f (b+1) ^g (c1) e (a+1) ^f (b) ^g (c1) e (a1) ^f (b) ^g (c+1) 2Cf0g; (2.20) where e (a 0 ) is a chosen generator for the a 0 -th exterior power a 0 (E (a 0 ) ) a 0 (C n ), f (b 0 ) is a chosen generator for the b 0 -th exterior power b 0 (F (b 0 ) ) b 0 (C n ), and g (c 0 ) is a chosen gen- erator for thec 0 -th exterior power c 0 (G (c 0 ) ) c 0 (C n ). Since the exterior powers a 0 (E (a 0 ) ), b 0 (F (b 0 ) ), c 0 (G (c 0 ) ) are lines, the triangle invariant abc (E;F;G) is independent of the choices of the generators e (a 0 ) , f (b 0 ) , and g (c 0 ) , respectively, and thus is well-dened. The Maximum Span Property for the ag triple (E;F;G) ensures that each generator e (a 0 ) ^f (b 0 ) ^g (c 0 ) is nonzero in n (C n ), and so the triangle invariant abc (E;F;G) is indeed valued inCf0g. The six numerators and denominators appearing in the expression for the triangle in- variant abc (E;F;G) can be visualized in the discrete triangle n as the six vertices of a 20 hexagon, as shown in Figure 2.1. A (+)-vertex of the hexagon corresponds to a numerator, and a ()-vertex corresponds to a denominator. Fact 2.7. The triangle invariants abc (E;F;G) satisfy the following symmetries under per- mutation of the input triple of ags: (1) abc (E;F;G) = cab (G;E;F ); (2) abc (E;F;G) = bac (F;E;G) 1 . Next, consider a maximum span quadruple of ags (E;G;F;F 0 )2 Flag(C n ) 4 . To each integer 16j6n1, Fock and Goncharov associated a nonzero scalar denoted j (E;G;F;F 0 ) called an edge invariant of the maximum span ag quadruple (E;G;F;F 0 ) which is dened by the formula j (E;G;F;F 0 ) = e (j) ^g (nj1) ^f (1) e (j) ^g (nj1) ^f 0(1) e (j1) ^g (nj) ^f 0(1) e (j1) ^g (nj) ^f (1) 2Cf0g: (2.21) We visualize the n 1 edge invariants as in Figure 2.2. There, we see two discrete triangles n (E;F 0 ;G) and n (G;F;E) \sharing" a common edge. Then1 edge invariants are associated to the n 1 non-corner vertices of that shared edge. Specically, we think of the j-th edge invariant j (E;G;F;F 0 ) as either associated to the edge vertex (j; 0;nj) of n (E;F 0 ;G) or equivalently to the edge vertex (nj; 0;j) of n (G;F;E). The ratio (2.21) dening the edge invariant involves two interior vertices in int( n (E;F 0 ;G)) and two interior vertices in int( n (G;F;E)), the four of which together form a signed rectangle, where the signs correspond to the numerator and denominator in the denition, as before. 2.13 Action of PGL n (C) on generic ag triples We saw earlier that the action of PGL n (C) on the space of ags Flag(C n ) has a single orbit, as does the diagonal action of PGL n (C) on the space of generic ag pairs (E;G)2 21 Figure 2.2: Edge invariants for a generic quadruple of ags Flag(C n ) 2 . One of Fock and Goncharov's main insights was that the diagonal action of PGL n (C) on the space of maximum span ag triples (E;F;G)2 Flag(C n ) 3 has uncountably many orbits, whenn> 2, and that these orbits can be parametrized by the triangle invariants abc (E;F;G)2Cf0g. For n = 2, there is still only a single orbit, which is equivalent to the fact that any ordered triple of distinct points in CP 1 can be taken to any other such triple of points by an element of PGL 2 (C). The rst and simplest observation is that if (E;F;G) is a maximum span ag triple and if ' :C n !C n is a linear isomorphism, then abc (E;F;G) = abc ('E;'F;'G) by denition of the triangle invariants, since the linear map n (') : n (C n )! n (C n ) induced by ' is multiplication by the determinant of '. 22 Theorem 2.8 (Fock-Goncharov). Two maximum span ag triples (E;F;G) and (E 0 ;F 0 ;G 0 ) in Flag(C n ) 3 have the same triangle invariants, namely abc (E;F;G) = abc (E 0 ;F 0 ;G 0 ) in Cf0g for all positive integers a;b;c> 0 such that a +b +c =n, if and only if there exists a linear isomorphism ' :C n !C n such that ('E;'F;'G) = (E 0 ;F 0 ;G 0 ). Conversely, for each choice of numbers x abc 2Cf0g associated to the interior points (a;b;c)2 int( n ) of the n-discrete triangle n , there exists a maximum span ag triple (E;F;G) such that abc (E;F;G) =x abc 2Cf0g for all (a;b;c). In summary, the triangle invariants abc (E;F;G) put coordinates on the set of orbits for the diagonal action of PGL n (C) on the space of maximum span ag triples, thereby identifying the set of orbits with (Cf0g) (n1)(n2) 2 . Proof. See [19, §9]. This uses the concept of snakes, due to Fock and Goncharov. We give a sketch of the proof in §2.19. 2.14 Snakes Snakes are combinatorial objects associated to the (n1)-discrete triangle n1 . Recall from §2.11 that n1 consists of vertices (;; ) inZ 3 such that;; > 0 and+ + =n1. We may also need to refer to the n-discrete triangle n consisting of vertices (a;b;c) in Z 3 such that a;b;c> 0 and a +b +c =n. Given a vertex v = (;; )2 n1 we may refer to its -, -, and -coordinates inZ. Recall from §2.11 that the discrete triangle n1 is given an orientation so that the sequence (n 1; 0; 0); (0;n 1; 0); (0; 0;n 1) ! 2C( n1 ) 3 3 n1 ; (2.22) of corner vertices of the discrete triangle n1 goes clockwise around its boundary. 23 Denition 2.9. A snake-head is a xed corner vertex of the (n 1)-discrete triangle 2 ( (n 1; 0; 0); (0;n 1; 0); (0; 0;n 1) ) =C( n1 ) n1 : (2.23) Given a snake-head , let the corner vertices left ; right 2 ( (n 1; 0; 0); (0;n 1; 0); (0; 0;n 1) ) fgC( n1 ); (2.24) be uniquely dened by the property that the corner vertex left (resp. right ) is the rst corner vertex reached when traveling clockwise (resp. counterclockwise) along the boundary of the discrete triangle n1 starting from the corner vertex . Figure 2.3: Head-, left-, and right-coordinates associated to a snake-head This is illustrated in Figure 2.3. In particular, left and right are distinct. In total, there are three possibilities: 24 • we have = (n 1; 0; 0) =) left = (0;n 1; 0) and right = (0; 0;n 1); • or, = (0;n 1; 0) =) left = (0; 0;n 1) and right = (n 1; 0; 0); • or, = (0; 0;n 1) =) left = (n 1; 0; 0) and right = (0;n 1; 0). A corner vertex v2C( n1 ) has a unique -, -, or -coordinate that is nonzero, which we say corresponds to the coordinate-axis attached to the corner vertex v. Having xed a snake-head 2C( n1 ), the coordinate axis attached to the corner vertex v2C( n1 ) is called the head-, left-, or right-coordinate axis depending on whetherv is equal to, left , or right , respectively. In this way, a snake-head 2 C( n1 ) denes head-, left, and right-coordinate axes, and it makes sense to speak of the head-, left-, and right-coordinates of any (not necessarily corner) vertex v2 n1 . These coordinates are denoted respectively by [v] head ; [v] left ; [v] right 2Z v2 n1 ! : (2.25) Once again, there are three possibilities: for all v = (;; )2 n1 , • we have = (n 1; 0; 0) =) [v] head =, [v] left =, and [v] right = ; • or, = (0;n 1; 0) =) [v] head =, [v] left = , and [v] right =; • or, = (0; 0;n 1) =) [v] head = , [v] left =, and [v] right =. Figure 2.3 falls into the second possibility. Moreover, the snake-head2C( n1 ) also denes head-, left-, and right- coordinates on the n-discrete triangle n . Just replace , , with a, b, c, respectively. Denition 2.10. An n-snake is a pair (; ( k ) k=1;:::;n ) consisting of the following data: (1) a snake-head 2C( n1 ) xed in the set of corner vertices of n1 ; 25 (2) an ordered sequence ( 1 ; 2 ;:::; n )2 ( n1 ) n of n vertices in the discrete triangle n1 , called snake-vertices, satisfying the following properties: for all k = 1;:::;n, (a) we have [ k ] head =k 1; (b) and [ k ] left > [ k+1 ] left ; (c) and [ k ] right > [ k+1 ] right . Figure 2.4: Snake This is illustrated in Figure 2.4. On the right side of the gure is pictured an n-snake = (; ( k ) k=1;:::;n ) in the case n = 7. Here the snake-head 2C( n ) is equal to the corner vertex (n 1; 0; 0). On the left side, we show how the snake-vertices k 2 n1 can be thought of as small triangles in the n-discrete triangle n . It follows from the denition that • always n =2C( n1 ) is the snake-head, and in particular satises [ n ] head =n 1; • and 1 2 n1 satises [ 1 ] head = 0 and lies on the boundary edge of n1 containing both left and right , where we have 1 = left if and only if [ 1 ] right = 0, and 1 = right if and only if [ 1 ] left = 0. 26 We also remark that given a snake = (; ( k ) k=1;:::;n ), then once the (k +1)-th snake-vertex k+1 2 n1 is chosen, there are only two possibilities for the k-th snake-vertex k 2 n1 . Specically, the two possibilities are given as follows: • we have [ k ] left = [ k+1 ] left + 1 and [ k ] right = [ k+1 ] right ; • we have [ k ] left = [ k+1 ] left and [ k ] right = [ k+1 ] right + 1. 2.15 Line decomposition of (C n ) associated to a triple of ags and a snake Fix a maximum span triple of ags (E;F;G)2 Flag(C n ) 3 inC n , as dened in §2.10. For every vertex (;; ) in the (n 1)-discrete triangle n1 , the dimension dim (E () F () G ( ) ) ? = 1; (2.26) by the Maximum Span Property, since + + =n 1. Here we have employed the dual subspace ofE () F () G ( ) , dened in§2.6. Consequently, we dene a lineL (;; ) (C n ) by L (;; ) = E () F () G ( ) ? (C n ) ; (2.27) for each vertex (;; ) in the discrete triangle n1 . Now, in addition, x a snake = (; ( k ) k=1;:::;n ) with snake-head in the set of corner verticesC( n1 ). Writing k in n1 as k = ( k ; k ; k ), we obtain n lines L k =L ( k ; k ; k ) (C n ) k = 1;:::;n ! ; (2.28) 27 and one checks by genericity that this gives a line decomposition of (C n ) , namely (C n ) = n M k=1 L k : (2.29) We discuss another bookkeeping matter. We say that the ags E;F;G are attached to the-,-, -coordinate axes, respectively. Recall from§2.14 that a snake-head2C( n1 ) determines a permutation of the -, -, -coordinate axes, re-naming them as the head-, left-, right-coordinate axes, not necessarily in this order. In the same spirit, dene E head ; F left ; G right 2fE;F;Gg Flag(C n ); (2.30) to be the three distinct ags infE;F;Gg uniquely determined by the property that the ags E head , F left , G right are attached to the head-, left-, right-coordinate axes, respectively. As before, there are three possibilities: • we have = (n 1; 0; 0) =)E head =E, F left =F , and G right =G; • or, = (0;n 1; 0) =)E head =F , F left =G, and G right =E; • or, = (0; 0;n 1) =)E head =G, F left =E, and G right =F . See Fig. 2.5 for an example, and compare Figure 2.3. In the case shown, if a vertex v in the (n 1)-discrete triangle n1 has coordinates v = (;; ), then its new coordinates with respect to the snake-head are [v] head =, [v] left = , and [v] right =. 2.16 Projective basis of (C n ) associated to a triple of ags and a snake It turns out that the data of a maximum span ag triple (E;F;G) together with a snake = (; ( k ) k=1;:::;n ) determines more than just a line decomposition of (C n ) . In fact, it determines a projective basis of (C n ) . See §2.4 for projective bases. 28 Figure 2.5: Re-naming a ag triple with respect to a snake-head More specically, we will associate to this data a projective basis [U] of (C n ) , where U =fu 1 ; u 2 ;:::; u n g is a linear basis of (C n ) , determined up to a single degree of freedom, and where the square brackets denote the projective class represented by U. All projective basis representatives U will have the property that the co-vector u k 2 (C n ) is an element of the line L k =L ( k ; k ; k ) (C n ) , where k = ( k ; k ; k )2 n1 . We begin the construction. The single degree of freedom appears right at the beginning, as an arbitrary choice of a co-vector u n in the line L n = L (C n ) , recalling from the denition of snakes that the n-th snake-vertex n is always equal to the snake-head 2C( n1 ). Having dened co-vectors u n ; u n1 ;:::; u k+1 , we will dene a co-vector u k 2L k = E ( k ) F ( k ) G ( k ) ? (C n ) : (2.31) 29 Since we have at the beginning xed a snake and in particular a snake-head2C( n1 ), we may take advantage of our notation developed in §§2.14, 2.15. Specically, we may re- express the line L k+1 as L k+1 = E (k) head F ([ k+1 ] left ) left G ([ k+1 ] right ) right ? (C n ) : (2.32) Recall from the very end of §2.14 that given the (k + 1)-snake vertex k+1 , then there are only two vertices in n1 that can possibly be the k-vertex k . Since by denition [ k ] head = k 1, these two possible choices for k , denoted left k+1 and right k+1 , are uniquely determined by their left- and right-coordinates, dened as follows: [ left k+1 ] left = [ k+1 ] left + 1; [ left k+1 ] right = [ k+1 ] right ; (2.33) [ right k+1 ] left = [ k+1 ] left ; [ right k+1 ] right = [ k+1 ] right + 1: (2.34) See Figure 2.6 for a depiction. Then, L left k+1 = E (k1) head F ([ k+1 ] left +1) left G ([ k+1 ] right ) right ? (C n ) ; L right k+1 = E (k1) head F ([ k+1 ] left ) left G ([ k+1 ] right +1) right ? (C n ) : (2.35) It follows by the Maximum Span Property of (E;F;G) that the three lines L k+1 , L left k+1 , L right k+1 in (C n ) are distinct and coplanar. Specically, they lie in the plane E (k1) head F ([ k+1 ] left ) left G ([ k+1 ] right ) right ? (C n ) ; (2.36) which is indeed 2-dimensional, sincek1+[ k+1 ] left +[ k+1 ] right =n2. Consequently, given a nonzero co-vector u k+1 in the lineL k+1 , there exist unique nonzero co-vectors u left k+1 2L left k+1 and u right k+1 2L right k+1 such that u k+1 + u left k+1 + u right k+1 = 0 2 (C n ) ; (2.37) 30 Figure 2.6: Three co-planar lines used to dene a projective basis Denition 2.11 (Dening the projective basis by induction). We are now nally prepared to dene u k 2L k (C n ) as follows: (1) if k = left k+1 2 n1 , dene u k = +u left k+1 2L left k+1 ; and, (2) if k = right k+1 2 n1 , dene u k =u right k+1 2L right k+1 . In Figure 2.6, the denition of k falls into the second case. Thus, by induction, we have dened co-vectors u n ; u n1 ;:::; u 1 2 (C n ) , u k 2 L k , starting with an arbitrary choice of u n 2L n =L . Since the linesL k for 16k6n form a line decomposition of (C n ) , the collection of co-vectorsU =fu 1 ; u 2 ;:::; u n g forms a linear basis of (C n ) . If instead of u n we choose u n , then by the construction u k becomes u k for all k. It follows that we have dened a projective basis [U] = [fu 1 ; u 2 ;:::; u n g] of (C n ) 31 independent of any choices, as desired. 2.17 Elementary snake moves for triangles Let (E;F;G) be a triple of ags satisfying the Maximum Span Property, and let • = (; ( k ) k=1;:::;n ), • 0 = (; ( 0 k ) k=1;:::;n ), be two n-snakes with the same snake-head n = 0 n = 2 C( n1 ). Together with the ag triple (E;F;G), the two snakes ; 0 determine two projective bases [U]; [U 0 ] of the dual linear space (C n ) . By construction of these projective bases, upon xing two base-co-vectors u n ; u 0 n 2L (C n ) we end up xing two linear bases of (C n ) • U =fu 1 ; u 2 ;:::; u n g, • U 0 =fu 0 1 ; u 0 2 ;:::; u 0 n g, representing the projective bases [U]; [U 0 ]. We now ask the following question: Question 2.12. What is the change of basis matrix B U!U 02 GL n (C) transforming from the linear basisU of (C n ) to the linear basisU 0 of (C n ) ? We will see that, if we make a mild normalization assumption, then the answer can be described completely in terms of the geometry of the maximum span ag triple (E;F;G). More precisely, we normalize the problem by requiring the base-co-vectors u n = u 0 n to be equal. Moreover, the problem can be reduced to studying certain elementary pairs of snakes ; 0 that are \adjacent" in an appropriate sense. Let us begin with a combinatorial denition. Denition 2.13. Let ; 0 be two snakes with the same snake-head 2C( n1 ). We say that the ordered pair (; 0 ) is an adjacent pair of snakes if it falls into one of the following two families: 32 (1) if, for some 26k6n 1, the pair (; 0 ) satises (a) j = 0 j (1jk 1, k + 1jn) , (b) k = right k+1 ; and 0 k = left k+1 , in n1 , then the pair (; 0 ) is called an adjacent pair of snakes of diamond-type; (2) if the pair (; 0 ) satises (a) j = 0 j (26j6n) , (b) 1 = right 2 ; and 0 1 = left 2 , in n1 , then the pair (; 0 ) is called an adjacent pair of snakes of tail-type. Here, the vertices right k+1 , left k+1 , right 2 , left 2 are dened by equations (2.33), (2.34). In Figures 2.7 and 2.8 we depict adjacent pairs of diamond- and tail-type. Figure 2.7: Diamond-move 33 Figure 2.8: Tail-move Until we arrive at the next proposition, suppose (; 0 ) is an adjacent pair of snakes of diamond-type, and consider the six snake-vertices k+1 = 0 k+1 , k , 0 k , and k1 = 0 k1 in n , as shown in Figure 2.7. By denition of snakes, the head-coordinates are determined: [ k+1 ] head =k; [ k ] head = [ 0 k ] head =k 1; [ k1 ] head =k 2: (2.38) Since, by denition of diamond-type adjacent snakes, k = right k+1 ; 0 k = left k+1 ; k1 = 0 k1 ; (2.39) it follows that k1 = left k = 0right k : (2.40) We gather that the left- and right- coordinates of the (k 1)-snake-vertex k1 are related 34 to the left- and right-coordinates of the (k + 1)-snake-vertex k+1 by [ k1 ] left = [ 0right k ] left = [ 0 k ] left = [ left k+1 ] left = [ k+1 ] left + 1; (2.41) [ k1 ] right = [ left k ] right = [ k ] right = [ right k+1 ] right = [ k+1 ] right + 1; (2.42) where the rst equality is by equation (2.40), the second and fourth equalities are by de- nition, equations (2.33), (2.34), of the vertices left k and 0right k , and the third equality is by equation (2.39). In summary, our calculations show in particular that the head-, left-, and right- coordi- nates of the three snake-vertices k , 0 k , k1 satisfy [ k ] head = [ 0 k ] head =k 1; [ 0 k ] left = [ k1 ] left = [ k+1 ] left + 1; [ k1 ] right = [ k ] right = [ k+1 ] right + 1: (2.43) See Figure 2.7 above for a depiction of these three vertices, together which form a \downward- facing" triangle in the discrete triangle n1 and which are associated to a single vertex in the discrete triangle n . Recall that the snake-head 2C( n1 ) not only denes head-, left-, and right- coordi- nates on the discrete triangle n1 , but also on the n-discrete triangle n . This discussion, and in particular Equation (2.43), motivates considering the vertex v = (a;b;c) in the n-discrete triangle n dened by the coordinates [v] head =k 1; [v] left = [ k+1 ] left + 1; [v] right = [ k+1 ] right + 1: (2.44) 35 Note that v is indeed in n =f(a;b;c); a +b +c =ng since [v] head + [v] left + [v] right = (k 1) + ([ k+1 ] left + 1) + ([ k+1 ] right + 1) = ([ k+1 ] head 1) + ([ k+1 ] left + 1) + ([ k+1 ] right + 1) = (n 1) 1 + 1 + 1 =n: (2.45) In fact, v2 int( n ) =f(a;b;c)2 n ; a;b;c> 0g, since [v] head =k 1> 2 1> 0; [v] left = [ k+1 ] left + 1> 0 + 1> 0; [v] right = [ k+1 ] right + 1> 0 + 1> 0; (2.46) where we have used that 2 k n 1, since we are assuming that the adjacent pair of snakes (; 0 ) is of diamond-type. According to §2.12, since v2 int( n ), we may therefore compute Fock and Goncharov's triangle invariant X := v (E;F;G)2Cf0g: We are now prepared to give the answer to Question 2.12. Proposition 2.14. Let (E;F;G) be a maximum span ag triple, and let (; 0 ) be an adjacent pair of snakes. As a normalization condition, x two equal base co-vectors u n = u 0 n 2 L , thus determining two linear bases U;U 0 of the dual space (C n ) , representing the projective bases [U], [U 0 ] associated to the ag triple (E;F;G) and the snakes ; 0 , respectively, as discussed in Section 2.16 and at the beginning of this section. In the case that the pair (; 0 ) is of diamond-type, then the change of basis matrix B U!U 02 GL n (C) equals B U!U 0 = diag 0 @ (k1) times z }| { X;X;:::;X; ( 1 1 0 1 ); 1; 1;:::; 1 1 A 2 GL n (C); (2.47) where the integer 2kn 1 is as in the denition of diamond-type adjacent pairs, and 36 where X := v (E;F;G)2 Cf0g is the Fock-Goncharov triangle invariant associated to the interior vertex v2 int( n ) of the n-discrete triangle n dened in equation (2.44). In this case, the change of basis matrix B U!U 0 is called a diamond move transforming from the snake to the adjacent snake 0 . In the case that the pair (; 0 ) is of tail-type, then the change of basis matrix B U!U 02 GL n (C) equals B U!U 0 = diag ( 1 1 0 1 ); 1; 1;:::; 1 ! 2 GL n (C): (2.48) In this case, the change of basis matrix B U!U 0 is called a tail move transforming from the snake to the adjacent snake 0 . This proposition is the main ingredient going into the proof of Theorem 2.8. Remark 2.15. The change of basis matrices appearing in Equations (2.47) and (2.48) are called left matrices because, intuitively, the \motion" moving from the snake to the snake 0 is \to the left". This will be made more precise in later chapters, where we associate these change-of-basis matrices to the \motion" along an oriented curve in the surface S. 2.18 Proof of Proposition 2.14 Before we prove the proposition, we need a general lemma. If (E;F;G) is a maximum span ag triple inC n , and if v = (a;b;c)2 int( n ), namely a;b;c > 0 and a +b +c = n, then there is induced a maximum span ag triple (E;F;G) in the 3-dimensional vector space V =C n =(E (a1) F (b1) G (c1) ) =:C n =V 0 : (2.49) More precisely, the property that the sums E (a 0 ) +F (b1) +G (c1) , E (a1) +F (b 0 ) +G (c1) , and E (a1) +F (b1) +G (c 0 ) are direct for a 0 =a;a + 1, b 0 =b;b + 1, and c 0 =c;c + 1 ensures 37 that E;F;G dene complete ags by E (1) =E (a) =V 0 =E (a) =E (a1) ; E (2) =E (a+1) =V 0 =E (a+1) =E (a1) ; F (1) =F (b) =V 0 =F (b) =F (b1) ; F (2) =F (b+1) =V 0 =F (b+1) =F (b1) ; (2.50) G (1) =G (c) =V 0 =G (c) =G (c1) ; G (2) =G (c+1) =V 0 =G (c+1) =G (c1) ; where V 0 = E (a1) F (b1) G (c1) . Then the fact that (E;F;G) satises the Maximum Span Property follows from the fact that (E;F;G) does as well. Lemma 2.16. For (a;b;c)2 int( n ) and for maximum span triples (E;F;G) and (E;F;G) of ags inC n and V , respectively, where V =C n =(E (a1) F (b1) G (c1) ) ? , we have an equality of Fock-Goncharov triangle invariants abc (E;F;G) = 111 (E;F;G) 2Cf0g: Proof. Write the generatorse (a1) =e,e (a) =e^e 0 , ande (a+1) =e^e 0 ^e 00 , and similarly for f and g. Then this follows from the denitions of V , abc (E;F;G), and 111 (E;F;G). Proof of Proposition 2.14. We begin by proving the easier tail-move, Equation (2.48). Arrange the basesU andU 0 of co-vectors into matrices U and U 0 in GL n (C), with each co-vector u k or u 0 k forming a row vector in the matrix, as described in§2.5. Then the desired Equation (2.48) is, by Equation (2.9), equivalent to U = B U!U 0U 0 2 GL n (C): (2.51) Since u n = u 0 n and k = 0 k fork =n;n 1;:::; 2, it follows by the construction of the bases U andU 0 that u k = u 0 k 2L k =L 0 k for k =n;n 1;:::; 2. It remains to show u 1 ? = u 0 1 + u 0 2 = u 0 1 + u 2 . By denition, u 2 + u left 2 + u right 2 = 0. Also by denition, 0 1 = left 2 implies u 0 1 = +u left 2 , and 1 = right 2 implies u 1 =u right 2 . It follows 38 that u 2 + u 0 1 u 1 = 0 as desired. The more dicult case, which actually involves the geometry, is the diamond-move Equa- tion (2.47). We refer to Figure 2.9 throughout the proof. Figure 2.9: Proof of Proposition 2.14 Again, by Equation (2.9), this is equivalent to U = B U!U 0U 0 2 GL n (C): (2.52) A proof identical to that of the tail-move shows u i = u 0 i for i = n;n 1;:::;k + 1 and u k = u 0 k +u 0 k+1 . It remains to show u i =Xu 0 i fori =k 1;k 2;:::; 1 where againX is the Fock-Goncharov triangle invariant v (E;F;G) where v2 int( n ) satises [v] head =k 1 = [ k+1 ] right 1, [v] left = [ k+1 ] left + 1, and [v] right = [ k+1 ] right + 1. The essential computation is showing u k1 = Xu 0 k1 . Once this is proved, it follows immediately that u i = Xu 0 i for i =k 2;:::; 2; 1 by construction of the projective basis. By assumption, k1 = 0 k1 and so there exists 2Cf0g such that u k1 =u 0 k1 2 L k1 = L 0 k1 . Recall by assumption that k = right k+1 , 0 k = left k+1 , and k1 = 0 k1 , and so k1 = 0 k1 = left k = 0right k . Then by construction of u k1 and u 0 k1 , we have 39 u k1 = +u left k , and u 0 k1 =u 0right k , and in addition the relations u k + u left k + u right k = 0 and u 0 k + u 0left k + u 0right k = 0. It follows that u k + u k1 =u right k 2L right k = E ([ k ] head 1) head F ([ k ] left ) left G ([ k ] right +1) right ! ? = E (k2) head F ([ k+1 ] left ) left G ([ k+1 ] right +2) right ! ? ; (2.53) and u 0 k u 0 k1 =u 0left k 2L 0left k = E ([ 0 k ] head 1) head F ([ 0 k ] left +1) left G ([ 0 k ] right ) right ! ? = E ([k2] head 1) head F ([ 0 k+1 ] left +2) left G ([ 0 k+1 ] right ) right ! ? : (2.54) Since u k+1 = u 0 k+1 2 E (k) head F ([ k+1 ] left ) left G ([ k+1 ] right ) right ! ? ; u k 2 E (k1) head F ([ k+1 ] left ) left G ([ k+1 ] right +1) right ! ? ; u 0 k 2 E (k1) head F ([ 0 k+1 ] left +1) left G ([ 0 k+1 ] right ) right ! ? ; u k1 = u 0 k1 2 E (k2) head F ([ k+1 ] left +1) left G ([ k+1 ] right +1) right ! ? ; (2.55) and since k+1 = 0 k+1 , it follows that u k+1 = u 0 k+1 ; u k ; u 0 k ; u k1 ; u 0 k1 2 E (k2) head F ([ k+1 ] left ) left G ([ k+1 ] right ) right ! ? (C n ) : (2.56) 40 Note that the six lines in (C n ) appearing in Equations (2.53), (2.54), (2.55) correspond to the six vertices shown in Figure 2.9. Consequently, these linear forms induce linear forms u k+1 = u 0 k+1 ; u k ; u 0 k ; u k1 ; u 0 k1 2V where V = C n = E (k2) head F ([ k+1 ] left ) left G ([ k+1 ] right ) right is the quotient of C n by a (n 3)- dimensional subspace, since (k 2) + [ k+1 ] left + [ k+1 ] right = [ k+1 ] head 2 + [ k+1 ] left + [ k+1 ] right = (n 1) 2. Applying Lemma 2.16 to the point v = (a;b;c)2 int( n ) dened above such that X = v (E;F;G) we gather X = 111 (E;F;G) = 111 (E head ;F head ;G head ) = f (2) left ^g (1) right e (2) head ^g (1) right e (1) head ^g (2) right e (1) head ^f (2) left e (2) head ^f (1) left f (1) left ^g (2) right 2Cf0g; (2.57) where E;F;G are the images of E;F;G under the quotient C n −! C n , E ([v] head 1) head F ([v] left 1) left G ([v] right 1) right ! = C n , E (k2) head F ([ k+1 ] left ) left G ([ k+1 ] right ) right ! ; (2.58) and where we have used the property (1) of Fact 2.7 that the triangle invariant 111 (E;F;G) is unchanged under cyclic permutation of the triple of ags (E;F;G). By genericity of (E head ;F left ;G right ), there exist linearly independent vectors f;g;h in V such thatf generatesF (1) left ,g generatesG (1) right , andh generatesF (2) left \G (2) right . Lete 1 inV be a generator of E (1) head . By the Maximum Span Property, e 1 has three nonzero coordinates in the basis f;g;h ofV . So, we may assumef,g, andh are chosen such thate 1 =f +g +h. Choose e 2 in V such thatfe 1 ;e 2 g is a basis for E (2) head and put e 2 = xf +yg +zh. Since e 2 ze 1 is nonzero and independent frome 1 , we may assumee 2 is of the forme 2 =xf +yg. 41 Using the above formula for X, we compute X = f^h^g e 1 ^e 2 ^g e 1 ^g^h e 1 ^f^h e 1 ^e 2 ^f f^g^h = y x : (2.59) In particular, x6= 0. Recall that we have linear forms u k+1 = u 0 k+1 ; u k ; u 0 k ; u k1 ; u 0 k1 2 V . So, we can write these forms in terms of the basis n f ;g ;h o for V dual to the basis f;g;h for V . We saw above that u k + u k1 2 E (k2) head F ([ k+1 ] left ) left G ([ k+1 ] right +2) right ? . Consequently, u k + u k1 vanishes onG (2) right , hence u k + u k1 =Af for someA2C. Similarly, we also saw u 0 k u 0 k1 2 E (k2) head F ([ k+1 ] left +2) left G ([ k+1 ] right ) right ? , hence u 0 k u 0 k1 =Bg for someB2C. Lastly, since u k1 =u 0 k1 2 E (k2) head F ([ k+1 ] left +1) left G ([ k+1 ] right +1) right ? , then u k1 =u 0 k1 vanishes on F (1) left +G (1) right , so u k1 = u 0 k1 = Ch for some C2C. Moreover, A;B;C are nonzero. Indeed, recall u k +u k1 =u right k 6= 02 (C n ) and u 0 k u 0 k1 =u 0left k 6= 02 (C n ) and also u k1 =u 0 k1 6= 02 (C n ) . Since these linear forms induce linear forms inV , their induced linear forms are also nonzero. It follows that u k =Af Ch ; u 0 k =Bg +C 1 h ; u k+1 = u k u 0 k =Af Bg C( 1 + 1)h ; (2.60) where we have used the dening relations u k+1 + u left k+1 + u right k+1 = 0, u 0 k = +u left k+1 , and u k = u right k+1 . Finally, since u k+1 2 E (k) head F ([ k+1 ] left ) left G ([ k+1 ] right ) right ? then u k+1 vanishes on E (2) head , since u k 2 E (k1) head F ([ k+1 ] left ) left G ([ k+1 ] right +1) right ? then u k vanishes onE (1) head +G (1) right , and since u 0 k 2 E (k1) head F ([ k+1 ] left +1) left G ([ k+1 ] right ) right ? then u 0 k vanishes on E (1) head +F (1) left . 42 Consequently, remembering e 1 =f +g +h and e 2 =xf +yg, 0 = u k+1 (e 2 ) =AxBy; 0 = u k (e 1 ) =AC; 0 = u 0 k (e 1 ) =B +C 1 : (2.61) Therefore, = C B = A B = y x =X; (2.62) as desired. 2.19 Sketch of proof of Fock and Goncharov's theorem In this section, based on [19, §9], we want to outline the main ideas behind the proof of Theorem 2.8. This is an application of the theory developed in §§2.14-2.18. Along the way, we will prove some handy facts that are useful when working with triples of ags in practice. There are two parts of the theorem: existence and uniqueness. We give a complete proof for uniqueness, and only brie y touch on the proof of existence. Uniqueness. Consider the rst statement of the theorem. We want to show that two maximum span triples of ags (E;F;G) and (E 0 ;F 0 ;G 0 ) (c.f. Denition 2.4) have the same triangle invariants abc (E;F;G) = abc (E 0 ;F 0 ;G 0 ) inCf0g for all (a;b;c)2 int( n ) if and only if there is a linear isomorphism ' of C n taking the ag triple (E;F;G) to the triple (E 0 ;F 0 ;G 0 ), i.e. ('(E);'(F );'(G)) = (E 0 ;F 0 ;G 0 ) 2 Flag(C n ) 3 : (2.63) One of these directions is obvious, since triangle invariants are preserved under applying a linear isomorphism to a maximum span ag triple (E;F;G). So, for the other direction, as- sume that we are given two triples (E;F;G) and (E 0 ;F 0 ;G 0 ) with the same triangle invariants abc . We need to construct the desired linear isomorphism '. 43 Step 1. For the time being, we consider only one of the ag triples, say (E;F;G). Let 2C( n1 ) be a snake-head. Consider the two snakes bot and top uniquely determined by the property that they both have snake-head , that bot has constant left-coordinate equal to zero, and that top has constant right-coordinate equal to zero (c.f. Equation (2.25)). These two snakes should be thought of as living on two edges of the discrete triangle n1 , with their snake-heads located at the common corner vertex . Let bot = 1 ; 2 ;:::; N1 ; N = top be a sequence of adjacent snakes \slithering" from bot to top (c.f. Denition 2.13). Each snake i determines a projective basis U i of (C n ) , which we may assume are all normalized so that u i n = u i 0 n for alli;i 0 . Namely, each projective basis has the same co-vector u i n located in the snake-head line L =L i (C n ) . By Proposition 2.14, for each i = 1; 2;:::;N 1, the change-of-basis matrix B U i !U i+1 is completely determined by the triangle invariants abc (E;F;G). Consequently, by Equation (2.10), this is also the case for the full change-of-basis matrix B U bot !U top = B U 1 !U 2B U 2 !U 3 B U N1 !U N 2 GL n (C); (2.64) transforming from the projective basis for the bottom snake to that for the top snake. To see what this all means for triples of ags, which after all live in C n not (C n ) , we need to pass to the dual basis, c.f. §2.3. By Equation (2.11), B U bot !U top = B V bot V top =: B 2 GL n (C); (2.65) so consequently the matrix B on the right hand side of this equation is also completely determined by the triangle invariants abc (E;F;G). In summary, given just the triangle invariants abc (E;F;G) and a snake-head , one can re-construct the matrix B. Recall that we think of a basisU of (C n ) as a matrix of row vectors U = u 1 u 2 . . . un ! 2 GL n (C) and we think of its dual basis V ofC n as a matrix of column vectors V = ( v 1 v 2 vn )2 GL n (C), where V = U 1 2 GL n (C). By the denition of B V bot V top, Equation (2.7), it is 44 immediate that V bot B V bot 1 ! V bot = V top 2 GL n (C): (2.66) We will interpret this equation momentarily. Step 2. Throughout this step, we keep the notation from Step 1. Given the snake-head , recall the ag re-indexing notation E head , G right , F left , see Equation (2.30). We have the following useful result: Lemma 2.17. Write V bot = (v bot 1 v bot 2 v bot n ) and V top = (v top 1 v top 2 v top n ) in GL n (C). Then, for all 1a;bn, E (a) head = Span v bot 1 ; v bot 2 ;:::; v bot a = Span v top 1 ; v top 2 ;:::; v top a ; G (b) right = Span v bot nb+1 ;:::; v bot n1 ; v bot n ; F (b) left = Span v top nb+1 ;:::; v top n1 ; v top n C n : (2.67) Proof. Fix k2f1;:::;ng. For each kk 0 n, u bot k 0 2 E (k 0 1) head G (nk 0 ) right ? E (k 0 1) head ? E (k1) head ? (C n ) : (2.68) Since the dimension of E (k1) head ? is n (k 1), it follows that E (k1) head ? = Span u bot k ; u bot k+1 ;:::; u bot n (C n ) : (2.69) Let the canonical isomorphismC n = (C n ) be denoted by v7!b v, whereb v(u) = u(v) for all u2 (C n ) . Then any subspace WC n corresponds to the subspace W ?? (C n ) under the canonical isomorphism. Now, by denition of the dual basis V bot dual to the basis U bot , for all k k 0 n and 1 ` k 1 we have b v bot ` (u bot k 0 ) = u bot k 0 (v bot ` ) = 0. Consequently, eachb v bot ` vanishes on E (k1) head ? (C n ) , that is b v bot ` 2 E (k1) head ?? (C n ) . Therefore, v bot ` 2E (k1) head for all 1`k 1. We conclude that E (a) head = Span v bot 1 ; v bot 2 ;:::; v bot a for all 1an 1 by the linear independence of the basis vectors v bot 1 ; v bot 2 ;:::; v bot n . 45 The proofs for the three other equations go the same way. For instance, the above expression implies u bot k 0 2 G (nk 0 )? right G (nk)? right for all 1 k 0 k. Continue from there as before, then at the end replace k with nk. Equation (2.66) together with Lemma 2.17 gives us the following two interesting facts: Fact 2.18. There exists a linear isomorphism ' ofC n such that '(E head ;G right ) = (E head ;F left ) 2 Flag(C n ) 2 : (2.70) We think of the mapping ' as a \local monodromy operator" that \maps ags forward". Proof. Choose the snake-head such that E = E head , F = F left , and G = G right . Then we can take ' := V bot B V bot 1 2 GL n (C): (2.71) Fact 2.19. Let E 0 and G 0 be the standard ascending and descending ags, respectively (c.f. Equation (2.13)). If the snake-head is chosen such that E = E head , F = F left , and G = G right , then there exists a linear isomorphism ' ofC n such that 'E =E 0 ; 'G =G 0 ; 'F = BG 0 2 Flag(C n ): (2.72) Proof. By the two equations V bot 1 V bot = Id n ; V bot 1 V top = B 2 GL n (C); (2.73) we can take ' := V bot 1 2 GL n (C): (2.74) Step 3. Consider once again two maximum span ag triples (E;F;G) and (E 0 ;F 0 ;G 0 ) with the same triangle invariants abc (E;F;G) = abc (E 0 ;F 0 ;G 0 ). Our goal is to produce a 46 linear isomorphism ofC n mapping one ag triple to the other. Choose a snake-head such thatE =E head ,F =F left , andG =G right . Then by denition E 0 =E 0 head ,F 0 =F 0 left , andG 0 =G 0 right as well. These two ag triples induce two matrices B and B 0 , as in Equation (2.65). Since the triangle invariants agree, B = B 0 in GL n (C). By Fact 2.19, there exist linear isomorphisms ' and ' 0 ofC n such that 'E =E 0 ; 'G =G 0 ; 'F = BG 0 ; ' 0 E 0 =E 0 ; ' 0 G 0 =G 0 ; ' 0 F 0 = BG 0 2 Flag(C n ): (2.75) Thus, the composition ' 01 ' sends the ag triple (E;F;G) to the ag triple (E 0 ;F 0 ;G 0 ). Existence. Given xed nonzero complex numbers x abc 2 Cf0g for each (a;b;c)2 int( n ), we want to construct a maximum span ag triple (E;F;G) satisfying the property that the triangle invariants abc (E;F;G) =x abc 2Cf0g (a;b;c)2 int( n ) ! ; (2.76) agree with the prescribed numbers x abc . In light of Lemma 2.17 and Fact 2.19, the answer is somewhat obvious. It is the proof that is more challenging. Supposing such a generic triple (E;F;G) existed, choose the snake- head such that E head =E, G right =G, andF left =F , and then consider the matrix B (c.f. Equation (2.65)), which is a function of the abc (E;F;G) = x abc . Even though we do not know the form of (E;F;G) we do know the exact form of B as a function ofx abc , ultimately due to Proposition 2.14. Now, starting with just the numbers x abc and no triple of ags, all we can do is dene a matrix B = B(x abc ) by the known formula. In accordance with Fact 2.19, we then dene E :=E 0 ; G :=G 0 ; F := B(x abc )G 0 2 Flag(C n ): (2.77) 47 Lemma 2.20. The ag triple (E;F;G) dened in Equation (2.77) satises the Maximum Span Property. Moreover, its triangle invariants abc (E;F;G) for (a;b;c)2 int( n ) are equal to the prescribed numbers x abc 2Cf0g. One fun corollary, which is more or less part of the proof, is the following observation: Fact 2.21. For the \standard form" maximum span ag triple (E;F;G) constructed in Equation (2.77), the bottom projective and dual projective bases U bot = 0 B @ u bot 1 u bot 2 . . . u bot n 1 C A = V bot = (v bot 1 v bot 2 v bot n ) = Id n 2 GL n (C); (2.78) are both equal to the identity matrix, assuming the normalization u bot n = e n (see Eq. (2.5)). This is expected from Equation (2.66) together with Lemma 2.17 and Equation (2.77), and can be easily deduced by direct calculation of the bottom projective basis once it is known that (E;F;G) is a maximum span ag triple. The consequence is that, when the ag triple is in \standard form" (c.f. Equation (2.77)), then the \local monodromy operator" that \maps ags forward" (c.f. Fact 2.18) can be computed entirely in terms of the Fock-Goncharov invariants abc (E;F;G) =x abc . This has profound implications for the geometric theory, and is the intuitive starting point for our algebraic work in Chapters 3 and 4. We omit a proof of Lemma 2.20. However, if one assumes that all of the numbersx abc > 0 are positive, then there are elementary positivity properties of the Fock-Goncharov matrices (c.f. Proposition 2.14) that make the proof almost trivial. For arbitrary x abc 2Cf0g, the a priori concern is that there might be \un-desirable cancellations" that ruin genericity. This completes our theoretical discussion of Theorem 2.8. We end with some examples. Example 2.22. Using Lemma 2.20, we will describe all maximum span triples of ags (E;F;G) for n = 2; 3; 4 by giving their \standard form". Recall that in all cases E = E 0 andG =G 0 are the standard ascending and descending ags. So we just need to provideF . 48 In order to describe a general ag F , we will give a generating basis B =ff 1 ;f 2 ;:::;f n g for the ag, meaning that the subspace F (b) is spanned by the vectors f 1 ;f 2 ;:::;f b . n=2. There are no triangle invariants. The matrix B and generating basisB for F are B = ( 1 1 0 1 ); B =f( 1 1 ); ( 1 0 )g: (2.79) n=3. There is a single triangle invariant X = x 111 . The matrix B(X) and generating basisB for the ag F are B(X) = X X+1 1 0 1 1 0 0 1 ; B =f 1 1 1 ; X+1 1 0 ; X 0 0 g: (2.80) Remark 2.23. This example already brings into play the subtlety of the notion of \gener- icity" for triples of ags. By Lemma 2.20, this example produces a maximum span ag triple (E;F;G) in the case X =1. On the other hand, we saw in the proof of Proposition 2.6 that this ag triple, namely when X =1, does not satisfy the Minimum Intersection Property (however, by Proposition 2.5, its dual ag triple does!). n=4. There are three triangle invariants X 1 = x 112 , X 2 = x 211 , and X 3 = x 121 . The matrix B(X 1 ;X 2 ;X 3 ) and generating basisB for the ag F are B(X) = X 1 X 2 X 3 X 1 X 2 X 3 +X 1 X 2 +X 2 X 1 X 2 +X 2 +1 1 0 X 2 X 2 +1 1 0 0 1 1 0 0 0 1 ; B =f 1 1 1 1 ; X 1 X 2 +X 2 +1 X 2 +1 1 0 ; X 1 X 2 X 3 +X 1 X 2 +X 2 X 2 0 0 ; X 1 X 2 X 3 0 0 0 g: (2.81) 2.20 Right snakes and right matrices If there are left matrices (see Remark 2.15), then there should also be a right matrices. The setup is similar to that for left matrices, but there are also important dierences coming from the \symmetry breaking" that occurred when we dened snakes. In the denition a snake that we have been using up to this point, there is a snake- 49 head 2 C( n1 ) and a sequence of snake-vertices ( 1 ;:::; n )2 ( n1 ) n such that the snake-head = n is equal to the n-th snake-vertex. But we could have equally dened a snake such that the snake-head = 1 is equal to the rst snake vertex (and the rest of the denition adjusted accordingly in the obvious way). To distinguish between these two denitions, we will refer to the second denition as a right snake ( = 1 ; ( k ) k=1;:::;n ). It turns out that when we make this second choice for the denition of a snake, so that the snake-head = 1 , then the algorithm dening the projective basis associated to a snake also needs to be adjusted. Specically, Denition 2.11 should be changed to read (1) if k = left k1 2 n1 , dene u k =u left k1 2L left k1 ; and, (2) if k = right k1 2 n1 , dene u k = +u right k1 2L right k1 . Notice that what was before (+;) has now become (; +), and we have also adjusted the indices accordingly. Here, the notion of \left" and \right" is dened with respect to the snake-head 2C( n1 ) as always, see §2.14. Preparing now to formulate an analogue of Proposition 2.14 but for right snakes, we dene an ordered pair (; 0 ) of right snakes having the same snake-head = 1 = 0 1 to be an adjacent pair of right snakes of diamond-or tail-type, respectively, if either (1) for some 26k6n 1, the pair (; 0 ) satises (a) j = 0 j (1jk 1, k + 1jn) , (b) k = left k1 ; and 0 k = right k1 ; or, (2) if the pair (; 0 ) satises (a) j = 0 j (16j6n 1) , (b) n = left n1 ; and 0 n = right n1 . 50 Given an adjacent pair of right snakes of diamond-type, there is a corresponding vertex v2 n in the discrete n-triangle carrying its associated Fock-Goncharov triangle invariant X2Cf0g, just as there was in §2.17. The following result is the analogue of Proposition 2.14 for right snakes. Proposition 2.24. For an adjacent pair of right snakes (; 0 ) of diamond- or tail-type, with associated projective bases U and U 0 of (C n ) , respectively, normalized so that u 1 = u 0 1 in L =L 1 =L 0 1 (C n ) , then either B U!U 0 = diag 1;:::; 1; 1; ( 1 0 1 1 ); (nk) times z }| { X 1 ;:::;X 1 ;X 1 ! 2 GL n (C); (2.82) in the case of the diamond-move, or B U!U 0 = diag 1;:::; 1; 1; ( 1 0 1 1 ) ! 2 GL n (C); (2.83) in the case of the tail-move. Proof. This is similar to the proof of Proposition 2.14. Remark 2.25. From now on, whenever we speak of a \snake", we always mean with respect to the original Denition 2.10, so that the snake-head = n . We will rarely if ever refer to \right snakes" again, and if we do we will make sure to alert the reader. 2.21 Triangle invariants as shears LetV be a complex vector space. By a line inV we mean a 1-dimensional complex subspace of V . By a plane we mean a 2-dimensional subspace. Denition 2.26. A shear from a line L 1 in V to another line L 2 in V is simply a linear isomorphism :L 1 !L 2 . 51 Let (E;F;G) be a maximum span ag triple, and let 2C( n1 ) be a snake-head. Let ; 0 be two snakes (using the original denition) having the same snake-head and forming an adjacent pair (; 0 ) of diamond-type. Let the integer k, the vertex v2 int( n ), and the nonzero complex number X =T v (E;F;G) be dened as in Proposition 2.14. In particular, consider the three snake-vertices k ; 0 k ; k1 = 0 k1 2 n1 ; (2.84) together forming a downward-facing triangle in the (n1)-discrete triangle n1 , and whose head-, left-, and right-coordinates, computed in equation (2.43), motivated the denition of the vertex v (c.f. Denition 2.44). We recall Figure 2.9. The proof of Proposition 2.14 hinged on a computation of the projective basis co-vectors u k ; u 0 k ; u k1 ; u 0 k1 2 (C n ) ; (2.85) where the co-vector u j lives in the line L j and the co-vector u 0 j lives in the line L 0 j . In particular, the co-vectors u k1 =Xu 0 k1 2L k1 dier by the ratio X. The computation of the projective basis co-vectors appearing in equation (2.85) involves not only the three lines determined by the snake-vertices appearing in equation (2.84), but in addition the three lines determined by the vertices k+1 = 0 k+1 ; 0left k ; right k 2 (C n ) ; (2.86) which together form a bigger upward-facing triangle in the (n 1)-discrete triangle n1 containing the smaller downward-facing triangle formed by the three vertices of equation (2.84). Refer once again to Figure 2.9 to see these triangles. There are three smaller upward- facing triangles contained in this bigger upward-facing triangle, namely those three smaller 52 triangles with vertex sets 1 = k+1 = 0 k+1 ; 0 k ; k ; 2 = 0 k ; 0left k ; k1 = 0 k1 ; 3 = n k ; k1 = 0 k1 ; right k o : (2.87) The crucial property used to dene the projective basis co-vectors in (2.85) is that the three vertices of each of these smaller upward-facing triangles correspond to three distinct lines in (C n ) , together which lie in a single plane in (C n ) . Consequently, each smaller upward- facing triangle i denes six shears 12 :L 1 !L 2 , 23 :L 2 !L 3 , and 31 :L 3 !L 1 , as well as their inverses, among the three co-planar lines, call them L 1 ;L 2 ;L 3 associated to i . For instance, the shear from L 1 to L 2 sends the point p 1 in L 1 to the point p 2 in L 2 such that p 1 +p 2 +p 3 = 0 2 (C n ) ; (2.88) where p 2 and p 3 are the unique points in the lines L 2 and L 3 , respectively Shears help us to understand conceptually the ratio u k1 =Xu 0 k1 appearing in Propo- sition 2.14. Once again, Figure 2.9 will be helpful for the following result: Proposition 2.27. Using the notation from the current section, let p 0 be an arbitrary point in the line L k1 = L 0 k1 . Now, let p 1 be the point in the line L 0 k resulting from the shear L k1 !L 0 k associated to the triangle 2 applied to the point p 0 , let p 2 be the point in the lineL k resulting from the shearL 0 k !L k associated to the triangle 1 applied to the point p 1 , and let p 3 be the point in the line L k1 = L 0 k1 resulting from the shear L k ! L k1 associated to the triangle 3 applied to the point p 2 . Then p 3 = +Xp 0 : (2.89) This was the case going counter-clockwise around the downward-facing triangle. If instead 53 one goes clockwise around the triangle, then the total shearing is +X 1 . Proof. This is more or less equivalent to the proof of Proposition 2.14. Indeed, using the notation from §2.18, one checks that, at the level of the quotient V , • starting with the point p 0 =h , the triangle 2 associates the point p 1 =g h ; • starting with the point p 1 , the triangle 1 associates the point p 2 =X(f h ); • starting with the point p 2 , the triangle 3 associates the point p 3 =Xh =Xp 0 . 2.22 Elementary snake moves for edges Let (E;G;F;F 0 ) be a maximum span ag quadruple, dened in§2.10. According to§2.12, for each j = 1;:::;n 1 we may consider the Fock-Goncharov edge invariant j (E;G;F;F 0 )2 Cf0g associated to the maximum span ag quadruple (E;G;F;F 0 ). In§2.17 we computed change-of-basis matrices for snake moves traveling across a triangle from one edge to another. Now we discuss snake moves associated to a single edge lying at the interface between two triangles. As depicted in Figure 2.10, we formalize each triangle as a copy of the discrete triangle n1 . One triangle n1 (G;F;E) has a maximum span ag triple (G;F;E) associated to the corner vertices ((n 1; 0; 0); (0;n 1; 0); (0; 0;n 1)), and the other triangle n1 (E;F 0 ;G) has a maximum span ag triple (E;F 0 ;G) associated to the corner vertices ((n 1; 0; 0); (0;n 1; 0); (0; 0;n 1)). Consider now two snakes and 0 in n1 (G;F;E) and n1 (E;F 0 ;G), respectively, dened as follows. The rst snake, , is dened by k := (nk; 0;k 1) 2 n1 (G;F;E) k = 1;:::;n ! ; (2.90) and the second snake, 0 , is dened by 0 k := (k 1; 0;nk) 2 n1 (E;F 0 ;G) k = 1;:::;n ! : (2.91) 54 Figure 2.10: Shearing snakes Associated to the snakes and 0 are line decompositions of (C n ) coming from the triples (G;F;E) and (E;F 0 ;G), respectively. Note that the line decompositions are the same: L k =L 0 k = E (k1) G (nk) ? (C n ) (k = 1;:::;n): (2.92) Moreover, these snakes and 0 determine projective bases [U] and [U 0 ], respectively, of (C n ) adapted to this line decomposition. If we normalize by xing two equal base co-vectors u n = u 0 n in L n =L 0 n , then we obtain two basesU;U 0 2 GL n (C) of (C n ) , and we may ask to determine the change of basis matrix B U!U 0. In the gure, the triangular \plates" shown attached to each snake indicate the lines involved in the denition of the projective bases. 55 Proposition 2.28. Given the above notation, the change of basis matrix is B U!U 0 = n1 Y j=1 diag j (E;G;F;F 0 )Id j ; Id nj ! 2 GL n (C); (2.93) where j (E;G;F;F 0 )2 Cf0g is the j-th Fock-Goncharov edge invariant associated to the maximum span quadruple of ags (E;G;F;F 0 ), and where Id j and Id nj denote identity matrices. Note that the order in which these diagonal matrices are multiplied does not matter. Proof. This follows along the same lines as does the proof of Proposition 2.14. First, one proves the proposition in the case n = 2. Namely, let V be a 2-dimensional vector space equipped with a generic quadruple of ags (E;G;F;F 0 ) having generators (e;g;f;f 0 ), and equipped with snakes = ( 1 ; 2 ) and 0 = ( 0 1 ; 0 2 ) dened as above. If u 2 = u 0 2 2hei ? , if u 1 := +u left 2 2hgi ? , and if u 0 1 :=u 0right 2 2hgi ? (c.f. 2.11), then u 1 =Zu 0 1 2V ; (2.94) where Z := 1 (E;G;F;F 0 ) = e^f e^f 0 g^f 0 g^f 2Cf0g; (2.95) is the single edge invariant for the common edge between the two triangles. To show this, choose e and g such that f =e +g and f 0 =xe +yg for nonzero numbers x andy. One computes thatZ =x=y. Letfe ;g g be the dual basis forV . Without loss of generality, assume u 2 = u 0 2 =g . Put u 1 =Ae and u 0 1 =Be . Using u 2 + u 1 2 f ? and u 0 2 u 0 1 2 D f 0 E ? , one computes A =1 and B =y=x. Equation (2.94) follows. Next, we proceed by induction. We want to show u k =Z k Z k+1 Z k+2 Z n1 u 0 k 2 (C n ) k = 1; 2;:::;n ! ; (2.96) where Z j := j (E;G;F;F 0 ) for j = 1;:::;n 1 is the j-th edge invariant. The base case is 56 trivial, since u n = u 0 n by denition. The induction hypothesis is that, fork = 1; 2;:::;n 1, then u k+1 =Z k+1 Z k+2 Z n1 u 0 k+1 . Dene V to be the 2-dimensional quotient vector space V :=C n , E (k1) G (nk1) ! : (2.97) (Figure 2.12 might be helpful, with k instead of j.) The maximum span quadruple of ags (E;G;F;F 0 ) inC n descends to a maximum span quadruple of ags (E;G;F;F 0 ) inV . Check that the forms u k+1 , u 0 k+1 , u right k+1 , u 0left k+1 , u k , u 0 k (c.f. 2.11) in (C n ) descend to forms in V , denoted u k+1 , u 0 k+1 , u right k+1 , u 0left k+1 , u k , u 0 k . In particular, we have the equivalence u k =Z k Z k+1 Z n1 u 0 k 2 (C n ) () u k =Z k Z k+1 Z n1 u 0 k 2V : (2.98) Trying to keep with the notation at the beginning of the proof for the case n = 2 (and possibly abusing notation as a result), dene u 2 := u k+1 and u 0 2 := Z k+1 Z k+2 Z n1 u 0 k+1 in V . Note by the induction hypothesis that u 2 = u 0 2 . These induce u 1 := +u left 2 and u 0 1 :=u 0right 2 in V as above. The key observation is that the two edge invariants Z k := k (E;G;F;F 0 ) = 1 (E;G;F;F 0 ) 2Cf0g; (2.99) are the same. By our work above, namely Equation (2.94), it follows that u 1 =Z k u 0 1 2V : (2.100) We claim that u 1 = u k and u 0 1 =Z k+1 Z n1 u 0 k (Warning: possible abuse of notation, see previous paragraph), which, if true, establishes Equation (2.96) via Equations (2.100) and (2.98). Indeed, this follows as, by denition, u k+1 +u k +u right k+1 = 0 and u 0 k+1 u 0 k +u 0left k+1 = 0 in (C n ) , and this equality descends toV , thereby determining u 1 and u 0 1 by uniqueness. 57 As described more fully in the next section, we think of this proposition as describing how the snake 0 is obtained from the snake by shearing according to the Fock-Goncharov edge invariants. Next, we once again consider only a single discrete triangle n (E;F;G). This time, let and 0 be two snakes dened by k = (nk; 0;k 1) 2 n1 k = 1;:::;n ! ; (2.101) and 0 k = (k 1; 0;nk) 2 n1 k = 1;:::;n ! ; (2.102) as shown in Figure 2.11. Figure 2.11: Clockwise U-turn In contrast to the previous case involving two triangles, the lines L k 6=L 0 k in (C n ) are not equal, and in fact L k =L 0 nk . So this time, we ask for our normalization condition to be u n = u 0 1 2L = (G (n1) ) ? , and we compute the corresponding change of basis matrix. 58 Proposition 2.29. Given the above notation, the change of basis matrix B U!U 02 GL n (C) is o-diagonal of the form B U!U 0 ! i(ni+1) = (1) (ni) 2C i = 1;:::;n ! : (2.103) Equivalently, the entry in the last row is always +1 and then the entries alternate between +1 and1 as we go up the rows. Proof. This is also done by induction, using Denition 2.11. 2.23 Edge invariants as shears Analogous to §2.21, we show that if (E;G;F;F 0 ) is a maximum span ag quadruple, then Fock and Goncharov's edge invariants k (E;G;F;F 0 ) can be conceptually understood as measuring the shearing from a line in (C n ) to itself. We recall that in the discrete triangle n1 , every small upward-facing triangle (viewed relative to a base of the discrete triangle) determines three distinct co-planar lines in (C n ) . A point in any one of these three lines determines unique points in the other two lines. As we did in§2.22, consider an edge at the interface of two triangles n1 (E;F 0 ;G) and n1 (G;F;E) as shown in Figure 2.12. Recall that for each k = 1;:::;n, the lines L (k1;0;nk) (E;F 0 ;G) =L (nk;0;k1) (G;F;E) = E (k1) G (nk) ! ? (C n ) ; (2.104) are equal. The edge at the interface between the two triangles is divided into n 1 small intervals I j bounded by two points (j; 0;nj 1) and (j 1; 0;nj) in n1 (E;F 0 ;G) on the left and the right, respectively. Equivalently, the interval I j is bounded by (nj 1; 0;j) and (nj; 0;j 1) in n1 (G;F;E) on the left and the right, respectively. This is shown in Figure 2.12, where instead of displaying the interval I j we display the edge invariant Z j 59 Figure 2.12: Edge invariants as shears which is associated to I j , as we will explain momentarily. This interval I j forms the horizontal diagonal of a small diamond, half of which lies in n1 (E;F 0 ;G) and half of which lies in n1 (G;F;E). Each half is a small triangle. Let us call the upward-facing triangle 0 = 0 (E;F 0 ;G) and the downward-facing (relative to the paper) triangler =r(G;F;E) living in n1 (E;F 0 ;G) and n1 (G;F;E), respectively. Then the small triangles 0 andr are constituted of vertices 0 =fv 0 0 ; v 0 1 ; v 0 2 g; r =fv 0 ; v 1 ; v 2 g; (2.105) where these vertices v 0 2 n1 (E;F 0 ;G) and v2 n1 (G;F;E) are dened in Figure 2.12. 60 We are now ready to prove an analogue of Proposition 2.27 for the Fock-Goncharov edge invariants Z j := j (E;G;F;F 0 ) for j = 1;:::;n 1. Beware of the un-avoidable sign. Proposition 2.30. Using the notation from the current section, let p 0 be an arbitrary point in the line L 0 := L v 0 0 (E;F 0 ;G). Now, let p 1 be the point in the line L 1 := L v 0 1 (E;F 0 ;G) = L v 1 (G;F;E) resulting from the shearL 0 !L 1 associated to the small upward-facing triangle 0 in n1 (E;F 0 ;G) applied to the point p 0 , and let p 2 be the point in the line L 2 := L v 0 (G;F;E) =L 0 resulting from the shearL 1 !L 2 associated to the small downward-facing triangler in n1 (G;F;E) applied to the point p 1 . Then p 2 =Z j p 0 2L 2 =L 0 ; (2.106) where Z j = j (E;G;F;F 0 ) is the j-th Fock-Goncharov edge invariant. This was the case going counter-clockwise around the diamond having horizontal diagonal I j . If instead one goes clockwise around the diamond, then the total shearing isZ 1 j . Proof. This is more or less equivalent to the proof of Proposition 2.28. Indeed, using the notation from the proof of that proposition, one checks that, at the level of the quotient V , • starting with the point p 0 =e , the triangle 0 associates the point p 1 =Zg ; • starting with the point p 1 , the triangler associates the point p 2 =Ze . 61 Part II Quantum aspects 62 Chapter 3 Points of quantum SL n coming from Fock-Goncharov snakes Conceptually, this chapter is motivated by Chapters 2 and 4. However, it makes sense for expository reasons to put it here. Throughout, we employ Fock and Goncharov's ideas appearing in [18, 21], see also [30]. For very related work, see also [56, 55, 14]. For the theory of quantum groups, see for instance [7, 37, 39, 47, 48]. Throughout the chapter, x an integer n> 2, a nonzero complex number q2Cf0g, and a n 2 -root ! =q 1 n 2 of q. The integerN plays a more uid role, and should be interpreted according to the context. 3.1 Quantum tori, matrix algebras, and Weyl ordering Let P be an integer NN antisymmetric matrix, for some positive integer N > 0 (here P stands for \Poisson"). Denition 3.1. The quantum torus (with n-th roots) T(P ) associated to P is the quotient of the free algebraC n X 1 n 1 ;X 1 n 1 ;:::;X 1 n N ;X 1 n N o in the indeterminatesX 1 n i by the two-sided 63 ideal generated by the relations X m i n i X m j n j =! P ij m i m j X m j n j X m i n i (m i ;m j 2Z); (3.1) and X m n i X m n i =X m n i X m n i = 1 (m2Z): (3.2) Dene X i := (X 1 n i ) n . By abuse of terminology, we refer to the X i as the generators of the quantum torusT(P ). Dene Q n := n m n ; m2Z o Q: (3.3) Written in terms of the generators X i and the fractionsQ n , the relations above become the more palatable X r i i X r j j =q P ij r i r j X r j j X r i i (r i ;r j 2Q n ); (3.4) and X r i X r i =X r i X r i = 1 (r2Q n ): (3.5) A vector space basis of the quantum torusT(P ) consists of ordered monomials B(T(P )) =fX r 1 1 X r N N ; r i 2Q n g: (3.6) Let T 1 ;T 2 ;:::;T M be quantum tori with generators X i1 ;X i2 ;:::;X iN i . Form the tensor product algebra e T = N M i=1 T i , with the usual algebra structure on e T dened by (p 1 p 2 p M ) (p 0 1 p 0 2 p 0 M ) =p 1 p 0 1 p 2 p 0 2 p M p 0 M ; (3.7) extended linearly to all tensors (here p stands for \polynomial"). In particular, we may 64 naturally viewT i as a subalgebraT i e T via the algebra embedding p i 7! 1 1 p i 1 1; (3.8) where in the tensor product the p i appears in thei-th place. It follows that the subalgebras T i ;T j e T commute if i6= j. The tensor product algebra e T is also a quantum torus, with vector space basis X r 11 11 X r 1N 1 1N 1 X r 21 21 X r 2N 2 2N 2 X r M1 M1 X r MN M MN M ; r ij 2Q n = X r 11 11 X r 1N 1 1N 1 X r 21 21 X r 2N 2 2N 2 X r M1 M1 X r MN M MN M ; r ij 2Q n ; (3.9) where the last equality implicitly uses the natural embeddingsT i e T. If the antisymmetric matrices deningT 1 ;:::;T M areP 1 ;:::;P M , then the antisymmetric matrix e P dening e T is determined by X ij X i 0 j 0 = 8 > > < > > : q (P i ) jj 0 X i 0 j 0X ij ; i =i 0 ; X i 0 j 0X ij ; i6=i 0 : (3.10) Denition 3.2. Let T be a, possibly-non-commutative, algebra, and let n be a positive integer. The matrix algebra with coecients in T, denoted M n (T), is the complex vector space of nn matrices, equipped with the usual \left-to-right" multiplicative structure. Namely, the product MN of two matrices M and N is dened entry-wise by (MN) ij = n X k=1 M ik N kj 2T 1i;jn ! : (3.11) Here, we use the usual convention that the entry M ij of a matrix M is the entry in the i-th row and j-th column. Note that, crucially, the order of M ik and N kj in the above equation matters since the algebraT may not be commutative. 65 IfT is a quantum torus, then there is a set function []:T!T; (3.12) called the Weyl quantum ordering, dened on possibly-non-ordered monomials m =aX r 1 i 1 X r K i K ; (3.13) where a2C, by the equation [m] :=q 1 2 P 1k<`K P i k i ` r k r ` m; (3.14) and dened on polynomials p = P i m i by [p] = P i [m i ]. The Weyl ordering is specially designed so that [X r 1 i 1 X r K i K ] = [X r (1) i (1) X r (K) i (K) ]; (3.15) for every permutation off1;:::;Kg. The Weyl ordering extends to a set function of matrix algebras []: M n (T)! M n (T); (3.16) by putting [M] ij := [M ij ]: (3.17) 3.2 Fock-Goncharov quantum torus for a triangle Recall the discrete triangle n and its interior int( n ) from§2.11. Recall thatC( n ) denotes the set of corner verticesf(n; 0; 0); (0;n; 0); (0; 0;n)g of the discrete triangle n . We think 66 Figure 3.1: Quiver dening relations in the Fock-Goncharov algebra of the discrete triangle n as overlayed on top of a topological triangle T. Dene a set function P : n C( n ) ! n C( n ) ! !f2;1; 0; 1; 2g; (3.18) using the quiver with vertex set n C( n ) illustrated in Figure 3.1. The function P is dened by sending the tuple (v 1 ;v 2 ) of vertices of n C( n ) to 2 (resp. 2) if there is a solid arrow pointing from v 1 tov 2 (resp. v 2 tov 1 ), to 1 (resp.1) if there is a dotted arrow pointing from v 1 to v 2 (resp. v 2 to v 1 ), and to 0 if there is no arrow connecting v 1 and v 2 . Notice that all the small downward-facing triangles are oriented clockwise, and all the small upward-facing triangles (except the three on the corners) are oriented counter-clockwise. By labeling the vertices of n C( n ) by their coordinates (a;b;c) we may think of the function P as a \matrix" P = (P abc;a 0 b 0 c 0), called the Poisson matrix associated to the quiver. Denition 3.3. Dene the Fock-Goncharov quantum torus FG ! SLn(C) (T) associated to the triangle T to be the quantum torusT(P ) dened by the Poisson matrix P , with generators X abc for all (a;b;c)2 n C( n ). For j = 1; 2;:::;n 1, we write Z j (resp. Z 0 j ) in place of X j0(nj) (resp. X j(nj)0 ). 67 3.3 Shearing and triangle matrices Let T be a quantum torus. For j = 1; 2;:::;n 1 and any element Z2 T such that Z r makes sense for allr2Q n (for instance, Z could be any generator ofT, or a product of two commuting elements ofT), dene matrices S j (Z) (whereS stands for \shear") in M n (T) by S j (Z) =Z j n 0 B B B B @ Z Z . . . Z 1 1 . . . 1 1 C C C C A 2 M n (T) Z appears j times ! : (3.19) Notice the normalizing factor Z j n multiplying the matrix on the left (or on the right). Similarly, for j = 1; 2;:::;n 1 and any element X2T such that X r makes sense for all r2Q n , dene matrices T left j (X) (where T stands for \triangle") in M n (T) by T left j (X) =X j1 n 0 B B B B B @ X X . . . X 1 1 1 1 . . . 1 1 C C C C C A 2 M n (T) X appears j 1 times ! : (3.20) Observe that the triangle matrix T left j (X) is precisely the normalization of the tail- or diamond-move change-of-basis matrix B U!U 02 GL n (C) appearing in Proposition 2.14, de- pending on whether j = 1 or j > 1, respectively, and where in the latter case X plays the role of the Fock-Goncharov triangle invariant associated to the diamond. Also observe that the shearing matrix S j (Z) is precisely the normalization of the j-th multiplicand for the edge change-of-basis matrix appearing in Proposition 2.28, where Z plays the role of the j-th Fock-Goncharov edge invariant. Remark 3.4. In the classical setting, normalizing the change-of-basis matrix in order to obtain either T left j (X) or S j (Z) requires choosing an n-th root of X or Z. However, in the current algebraic setting, this choice is built into the denition of the quantum torus. Even 68 so, classically speaking, and in particular when studying the geometry, often X and Z are taken to be positive, in which case there is a canonical choice of positive n-th root. We also need a matrix T right j (X) corresponding to the change-of-basis matrix associated to diamond-moves for right snakes, as described in Proposition 2.24. This is T right j (X) =X j1 n 0 B B B B B B @ 1 . . . 1 1 1 1 X 1 . . . X 1 X 1 1 C C C C C C A 2 M n (T) X appears j 1 times ! ; (3.21) dened for j = 1; 2;:::;n 1. 3.4 Classical left, right, and edge matrices Recall from Chapter 2 that, for the discrete triangle n1 equipped with a maximum span ag triple (E;F;G), if 2 C( n1 ) is a snake-head, if bot is the bottom snake dened by [ k ] left = 0 for all k = 1;:::;n, and if top is the top snake dened by [ k ] right = 0 for all k = 1;:::;n, then the change-of-basis matrix B U bot !U top from the projective basis U bot associated to the bottom snake bot to the projective basisU top associated to the top snake top , normalized so that u bot n = u top n in the lineL (C n ) , is completely determined by the triangle invariants abc (E;F;G)2Cf0g associated to interior vertices (a;b;c)2 int( n ) of the discrete triangle n . Actually writing down the matrix B U bot !U top requires choosing a sequence of adjacent snakes bot = 1 ; 2 ;:::; N = top going from the bottom snake to the top snake. For then B U bot !U top = N1 Y i=1 B U i !U i+1 = B U 1 !U 2 B U 2 !U 3 B U N1 !U N 2 GL n (C); (3.22) where the matrices B U i !U i+1 on the right hand side are the change of basis matrices between the projective bases U i associated to the snakes i (all normalized to have the same n-th 69 co-vector in the line L ). By Proposition 2.14, these matrices are all of the form T left j (X i ) before normalization, as discussed in the previous section. The normalized matrix B U bot !U top=Det(B U bot !U top) 1 n is called the left matrix, denoted M left . See Remark 3.4. So far, the variables X i have been non-zero complex numbers. If, instead, we think of them as indeterminates X i in a polynomial ring, then the left matrix can be thought of as M left = M left (X i 's) = Y T left j (X i ) 2 SL n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]) = SL n (FG 1 SLn(C) (T)); (3.23) observing that the Fock-Goncharov algebra FG 1 SLn(C) (T) evaluated at q = 1 is the same as the polynomial ringC[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]. Similarly, in the setting of right snakes, as described in §2.20, the normalized matrix B U bot !U top=Det(B U bot !U top) 1 n is called the right matrix, denoted M right . Here,U bot andU top are dened by the appropriate choices of right snakes, which we have not specied precisely, but which should be fairly clear from the discussion in §2.20 (more examples below). Thus, M right = M right (X i 's) = Y T right j (X i ) 2 SL n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]) = SL n (FG 1 SLn(C) (T)): (3.24) We also saw in Proposition 2.28 how if two triangles share a common edge, say the bottom edge, then there is associated a normalized edge matrix E bot = E bot (Z bot j 's) = B U bot !U bot0 , Det B U bot !U bot0 !1 n = n1 Y j=1 S j (Z bot j ) 2 SL n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]) = SL n (FG 1 SLn(C) (T)); (3.25) obtained as the product of n 1 shearing matrices S j (Z bot j ), and transforming between the two coordinate systems dened by two opposite-facing snakes on the common bottom edge 70 between the two triangles, one snake corresponding to one triangle and the other snake corre- sponding to the other triangle. Similarly, there is an edge matrix E top (Z top j 's) corresponding to the top (left or right) edge. Remark 3.5. Note that the shearing matrices S j (Z j ) commute with each other, and so the order in which they are multiplied is immaterial for the denition of the edge matrix E(Z j 's). On the other hand, the order in which the triangle matrices T left j (X i ) are multiplied in order to dene the left matrix M left (X i 's) is important, and depends on the sequence of snakes used to go from the bottom snake to the top snake. Note that M left (X i 's), however, is dened independently of which sequence of snakes is chosen. 3.5 Quantum left and right matrices Recall, given a quantum torusT with Poisson matrix P , the denition of the Weyl ordering [], c.f. Equations (3.12)-(3.17). In particular, in Equation (3.17), we showed how the Weyl ordering can be thought of as a set function of matrix algebras [] : M n (T)! M n (T). We want to consider a slightly dierent mapping of matrix algebras. Note that there is a multi-valued function C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] m.v. ! T that sends a \commutative polynomial" p2C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] to the set of all \non-commutative polynomials"fP p gT that can be formed fromp by treating its variables as non-commuting. Rigorously, this multi-valued function is the inverse of the projection of the free algebra C n X 1 n 1 ;X 1 n 1 ;:::;X 1 n N ;X 1 n N o ontoC[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ], namely the usual (commutative) polynomial ring, followed by the projection of the free algebra onto the quantum torusT. There is then induced a multi-valued function M n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]) m.v. ! M n (T). By the symmetric property of the Weyl ordering, Equation (3.15), this multi-valued function of matrix algebras induces a bona de single-valued set function by the composition [] : M n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]) m.v. −! M n (T) [] −! M n (T): (3.26) 71 We wish to apply this function of matrix algebras to the classical matrices of the previous sec- tion, with coecients in the classical polynomial ringC[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] =FG 1 SLn(C) (T), when the quantum torusT is taken to be the Fock-Goncharov algebraT =FG ! SLn(C) (T). Denition 3.6. The quantum left matrix L ! = L ! (X i ;Z bot j ;Z top j 's) is dened by L ! = L ! (X i ;Z bot j ;Z top j 's) = " E bot (Z bot j 's)M left (X i 's)E top (Z top j 's) # 2 M n FG ! SLn(C) (T) ! ; (3.27) is the Weyl ordering of the product of the three classical matrices, namely the shearing matrix associated to the bottom edge, the left matrix associated to the movement of snakes from the bottom edge to the top edge, and the shearing matrix associated to the top edge. Similarly, the quantum right matrix R ! = R ! (X i ;Z bot j ;Z top j 's) is dened by R ! = R ! (X i ;Z bot j ;Z top j 's) = " E bot (Z bot j 's)M right (X i 's)E top (Z top j 's) # 2 M n FG ! SLn(C) (T) ! : (3.28) 3.6 Quantum SL n and its points Let M n (FG ! SLn(C) (T)) denote the non-commutative algebra ofnn matrices with coecients in the Fock-Goncharov algebraFG ! SLn(C) (T). See Denition 3.2. Denition 3.7. We say 2 2 matrix M = ( a b c d ) in M 2 (T), for some complex, possibly-non- commutative, algebraT, is aT-point of M q 2 , denoted M2 M q 2 (T), if ba =qab; dc =qcd; ca =qac; db =qbd; bc =cb; daad = (qq 1 )bc 2T: (3.29) We say a matrix M2 M 2 (T) is aT-point of SL q 2 , denoted M2 SL q 2 (T), if M2 M q 2 (T); (3.30) 72 and in addition the quantum determinant Det q (M) :=daqbc =adq 1 bc = 1 2T; (3.31) is equal to 1. Denition 3.8. LetT be as in the previous denition. We say a nn matrix M2 M n (T) is aT-point of M q n , denoted M2 M q n (T), if every 2 2 sub-matrix of M is aT-point of M q 2 . Namely, M im M ik =qM ik M im ; M jm M im =qM im M jm ; (3.32) M im M jk =M jk M im ; M jm M ik M ik M jm = (qq 1 )M im M jk ; for all i<j and k<m, where 1i;j;k;mn. There also exists a notion of the quantum determinant Det q (M)2 T of a matrix M2 M n (T). Note that it follows from the denition of the quantum determinant that if M is a triangular matrix in M q n (T) M n (T), then the diagonal entries M ii pairwise commute and Det q (M) = n Y i=1 M ii 2T: (3.33) We say M2 M n (T) is aT-point of the dual quantum group SL q n , denoted M2 SL q n (T), if both M2 M q n (T) and the quantum determinant Det q (M) = 1. Remark 3.9. Observe that the subsets M q n (T) M n (T) and SL q n (T) M n (T) are not necessarily even linear subspaces (hence, nor subalgebras) of M n (T). 3.7 First theorem Let T =FG ! SLn(C) (T) be the Fock-Goncharov algebra associated to a triangle T, as dened in§3.2. Let L ! and R ! be the quantum left and right matrices, respectively, dened in§3.4. 73 We now prove a precise version of Theorem 2 from the Introduction. Theorem 3.10. The quantum left and right matrices L ! ; R ! 2 SL q n FG ! SLn(C) (T) ! M n FG ! SLn(C) (T) ! ; (3.34) areFG ! SLn(C) (T)-points of SL q n . First, we will summarize the proof, then give some concrete examples of the theorem in §3.8, and nish by providing more details for the proof in §§3.9-3.13. Sketch of proof. In the case n = 2, this is an enjoyable calculation. We outline the proof in the case n> 3, which requires a more conceptual approach. The argument hinges on the following well-known fact: If T is an algebra with subalgebras T 0 ;T 00 T that commute in the sense that a 0 a 00 = a 00 a 0 for all a 0 2 T 0 and a 00 2 T 00 , and if M 0 2 M n (T 0 ) M n (T) and M 00 2 M n (T 00 ) M n (T) areT-points of SL q n , then the matrix product M 0 M 00 2 M n (T 0 T 00 ) M n (T) is also aT-point of SL q n . Let us put M FG := L ! , the quantum left matrix, say. The proof will go the same for the quantum right matrix. The strategy is to see M FG 2 M n (FG ! SLn(C) (T)) as the product of simpler mutually-commuting matrices that are themselves points of SL q n . More precisely, for a xed sequence of snakes bot = 1 ; 2 ;:::; N = top moving left across the triangle from the bottom edge to the top-left edge, we will dene for each i = 1;:::;N 1 an auxiliary algebra S ! i , called the snake-move algebra, corresponding to the adjacent pair ( i ; i+1 ) of snakes. Now, as a technical point, there is a distinguished minimal sub-algebra A FG ! SLn(C) (T) such that M FG 2 M n (A) M n (FG ! SLn(C) (T)). We construct an algebra embedding A ,! N i S ! i . By virtue of this embedding, we may view M FG 2 M n (A) M n ( N i S ! i ). Following, we construct, for each i, a matrix M i 2 M n (S ! i ) M n ( N i S ! i ) with the property that M i is aS ! i -point of SL q n , in other words M i 2 SL q n (S ! i ) SL q n ( N i S ! i ). Since the subalgebrasS ! i ;S ! i 0 N i S ! i commute ifi6=i 0 , as they constitute dierent tensor factors 74 of N i S ! i , it follows from the fact mentioned above that M := M 1 M 2 M N1 2 M n ( N i S ! i ) is a ( N i S ! i )-point of SL q n , in other words M2 SL q n ( N i S ! i ). Now, since this matrix product M, as well as the quantum left matrix M FG , are being viewed as elements of M n ( N i S ! i ), it makes sense to ask whether M FG = M in M n ( N i S ! i ). Indeed, this turns out to be true, implying that M FG is in SL q n ( N i S ! i ). Since, moreover, M FG 2 M n (A) M n ( N i S ! i ), we obtain that M FG is in SL q n (A) SL q n (FG ! SLn(C) (T)). 3.8 Concrete formulas and examples The quantum left- and right-matrices L ! and R ! are, in theory, well-dened by Equations (3.27) and (3.28). But, in our experience, it is useful to see some explicit formulas. We recall that, in order to compute the matrices M left (X i 's) and M right (X i 's) appearing in the denitions of L ! and R ! , respectively, one needs to choose a sequence of adjacent snakes bot = 1 ; 2 ;:::; N = top going from the bottom edge to either the top-left or top-right edge, respectively. However, the resulting matrices M left (X i 's) and M right (X i 's) are independent of the choice of sequence, since by Propositions 2.14 and 2.24 they are just the change-of-basis matrices from the bottom snake to the top snake. We remind that for left matrices one uses our usual denition of snakes as in §2.14, and that for right matrices one uses the denition of right snakes as in §2.20. The dierence is that for left snakes the snake-head is = n , and for right snakes the snake-head is = 1 . In the spirit of concreteness, for the rest of this chapter, for left matrices we will take the snake-head, bottom snake, and top snake to be = (n 1; 0; 0) 2C( n1 ); bot k = (k 1; 0;nk) 2C( n1 ) k = 1;:::;n ! ; top k = (k 1;nk; 0) 2C( n1 ) k = 1;:::;n ! ; (3.35) 75 and for right matrices we will take the snake-head, bottom snake, and top snake to be = (0; 0;n 1) 2C( n1 ); bot k = (k 1; 0;nk) 2C( n1 ) k = 1;:::;n ! ; top k = (0;k 1;nk) 2C( n1 ) k = 1;:::;n ! : (3.36) (a) Left sequence (b) Right sequence Figure 3.2: Snake sequences for n = 5 To explicitly write down the matrices M left (X i 's) and M right (X i 's) we will choose the sequence of adjacent snakes bot = 1 ; 2 ;:::; N = top illustrated in the case n = 5 in Figure 3.2 and generalized in the obvious way to the case of arbitrary n. Then, explicit expressions for L ! and R ! are given by (see below in this paragraph for notation explanation) L ! = " n1 Y j=1 S j (Z j ) ! 1 a i=n1 T left 1 i Y j=2 T left j X (j1)(ni)(ij+1) !!! n1 Y j=1 S j (Z 0 j ) !# ; (3.37) 76 and R ! = " n1 Y j=1 S j (Z j ) ! 1 a i=n1 T right 1 i Y j=2 T right j X (ij+1)(ni)(j1) !!! n1 Y j=1 S j (Z 00 j ) !# ; (3.38) where we have used the shearing and triangle matrices S j (Z), T left j (X), and T right j (X) dened in Equations (3.19), (3.20), and (3.21). Notice that, by its denition, T left 1 (X) and T right 1 (X) do not actually involve the variable X, and so we have simply denoted these particular matrices by T left 1 and T right 1 , respectively, in the equations above. Here, we need to be clear about our notational conventions for multiplying matrices. We dene N Y i=M M i := M M M M+1 M N ; M Y i=N+1 M i := 1 MN ! ; M a i=N M i := M N M N1 M M ; N a i=M1 M i := 1 MN ! : (3.39) This multiplicative convention is only relevant for the triangle matrices T left j and T right j appearing in Equations (3.37) and (3.38), because, since L ! and R ! are dened using the Weyl ordering (Equation (3.17)), the order of the diagonal shearing matrices S j within the Weyl ordering is immaterial. We also have to be clear about our notation for the variablesZ, Z 0 ,Z 00 ,X appearing in these two equations. ByX abc we mean the usual indexing of the Fock- Goncharov triangle invariants by interior points of the discrete n-triangle (a;b;c)2 int( n ). (Warning: We remind that snakes live in the discrete (n 1)-triangle n1 , such as those triangles shown in Figure 3.2.) As for the shearing variables Z, Z 0 , Z 00 , in the case of • both the left and right matrices, the variable Z j is associated to the segment of the discrete (n 1)-triangle n1 between the vertices (j; 0;nj 1) and (j 1; 0;nj); • the left matrix, the variable Z 0 j is associated to the segment between the vertices (j;nj 1; 0) and (j 1;nj; 0); • the right matrix, the variable Z 00 j is associated to the segment between the vertices 77 (0;j 1;nj) and (0;j;nj 1). For a picture of the Z j 's, compare the top triangle in Figure 2.12. In the left (resp. right) case, the positions of the Z 0 j 's (resp. Z 00 j 's) are obtained from the positions of the Z j 's by rigid rotations about the snake-head = (n 1; 0; 0) (resp. = (0; 0;n 1)), see Figure 3.3. (a) Left and right matrices (b) Quiver Figure 3.3: The case n = 4 As an example in the case n = 4, see Figure 3.3. On the right hand side of the gure we show the commutation relations in the Fock-Goncharov algebraFG ! SL 4 (C) (T), recalling Figure 3.1 and the denitions given in Equations (3.1) and (3.4). For instance, the following are some sample commutation relations: X 3 Z 00 2 =q 2 X 3 Z 00 2 ; X 3 X 1 =q 2 X 1 X 3 ; Z 3 Z 2 =qZ 2 Z 3 ; Z 3 Z 0 3 =q 2 Z 0 3 Z 3 : (3.40) Using the particular snake sequences described above, recalling Figure 3.2 (but adjusted for the case n = 4) the quantum left and right matrices are computed as L ! = " Z 1 4 1 Z 2 4 2 Z 3 4 3 Z 1 Z 2 Z 3 Z 2 Z 3 Z 3 1 1 1 1 1 1 X 1 4 1 X 1 1 1 1 1 X 2 4 2 X 2 X 2 1 1 1 1 1 1 1 1 X 1 4 3 X 3 1 1 1 1 1 1 1 1 1 Z 0 1 4 1 Z 0 2 4 2 Z 0 3 4 3 Z 0 1 Z 0 2 Z 0 3 Z 0 2 Z 0 3 Z 0 3 1 !# ; (3.41) 78 and R ! = " Z 1 4 1 Z 2 4 2 Z 3 4 3 Z 1 Z 2 Z 3 Z 2 Z 3 Z 3 1 1 1 1 1 1 X + 1 4 2 1 1 1 1 X 1 2 X + 2 4 1 1 1 1 X 1 1 X 1 1 ! 1 1 1 1 1 X + 1 4 3 1 1 1 1 X 1 3 1 1 1 1 1 Z 00 1 4 1 Z 00 2 4 2 Z 00 3 4 3 Z 00 1 Z 00 2 Z 00 3 Z 00 2 Z 00 3 Z 00 3 1 !# : (3.42) The theorem says that these two matrices are elements of SL q 4 (FG ! SL 4 (C) (T)). For instance, the entries a;b;c;d of the 2 2 sub-matrix (arranged as a 4 1 matrix) of L ! a b c d = L ! 13 L ! 14 L ! 23 L ! 24 ! = 0 B B B B B @ [Z 1 4 3 Z 2 4 2 Z 3 4 1 Z 0 1 4 3 Z 0 2 4 2 Z 0 1 4 1 X 1 4 1 X 2 4 2 X 1 4 3 ]+[Z 1 4 3 Z 2 4 2 Z 3 4 1 Z 0 1 4 3 Z 0 2 4 2 Z 0 1 4 1 X 1 4 1 X 2 4 2 X 1 4 3 ] +[Z 1 4 3 Z 2 4 2 Z 3 4 1 Z 0 1 4 3 Z 0 2 4 2 Z 0 1 4 1 X 3 4 1 X 2 4 2 X 1 4 3 ] [Z 1 4 3 Z 2 4 2 Z 3 4 1 Z 0 3 4 3 Z 0 2 4 2 Z 0 1 4 1 X 1 4 1 X 2 4 2 X 1 4 3 ] [Z 1 4 3 Z 2 4 2 Z 1 4 1 Z 0 1 4 3 Z 0 2 4 2 Z 0 1 4 1 X 1 4 1 X 2 4 2 X 1 4 3 ]+[Z 1 4 3 Z 2 4 2 Z 1 4 1 Z 0 1 4 3 Z 0 2 4 2 Z 0 1 4 1 X 1 4 1 X 2 4 2 X 1 4 3 ] [Z 1 4 3 Z 2 4 2 Z 1 4 1 Z 0 3 4 3 Z 0 2 4 2 Z 0 1 4 1 X 1 4 1 X 2 4 2 X 1 4 3 ] 1 C C C C C A ; (3.43) satisfy Equation (3.29). For a computer verication of this, see Appendix 5.16. We also check, in the same appendix, that Equation (3.29) is satised by the entries a;b;c;d of the 2 2 sub-matrix (arranged as a 4 1 matrix) of R ! a b c d = R ! 31 R ! 32 R ! 41 R ! 42 ! = 0 B B B B B @ [Z 1 4 3 Z 1 2 2 Z 1 4 1 X 1 4 2 X 1 2 1 X 1 4 3 Z 00 1 4 3 Z 00 1 2 2 Z 00 3 4 1 ] [Z 1 4 3 Z 1 2 2 Z 1 4 1 X 1 4 2 X 1 2 1 X 1 4 3 Z 00 1 4 3 Z 00 1 2 2 Z 00 1 4 1 ]+[Z 1 4 3 Z 1 2 2 Z 1 4 1 X 1 4 2 X 1 2 1 X 1 4 3 Z 00 1 4 3 Z 00 1 2 2 Z 00 1 4 1 ] [Z 3 4 3 Z 1 2 2 Z 1 4 1 X 1 4 2 X 1 2 1 X 1 4 3 Z 00 1 4 3 Z 00 1 2 2 Z 00 3 4 1 ] [Z 3 4 3 Z 1 2 2 Z 1 4 1 X 3 4 2 X 1 2 1 X 1 4 3 Z 00 1 4 3 Z 00 1 2 2 Z 00 1 4 1 ]+[Z 3 4 3 Z 1 2 2 Z 1 4 1 X 1 4 2 X 1 2 1 X 1 4 3 Z 00 1 4 3 Z 00 1 2 2 Z 00 1 4 1 ] +[Z 3 4 3 Z 1 2 2 Z 1 4 1 X 1 4 2 X 1 2 1 X 1 4 3 Z 00 1 4 3 Z 00 1 2 2 Z 00 1 4 1 ] 1 C C C C C A : (3.44) 3.9 Snake-move quantum tori After these empirical observations, we return to the proof of Theorem 3.10. We will focus on the case of the quantum left matrix L ! . 79 In the proof sketch of the theorem, we said that, given a sequence bot = 1 ; 2 ;:::; N = top going from the bottom snake to the top-left snake, then to every i = 1;:::;N 1 we will construct an algebra S ! j i corresponding to the adjacent pair of snakes ( i ; i+1 ). More specically, for each j = 1;:::;n 1, where n 1 is the rank of the Lie group SL n (C), we will construct an algebraS ! j called thej-th snake-move algebra, and each algebraS ! j i will be a copy of one of these n 1 snake-move algebrasS ! j . Denition 3.11. For j = 1;:::;n 1, the j-th snake-move quantum torus S ! j := T(P ) is the quantum torus with Poisson matrix P dened by the quiver shown in Figure 3.4 in the case j = 2;:::;n 1, and shown in Figure 3.5 in the case j = 1. The quantum torus has one generator per edge of the quiver. As usual, the quiver denes the anti-symmetric structure matrixP controlling theq-commutation relations in the quantum torusS ! j , where solid arrows carry a weight 2, and dotted arrows carry a weight 1, c.f. Equation (3.4). In Figure 3.4, we conceptualize the three generators x j1 ;z j1 ;z 0 j1 living in the snake- diamond-move algebraS ! j as quantum versions of the three shears appearing in Proposition 2.27, together which constitute the triangle invariant X associated to the downward-facing triangle. We will further exploit this intuition in§3.11, as it suggests a natural way to embed the (sub-algebraA of the) Fock-Goncharov algebraFG ! SLn(C) (T) into a tensor product N i S ! j i of snake-move algebras. Figure 3.4: Diamond snake-move algebra (j = 2;:::;n 1) 80 Figure 3.5: Tail snake-move algebra (j = 1) Remark 3.12. As indicated in the two gures, we think of each snake-move quiver as divided into two snake halves (except for possibly a joining diagonal generator). If we consider a snake sequence i moving left from the bottom snake bot to the top-left snake top , then intuitively we will insert a snake-move algebra S ! j i between each adjacent pair ( i ; i+1 ) of snakes. But, even better, we rst slice each snake i in half down the length of the snake, and then we associate the snake-move algebra S ! j i to the top-half 1=2 of i and to the bottom- half 0 1=2 of i+1 . In this way, each half-snake is associated to a unique snake-move algebra. (Except for the bottom-most and top-most half-snakes 0bot 1=2 and top 1=2 . But if we imagine that the triangle is embedded in a triangulation of a surface, then these un-paired bottom-most and top-most half-snakes 0bot 1=2 and top 1=2 , which lie on the bottom and top-left edges of the triangle, respectively, will in fact be paired with other snake-move algebras living in adjacent triangles.) See Figure 3.6, which uses the particular snake sequence shown in Figure 3.2a. 3.10 Quantum elementary snake-move matrices For each j = 1; 2;:::;n 1, we are now going to produce a matrix M j that is aS ! j -point of SL q n , valued in the j-th snake-move algebraS ! j . Lemma 3.13. Let j = 1;:::;n 1. Consider the matrix M j = " n1 Y j=1 S j (z j ) ! T left j (x j1 ) n1 Y j=1 S j (z 0 j ) !# 2 M n (S ! j ); (3.45) 81 Figure 3.6: Quantum snake sweep, shown in the case n = 4 recalling the denitions of S j (z) and T left j (x) appearing in Equations (3.19) and (3.20). Then, M j is an element of SL q n (S ! j ), namely, M j is aS ! j -point of SL q n . Note that whenj = 1, then the matrix T left 1 (x 0 ) = T left 1 does not involve any variablesx 0 , and so is well-dened despite the fact that x 0 is not dened. Proof. This is a direct calculation, checking that all of the entries of the matrix M j satisfy the relations of the dual quantum group SL q n in the j-th snake-move algebraS ! j . Example 3.14. In the case n = 4, the lemma says that the 4 4 matrix M 3 = " z 1 4 1 z 2 4 2 z 3 4 3 z 1 z 2 z 3 z 2 z 3 z 3 1 x 2 4 2 x 2 x 2 1 1 1 z 0 1 4 1 z 0 2 4 2 z 0 3 4 3 z 0 1 z 0 2 z 0 3 z 0 2 z 0 3 z 0 3 1 !# ; (3.46) with coecients in the snake-move algebraS ! 3 is an element of SL q 4 (S ! 3 ), where the variables q-commute according to the quiver appearing in Figure 3.4 taking n = 4 and j = 3. 82 3.11 Embedding a Fock-Goncharov sub-algebra into a tensor product of snake-move algebras Let bot = 1 ; 2 ;:::; N = top be a sequence of adjacent snakes going from the bottom snake to the top-left snake, such as our standard example, c.f. Figure 3.2a. To the snake pairs ( i ; i+1 ) constituting this snake sweep, we associate snake-move algebras S ! j i , recall Remark 3.12, and also Figure 3.6 showing the case n = 4. Recall also the Fock-Goncharov algebra FG ! SLn(C) (T) associated to the whole triangle T, c.f. Figure 3.3b for the quiver dening theq-commutation relations in the algebraFG ! SLn(C) (T) in the case n = 4. We are now forced into a technicality (see Remark 3.15 below). Consider the \minimal" sub-algebraA ofFG ! SLn(C) (T) such that the full quantum left matrix L ! (c.f. Denition 3.6 and Theorem 3.10) can be viewed as a matrix with coecients inA, i.e. L ! 2 M n (A). More precisely, dene A FG ! SLn(C) (T) to be the sub-algebra generated by all of the generators (and their inverses) ofFG ! SLn(C) (T) except for Z 00 1 n 1 ;:::;Z 00 1 n n1 , c.f. Figure 3.3. We claim that this data of a snake sweep (f i ; i+1 g) N1 i=1 denes an embedding of algebras A,−! N1 O i=1 S ! j i ; (3.47) realizing the algebra A as a sub-algebra of a tensor product of snake-move algebras S j i (tensored from left to right), eachS ! j i being associated to the adjacent pair of snakes ( i ; i+1 ). Remark 3.15. After much eort, we could not successfully put all of the puzzle pieces together in such a way that the whole Fock-Goncharov algebra FG ! SLn(C) (T) embeds into the tensor product, hence the necessity to restrict to the sub-algebra A. However, it might be natural in the theory to embed the whole algebra FG ! SLn(C) (T) into a \braided tensor product" of some sort, c.f. the discussion in §4.14. But we are not sure at this point. We explain the mapping through an example, such as that appearing in Figure 3.7 in the 83 case n = 4, compare also Figure 3.6. For instance, in the example shown in Figure 3.7, the generators X 2 (emphasized in the gure), Z 1 , Z 0 3 2A are mapped to X 2 7! 1 z 0 2 z 2 x 2 z 0 2 z 2 z 0 2 z 2 1; Z 1 7!z 1 1 1 1 1 1; Z 0 3 7! 1 1 z 0 3 z 3 z 0 3 z 3 z 0 3 z 3 z 0 3 2S ! 1 S ! 2 S ! 3 S ! 1 S ! 2 S ! 1 ; (3.48) Notice that monomials (say, z 2 x 2 z 0 2 ) appearing in a single tensor factor of the image consist of mutually commuting generators, and so the order in which they are written is irrelevant. From the gure, we see that this denes an algebra homomorphism, Equation (3.47). This is partly due to the fact that most of the dotted arrows, c.f. Figures 3.4 and 3.5, namely those lying at the interface between two snake-move algebras, \cancel each other out". Injectivity is due primarily to the property that every generator (i.e. edge) appearing on the right hand side of Figure 3.7 corresponds to a unique generator on the left hand side. 3.12 Finishing the proof of the theorem We return to the proof of Theorem 3.10, and in particular we refer back to the sketch of the proof discussed immediately following the statement of the theorem. We are at the stage where we have proved • M FG := L ! 2 M n (A) M n N N1 i=1 S ! j i ! ; • M := M 1 M 2 M N1 2 SL q n N N1 i=1 S ! j i ! M n N N1 i=1 S ! j i ! . To nish the proof, it remains to show M FG = M 2 M n N1 O i=1 S ! j i ! : (3.49) This is not dicult, however it is a bit tricky to write down the combinatorics (for this, Figure 3.7 is helpful). The key idea is to commute the many variables (e.g. Equation (3.48)) 84 Figure 3.7: Embedding into the tensor product of snake-move algebras appearing on the right hand side M of Equation (3.49), until M, dened via Equation (3.45) applied to the various j i , has been put into the form of the left hand side M FG , as dened by Equation (3.27) (followed by applying the embeddingA,−! N i S ! j i ). To do this, we use two simple facts. We omit the rest of the details of the proof. Lemma 3.16. (1) If M 1 ; M 2 ;:::; M N1 are matrices with coecients in arbitrary quan- 85 tum toriT 1 ;T 2 ;:::;T N1 , respectively, viewed as factors inT 1 T 2 T N1 , then [M 1 ][M 2 ] [M N1 ] = " M 1 M 2 M N1 # 2 M n T 1 T 2 T N1 ! : (3.50) That is, \the product of the Weyl orderings is the Weyl ordering of the product". (2) For the matrices S j (z) and T left j (x) appearing in Equations (3.19), (3.20), and (3.45), if we assume in addition that the variables z and x commute, then S j (z)T left j 0 (x) = T left j 0 (x)S j (z) if and only if j6=j 0 : (3.51) 3.13 Setup for the quantum right matrix We end with a few words about the proof for the quantum right matrix M FG := R ! , which essentially goes the same as for the left matrix. (1) The right version of the j-th snake algebra S ! j for j = 1; 2;:::;n 1 is given by replacing the quivers of Figures 3.4 and 3.5 by the quivers shown in Figures 3.8 and 3.9, respectively. (2) The j-th quantum elementary matrix M j of Lemma 3.13 is replaced by M j = " n1 Y j=1 S j (z j ) ! T right j (x nj+1 ) n1 Y j=1 S j (z 0 j ) !# 2 M n (S ! j ): (3.52) Compare Equation (3.42). Note T right 1 (x n ) = T right 1 is independent ofx n , which is un-dened. (3) The sub-algebraAFG ! SLn(C) (T) is generated by all but the Z 0 j 's, c.f. Figure 3.3. 86 Figure 3.8: Right diamond snake-move algebra (j = 2;:::;n 1) Figure 3.9: Right tail snake-move algebra (j = 1) 87 Chapter 4 Quantum traces for SL n (C): the case n = 3 The primary reference for this chapter is [5]. See also [27, 49] and [45, 44, 15]. For the theory of quantum groups, see for instance [7, 37, 39, 47, 48]. As in the previous chapter, we x an integer n> 2, a nonzero complex number q2 Cf0g, and a n 2 -root ! =q 1 n 2 of q. The integerN plays a more uid role, and should be interpreted according to the context. 4.1 Topological setup For this chapter, the topological setting in which we work mimics that appearing in [5, §1.1]. Specically, let S be a possibly-disconnected oriented punctured surface possibly- with-boundary, obtained by removing nitely many points called punctures from a compact surfaceS. We require that every boundary component of S has a puncture, that there is at least one puncture, and that the Euler characteristic (S)<d=2 of the resulting punctured surface S is less than half the number of components d of the boundary @S. In particular, every boundary component of the punctured surface S is homeomorphic to an open interval (0; 1). These topological conditions guarantee the existence of an ideal triangulation of 88 the punctured surface S, namely a triangulation of the compact surface S whose vertex set is equal to the set of punctures. For some examples, see Figure 4.1. (a) Four-times punctured sphere (b) Once punctured torus Figure 4.1: Examples of ideal triangulations Remark 4.1. For the remainder of the thesis, we always assume that ideal triangulations never contain any self-folded triangles. In other words, all triangles in the triangulation have three distinct edges. 4.2 Two classical trace polynomials In§1.1, we explained that an oriented closed curve on the surfaceS induces a trace function Tr on the character variety X SLn(C) (S) by the natural assignment 7! Tr(( ))2 C for every group homomorphism : 1 (S) ! SL n (C) from the fundamental group of S to SL n (C). Denote by : 1 (S)! PSL n (C) the projectivization of the representation . We also discussed how, given an ideal triangulation of S, Fock and Goncharov [19] as- sociate to certain projectivized representations (equipped with boundary data)N complex coordinates X i 2Cf0g, where the number N of coordinates depends only on the integer n and the topology of the surface S. Moreover, Fock and Goncharov provide a recipe that reconstructs the (conjugacy class of the) projective representation from its coordinatesX i . 89 In particular, for representations : 1 (S)! SL n (C) projecting to the projective represen- tation, and for an oriented closed curve immersed in the surfaceS, Fock and Goncharov give an explicit formula expressing, up to sign, the evaluation of the trace function Tr ()2C on the representation as a Laurent polynomial Tr () = Tr ()(X i ) = c Tr (X i ) in (n-th roots of) the coordinates X i associated to the projective representation . In other words, given an oriented closed curve immersed in S, Fock and Goncharov associate to such a curve a well-dened polynomial c Tr (X i ) := Tr b ( ) ! 2C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]; (4.1) dened as the usual matrix trace of a certain matrix b ( ) 2 M n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]); (4.2) with coecients in the commutative polynomial ringC[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]. By focusing on Fock and Goncharov's polynomial c Tr (X i ) instead of the rst trace polynomial Tr ()(X i ), we can circumvent the sign ambiguity mentioned in the previous paragraph. Later on, we will see that this second classical trace polynomial c Tr (X i ) carries a \natural choice of sign". The reason for this naturality, in the case n = 2, originates from the underlying hyperbolic geometry, as discussed in [5]. More precisely, we will see that the second classical trace polynomial c Tr (X i ) = Tr 1 (X i ) 2C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]; (4.3) is equal to the evaluation of the quantum trace map from Conjecture 1 at q = 1. Summarizing what we have said, the well-dened second classical trace polynomial c Tr (X i ) =Tr ()(X i ) 2C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]; (4.4) 90 is only equal to the rst classical trace polynomial Tr ()(X i ) up to a sign. The details vary depending on the parity of n, which controls the possible underlying spin structures. 4.3 Classical and quantum Fock-Goncharov coordinates on a triangulated surface Let the surface S be equipped with an ideal triangulation . Onto each edge E S in the ideal triangulation (including boundary edges, see Remark 4.3) we attach n 1 dots, and for each triangle T S in the triangulation we attach (n 1)(n 2)=2 dots onto the interior of the triangle. We do this in such a way that for each triangle T (meaning its interior plus its three boundary edges) there is a dot located at every vertex of the discrete triangle n C( n ) minus its corner vertices, thought of as overlayed on top of the triangle T. We illustrate this in Figure 4.2 for an ideal pentagon in the case n = 4. Solid lines and dotted lines indicate interior and boundary edges of the triangulated surface, respectively. Figure 4.2: Fock-Goncharov coordinates on a triangulated surface, shown in the case n = 4 In Chapter 2, we saw how Fock and Goncharov assign complex \coordinates" X i to the non-corner vertices of either a single discrete triangle or two discrete triangles glued together along a common edge. They dened these coordinatesX i using cross-ratios of congurations of ags, where each ag is attached to a corner of a triangle. In our current setting, these corners correspond to the punctures of an ideal triangulation. It turns out that there is a precise way to attach ag data to the entire triangulated surface S (up to passing to the 91 universal cover) such that complex coordinates X i assigned to the dots on the triangulation describe cross-ratios among these ags. However, since our theorems are algebraic in nature, we will not need precise statements about the geometry. For more details, see [42, 19, 4]. Let us now discuss how we formally think of these coordinates. Recall that Fock and Goncharov also dened quantum versions, also denoted X i , of their coordinates, which q- commute. For a single triangle T of the ideal triangulation , these quantum coordinates are modeled as (n-th powers of) generators X i = (X 1 n i ) n of the Fock-Goncharov algebra FG ! SLn(C) (T) dened in §3.2 associated to the triangle T. For a triangleT, recall that the generatorsX i of the Fock-Goncharov algebraFG ! SLn(C) (T) are indexed by non-corner vertices (a;b;c) of the discrete triangle n . We distinguish between interior generators X i := X abc indexed by points (a;b;c)2 int( n ) of the interior of the discrete triangle, and edge generators Z j :=X abc indexed by points (a;b;c)2 n int( n ) C( n ) on interiors of the boundary edges of the discrete triangle n . If E is a common edge lying at the interface between two triangles T 0 and T 00 of the ideal triangulation , then this edge corresponds to n 1 Fock-Goncharov coordinates Z, which we think of as attached onto the edge as previously discussed. For each one of these edge coordinates Z, there is associated a generator Z 0 2FG ! SLn(C) (T 0 ) lying on the edge E 0 of T 0 corresponding to the edge E 2 , and there is associated a paired generator Z 00 2 FG ! SLn(C) (T 00 ) lying on the edgeE 00 ofT 00 corresponding to the edgeE2. It is natural then to intuitively identify the Fock-Goncharov coordinate Z with the tensor product Z 0 Z 00 2 FG ! SLn(C) (T 0 ) FG ! SLn(C) (T 00 ). If, on the other hand,Z is a coordinate on a boundary edgeE of the ideal triangulation , then it is only associated to a single generator Z, namely that associated generator Z in the Fock-Goncharov algebra FG ! SLn(C) (T) of the triangle T in having E as a boundary edge. For a depiction, see Figure 4.3, where solid and dotted lines indicate interior and boundary edges, respectively. Denition 4.2. The Fock-Goncharov algebraFG ! SLn(C) (S;) of the surface S equipped with 92 (a) Before gluing (b) After gluing Figure 4.3: Interior edge coordinates as tensor products of generators, shown for n = 3 the ideal triangulation is the sub-algebra FG ! SLn(C) (S;) O triangles T FG ! SLn(C) (T); (4.5) of the tensor product of all the Fock-Goncharov algebras FG ! SLn(C) (T) associated to the triangles T of the ideal triangulation , generated: • by (n-th roots of) interior generators X 1 n 2FG ! SLn(C) (T) of triangles T in ; • by tensor products Z 0 1 n Z 00 1 n 2FG ! SLn(C) (T 0 ) FG ! SLn(C) (T 00 ) of (n-th roots of) paired edge generators associated to a common edge E lying at the interface between two triangles T 0 and T 00 in ; • and by (n-th roots of) edge generatorsZ 1 n 2FG ! SLn(C) (T) associated to boundary edges E of the ideal triangulation . In particular, when q =! = 1, the Fock-Goncharov algebra of the surfaceFG 1 SLn(C) (S;) is isomorphic to the algebra of commuting polynomialsC[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] where there is one generator X i per dot on the (glued) ideal triangulation (as in Figures 4.2 and 4.3b). Remark 4.3. Classically, or geometrically speaking, in order to assign coordinatesZ, dened as cross-ratios among congurations of ags, to a boundary edge of the surface, it may be required to x additional boundary data such as additional ags. Indeed, we have seen that 93 edge coordinates require a quadruple of ags in order to be dened, which always exists if an edge is interior and lies at the interface between two triangles. Nevertheless, at the quantum level, or algebraically speaking, we always assume there are coordinates attached to every edge, either interior or boundary. Intuitively, the dierence is that a boundary edge only receives \part of" of the coordinate compared to an interior edge. 4.4 Local monodromy matrices In Chapter 2, we saw a few dierent families of change-of-basis matrices expressing the coordinate changes between two (normalized) projective bases associated to two snakes, respectively. This section has a sizable overlap with §§3.3, 3.4. The rst family, coming in two types, consists of the tail- or diamond-move change- of-basis matrices B U!U 0 2 GL n (C) appearing in Proposition 2.14, depending on whether k = 1 or k > 1, respectively. In the case of the diamond-move, X plays the role of the Fock-Goncharov triangle invariant associated to the diamond. In the case of the tail-move, there is no variable X. We alternatively denote these two types of triangle matrices, after normalizing by their determinant, by T left j (X) := B U!U 0 , Det B U!U 0 !1 n 2 SL n C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] ! ; (4.6) where instead of thinking of the change-of-basis matrix B U!U 0 = B U!U 0(X) over the complex numbers, we think of it over the polynomial ring C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] by viewing the variable X as an indeterminate. We recall also Equation (3.20). Recall from Chapter 2 that, for the discrete triangle n1 equipped with a maximum span ag triple (E;F;G), if2C( n1 ) is a snake-head, if bot is the bottom snake dened by [ k ] left = 0 for all k = 1;:::;n, and if top is the top snake dened by [ k ] right = 0 for all k = 1;:::;n, then the change-of-basis matrix B U bot !U top from the projective basis U bot associated to the bottom snake bot to the projective basisU top associated to the top snake 94 top (normalized so that u bot n = u top n in the lineL (C n ) ) is completely determined by the triangle invariants abc (E;F;G)2Cf0g associated to interior vertices (a;b;c)2 int( n ) of the discrete triangle n . Actually computing down the matrix B U bot !U top requires choosing a sequence of adjacent snakes bot = 1 ; 2 ;:::; N = top going from the bottom snake to the top snake. For then B U bot !U top = B U 1 !U 2 B U 2 !U 3 B U N1 !U N 2 GL n (C); (4.7) where the matrices B U i !U i+1 on the right hand side are the change of basis matrices between the projective bases U i associated to the snakes i (all normalized to have the same n-th co-vector in the lineL ), and where we have used Equation (2.10). As we saw in the previous paragraph, these matrices B U i !U i+1 are all, after normalization, of the form T left j (X i ). We dene the left matrix M left (X i 's) by the normalization M left (X i 's) := B U bot !U top , Det B U bot !U top !1 n = Y T left j (X i ) 2 SL n C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] ! : (4.8) We think of the left matrix M left (X i 's) as associated to the local motion of a particle moving \left" across the interior of a triangle, as shown in Figure 4.4, which shows the case n = 3. We will make this more precise later on in §4.5. There is similarly a right matrix M right (X i 's) dened by M right (X i 's) := B U bot !U top , Det B U bot !U top !1 n = Y T right j (X i ) 2 SL n C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] ! : (4.9) where the form of the general change-of-basis matrix B U i !U i+1 is provided in Proposition 95 Figure 4.4: Left matrix 2.24 in the situation when the top snake is not the top-left snake as before, but the top-right snake. We recall also Equation (3.21). We think of the right matrix M right (X i 's) as associated to the local motion of a particle moving \right" across the interior of a triangle, as shown in Figure 4.5, which shows the case n = 3. We will make this more precise later on in §4.5. Figure 4.5: Right matrix The second family of change-of-basis matrices arises when there are two triangles T and T 0 in the triangulation having a common edgeE. In this situation, the matrix transforms between the projective bases associated to two same-oriented snakes, both living on the edge E, where one of the projective bases is computed with respect to one triangle and the other projective basis is computed with respect to the other triangle. The edge matrix E(Z j 's) is 96 dened as the normalized change-of-basis matrix E(Z j 's) := B U!U 0 , Det B U!U 0 !1 n = n1 Y j=1 S j (Z j ) 2 SL n C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ] ! ; (4.10) is computed in Proposition 2.28 and is the product ofn 1 shearing matrices S j (Z j ), where Z j plays the role of thej-th Fock-Goncharov edge invariant, also called a shear, associated to thej-th diamond lying in both triangles and straddling the edgeE. Recall Equation (3.19). We think of the edge matrix E(Z j 's) as associated to the local motion of a particle moving across an edge, as shown in Figure 4.6, which is shown in the case n = 3. We will make this more precise later on in §4.5. Notice that the labels Z 1 ;:::;Z n1 for the edge coordinates are not set-in-stone, but depend on the orientation of the curve as it passes that edge. The same edge coordinate Z might be called Z 1 at one point in time, and Z n1 at another time. The dierence is whether the shearing matrix S 1 (Z) or S n1 (Z), respectively, appears in the edge matrix E(Z j 's) at a given moment in time. The third and last family of change-of-basis matrices occurs for a single triangle with edge E and transforms between two opposite-facing snakes lying on the same edge E that we discussed for the second family. This matrix appears in Proposition 2.29 and does not involve any cross-ratios, indeed its entries only contain 0, 1, or1. We denote this matrix by U. Explicitly, the matrix U looks like U = 0 B B B B B B B B B B B B B B @ 1 1 . . . +1 1 +1 1 C C C C C C C C C C C C C C A 2 SL n (C): (4.11) We think of the U-turn matrix U as associated to the local motion of a particle making 97 Figure 4.6: Edge matrix a clockwise U-turn in the interior of the triangle, as shown in Figure 4.7, which is shown in the case n = 3. If the particle instead makes counter-clockwise U-turn, then we associate the transpose matrix U T . We will make this more precise later on in §4.5. Remark 4.4. Notice that U T = U 1 (and in particular U T equalsU whenn is even and equals +U when n is odd). So one might be tempted to prefer thinking of the dierence as by taking the inverse instead of the transpose. However, later on, when we consider the corresponding quantum U-turn matrices, which involve the parameter q and which recover the classical U-turn matrices when we setq = 1, then the quantum counter-clockwise U-turn will be the transpose of the quantum clockwise U-turn, which it turns out is distinct from the inverse. 98 (a) Clockwise U-turn (b) Counter-clockwise U-turn Figure 4.7: U-turn matrices 4.5 Computing the classical trace polynomial Let be an immersed oriented closed curve in the triangulated surface S such that is transverse to the ideal triangulation . We want to compute the classical trace polyno- mial c Tr (X i ), namely the second version of the trace polynomial discussed in §4.2. This is computed as a \state-sum formula" via a \local-to-global" or \transfer matrices" method. More precisely, as we travel along the curve according to its orientation, assume crosses the edges E i fori = 1; 2;:::;N 1 in that order, and assume crosses the triangles T j for j = 1; 2;:::;N 2 in that order. As the curve crosses the edge E i going from triangle T := T j to triangle T 0 := T j+1 , we associate to this local movement across the edge E i the edge matrix E i := E i (Z k 's) 2 SL n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]); (4.12) 99 dened in Equation (4.10). We recall the discussion in§4.3 and in particular Figure 4.6. We emphasize, as depicted in the gure, that the labeling of then1 coordinates located on the edge E i , which are needed to compute the edge matrix E i , is not set-in-stone, but depends on how, at the given moment in time, the curve is passing the edge. The labeling is always such that Z 1 is the right-most coordinate and Z n1 is the left-most coordinate relative to the oriented curve. As crosses a triangle T j between two edges E i and E i+1 , it does one of three things: • the curve turns left; • or turns right; • or does a U-turn (namely, it returns to the same edge). We also keep track of: for the rst and second items above, the number of full turns t j to the left that the curve makes while crossing the triangle T j ; and for the third item above, the number of half turns 2t j 1 to the left that the curve makes before coming back to the same edge E i . Note t j 2Z. We will see that the turning integer t j associated to the curve will only be relevant when the parity of the xed integer n of SL n (C) is even. If while crossing the triangle T j from the edge E bot :=E i to the edge E top :=E i+1 , the curve turns left, then we associate to this local movement across the triangle T j the left matrix M j := (1) (n1)t j M left (X k 's) 2 SL n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]); (4.13) dened in Equation (4.8), modied by a sign. If instead the curve turns right, then we associate to this local movement the signed right matrix M j := (1) (n1)t j M right (X k 's) 2 SL n (C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]); (4.14) dened in Equation (4.9). We recall the discussion in §4.3 and in particular Figures 4.4 and 4.5. Notice, for the curve shown in those two gures, that M j = +M left (X k 's) or 100 M j = +M right (X k 's), respectively, since the curve has zero winding number as it crosses the triangles, and so no sign adjustment is necessary for the examples in those two gures. Lastly, if instead the curve makes a U-turn, then we associate to this local movement the sign-adjusted U-turn matrix M j := (1) (n1)t j U 2 SL n (C); (4.15) where the matrix U is dened in Equation (4.11). For instance, if the motion is clockwise (and with no extra winding) thent j = 0 and M j = U, and if the motion is counter-clockwise (and with no extra winding) then t j = 1 and M j =U = U T , depending on the parity of n. We recall the discussion in §4.4 and in particular Figure 4.7. Denition 4.5. Using the above notation, the second classical trace polynomial c Tr (X i ) associated to the immersed closed oriented curve is dened by c Tr (X i ) = Tr M 1 E 1 M 2 E 2 M k E k ! 2C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]; (4.16) where on the right hand side we have taken the usual matrix trace. Note that this is inde- pendent of where we start along the curve , since the trace is invariant under conjugation. 4.6 Framed oriented links in the thickened surface So far, we have been working in the 2-dimensional setting of the surface S. Now, we begin working in the 3-dimensional setting of the thickened surface S [0; 1]. Here we follow [5, §3.1], one major dierence being that we consider oriented links. Denition 4.6. A framed oriented link K in the thickened surface S [0; 1] is a compact oriented one-dimensional manifold possibly-with-boundaryKS [0; 1] that is embedded in S [0; 1], equipped with a framing, satisfying the following properties: 101 (1) we have @K =K\ (S [0; 1]); (2) the framing at a boundary point of K is vertical, meaning parallel to the [0; 1] axis, and points in the 1 direction; (3) for each boundary component k of S, the nitely many points K\ (k [0; 1]) have distinct heights, meaning the coordinates with respect to [0; 1]. Here, by a framing, we mean a choice of continuous non-zero vector eld transverse to K. A framed oriented knot is a closed framed oriented link with one connected component. Two framed oriented links K and K 0 are isotopic if there is an ambient isotopy of the thickened surface S [0; 1] taking K to K 0 through the class of framed oriented links. By possibly introducing kinks, it is always possible to isotope a framed oriented link so that it has blackboard framing, meaning constant vertical framing in the 1 direction. It is common to think of a framed link K as a \ribbon", namely a \thin" annulus embedded in S [0; 1], in which case we think of the framing as a continuous vector eld normal to the embedded annulus K. From now on, we will use the picture conventions of [5,§3.5] to represent framed oriented links in S [0; 1], in particular all link diagrams have blackboard framing. 4.7 Stated links To deal algebraically with links-with-boundary, we need the following concept: Denition 4.7. An-stated framed oriented link (K;s) is the data of a framed oriented link K equipped with a set function s :@K!f1; 2;:::;ng; (4.17) called the state function, assigning to each element of the boundary of the link a state, namely a number inf1; 2;:::;ng. 102 The notion of isotopy for the underlying links induces a notion of isotopy for stated links. Notice that a stated closed link is the same thing as a closed link. 4.8 Second theorem We now restrict to the casen = 3. Let the surfaceS be equipped with an ideal triangulation . Recall the Fock-Goncharov algebraFG ! SL 3 (C) (S;) associated to this data, Denition 4.2. Note that if K S [0; 1] is a framed oriented knot (meaning, in particular, that it is closed), and if : S [0; 1]! Sf 1 2 g = S is the natural projection, then, possibly after an arbitrarily small perturbation of the knotK, we have that :=(K) is an oriented closed curved immersed in S, and so we may consider the second classical trace polynomial c Tr (X i ) associated to , dened in Denition 4.5. We now give a precise version of Theorem 1 from the Introduction. Theorem 4.8. Let q2Cf0g be a nonzero complex number, and let ! = q 1 9 be a ninth root of q. In the case n = 3, there is a function of sets Tr q : ( isotopy classes of stated framed oriented links ) −!FG ! SL 3 (C) (S;) O triangles T FG ! SL 3 (C) (T); (4.18) sending a stated framed oriented link (K;s) in the thickened surface S [0; 1] to a Laurent polynomial Tr q (K;s) = Tr q (K;s) (X i ) 2 FG ! SL 3 (C) (S;) in the q-deformed (third roots of the) SL 3 (C)-Fock-Goncharov coordinates X i . This assignment satises the property that when q =! = 1, and when K =K 1 tK 2 ttK m is a closed framed oriented link thought of as a disjoint union of knots K ` , then Tr 1 K (X i ) = m Y `=1 c Tr ` (X i ) 2C[X 1 n 1 ;X 1 n 2 ;:::;X 1 n N ]; (4.19) 103 (c.f. the end of Denition 4.2) where ` is the immersed oriented closed curve obtained by projecting the knot componentK ` of the closed linkK to S, and where the (second) classical trace polynomial c Tr ` (X i ) is dened in Denition 4.5 (see also §4.2). This construction also satises the State Sum Property, which is the obvious generaliza- tion to the case n = 3 of Property (1) of Theorem 11 in [5, §3.4]. 4.9 Matrix conventions From now on, we only work in the case n = 3. We will need to display 3 3 and 9 9 matrices. Lower indices will indicate rows and upper indices will indicate columns. A 3 3 matrix M = (M j i ) will be displayed in the general form M = 0 B B B B @ M 1 1 M 2 1 M 3 1 M 1 2 M 2 2 M 3 2 M 1 3 M 2 3 M 3 3 1 C C C C A : (4.20) A 9 9 matrix M = (M j 1 j 2 i 1 i 2 ) will be displayed in the general form M = 0 B B B B B B B B B B B B B B B B B B B B B B B B @ M 11 11 M 12 11 M 13 11 M 21 11 M 22 11 M 23 11 M 31 11 M 32 11 M 33 11 M 11 12 M 12 12 M 13 12 M 21 12 M 22 12 M 23 12 M 31 12 M 32 12 M 33 12 M 11 13 M 12 13 M 13 13 M 21 13 M 22 13 M 23 13 M 31 13 M 32 13 M 33 13 M 11 21 M 12 21 M 13 21 M 21 21 M 22 21 M 23 21 M 31 21 M 32 21 M 33 21 M 11 22 M 12 22 M 13 22 M 21 22 M 22 22 M 23 22 M 31 22 M 32 22 M 33 22 M 11 23 M 12 23 M 13 23 M 21 23 M 22 23 M 23 23 M 31 23 M 32 23 M 33 23 M 11 31 M 12 31 M 13 31 M 21 31 M 22 31 M 23 31 M 31 31 M 32 31 M 33 31 M 11 32 M 12 32 M 13 32 M 21 32 M 22 32 M 23 32 M 31 32 M 32 32 M 33 32 M 11 33 M 12 33 M 13 33 M 21 33 M 22 33 M 23 33 M 31 33 M 32 33 M 33 33 1 C C C C C C C C C C C C C C C C C C C C C C C C A : (4.21) 104 4.10 Biangles and the Reshetikhin-Turaev invariant A biangle B is a closed disk with two punctures on its boundary. Biangles do not admit ideal triangulations, and so cannot be chosen for our general surface S. Nevertheless, we may still consider stated framed oriented links (K;s) in the thickened biangle B [0; 1], dened just as before, and it turns out there is a famous construction of Reshetikhin and Turaev [54] that gives us a version of Theorem 4.8 for biangles that is valued in the complex numbersC. In order to state the result, we need to dene some elementary matrices associated to certain link diagrams, namely various U-turns and crossings. We begin with the U-turns. In Figures 4.8 and 4.9, we show the four possible U-turns, which are in particular stated oriented framed links with the blackboard framing. In accor- dance with our picture conventions from §4.6, the boundary point of the link that is closer to the arrow on the boundary of the biangle is higher, with respect to the [0; 1] direction, than the boundary point of the link that is farther away from the arrow. Dene a 3 3 matrix U q by U q := 0 B B B B @ 0 0 +q 7=3 0 q 4=3 0 +q 1=3 0 0 1 C C C C A 2 M 3 (C): (4.22) Notice that when q = 1 and n = 3 then the matrix U 1 agrees with the matrix U from Equation (4.11). Here, as usual, by q 1=3 we mean ! 3 . Now, dene for each pair of states s 1 ;s 2 2f1; 2; 3g four complex numbers Tr q (U cw dec ) s 2 s 1 ; Tr q (U ccw inc ) s 2 s 1 ; Tr q (U ccw dec ) s 2 s 1 ; Tr q (U cw inc ) s 2 s 1 2C; (4.23) 105 (a) Clockwise (b) Counter- clockwise Figure 4.8: Decreasing U-turns (a) Counter- clockwise (b) Clockwise Figure 4.9: Increasing U-turns by the matrix equations Tr q (U cw dec ) s 2 s 1 ! := U q 2 M 3 (C); (4.24) Tr q (U ccw inc ) s 2 s 1 ! := (U q ) T 2 M 3 (C); (4.25) Tr q (U ccw dec ) s 2 s 1 ! := +q 8=3 Tr q (U cw dec ) s 2 s 1 ! 2 M 3 (C); (4.26) 106 and Tr q (U cw inc ) s 2 s 1 ! := +q 8=3 Tr q (U ccw inc ) s 2 s 1 ! 2 M 3 (C): (4.27) Next, we do something similar for crossings. In Figures 4.10 and 4.11 we show the eight possible crossings, with blackboard framing and adhering to the usual picture conventions. (a) positive crossing, over strand higher to lower (b) negative crossing, over strand lower to higher (c) positive crossing, over strand lower to higher (d) negative crossing, over strand higher to lower Figure 4.10: Same direction crossings (a) negative crossing, over strand higher to lower (b) positive crossing, over strand lower to higher (c) negative crossing, over strand lower to higher (d) positive crossing, over strand higher to lower Figure 4.11: Opposite direction crossings 107 Dene two 9 9 matrices C q same and C q opp by C q same :=q +1=3 0 B B B B B B B B B B B B B B B B B B B B B B B B @ q 1 0 0 0 0 0 0 0 0 0 q 1 q 0 1 0 0 0 0 0 0 0 q 1 q 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 q 1 0 0 0 0 0 0 0 0 0 q 1 q 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 q 1 1 C C C C C C C C C C C C C C C C C C C C C C C C A 2 M 9 (C); (4.28) and C q opp :=q +2=3 0 B B B B B B B B B B B B B B B B B B B B B B B B @ q 1 0 0 0 0 0 0 0 0 0 0 0 q 1 0 0 0 0 0 0 0 q 2 1 0 q 1 q 0 1 0 0 0 q 1 0 0 0 0 0 0 0 0 0 q 1 q 0 1 0 0 0 0 0 0 0 0 0 0 0 q 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 q 1 0 0 0 0 0 0 0 0 0 0 0 q 1 1 C C C C C C C C C C C C C C C C C C C C C C C C A 2 M 9 (C): (4.29) Notice that when q = 1 then the two matrices C 1 same and C 1 opp are identical. For another property of these so-called R-matrices, see §4.13. 108 Now, dene for each quadruple of states s 1 ;s 2 ;s 3 ;s 4 2f1; 2; 3g eight complex numbers Tr q (C over-to-lower pos-same ) s 3 s 4 s 1 s 2 ; Tr q (C over-to-higher neg-same ) s 3 s 4 s 1 s 2 ; Tr q (C over-to-higher pos-same ) s 3 s 4 s 1 s 2 ; Tr q (C over-to-lower neg-same ) s 3 s 4 s 1 s 2 ; Tr q (C over-to-lower neg-opp ) s 3 s 4 s 1 s 2 ; Tr q (C over-to-higher pos-opp ) s 3 s 4 s 1 s 2 ; Tr q (C over-to-higher neg-opp ) s 3 s 4 s 1 s 2 ; Tr q (C over-to-lower pos-opp ) s 3 s 4 s 1 s 2 ; (4.30) by the matrix equations Tr q (C over-to-lower pos-same ) s 3 s 4 s 1 s 2 ! := C q same 2 M 9 (C); (4.31) Tr q (C over-to-higher neg-same ) s 3 s 4 s 1 s 2 ! := (C q same ) 1 2 M 9 (C); (4.32) Tr q (C over-to-higher pos-same ) s 3 s 4 s 1 s 2 ! := C q same 2 M 9 (C); (4.33) Tr q (C over-to-lower neg-same ) s 3 s 4 s 1 s 2 ! := (C q same ) 1 2 M 9 (C); (4.34) Tr q (C over-to-lower neg-opp ) s 3 s 4 s 1 s 2 ! := C q opp 2 M 9 (C); (4.35) Tr q (C over-to-higher pos-opp ) s 3 s 4 s 1 s 2 ! := (C q opp ) 1 2 M 9 (C); (4.36) Tr q (C over-to-higher neg-opp ) s 3 s 4 s 1 s 2 ! := C q opp 2 M 9 (C); (4.37) and Tr q (C over-to-lower pos-opp ) s 3 s 4 s 1 s 2 ! := (C q opp ) 1 2 M 9 (C): (4.38) Finally, consider a single strand crossing from one boundary edge of the biangle to the other boundary edge, as shown in Figure 4.12. Notice that the height of the strand with respect to the [0; 1] component does not play a role in this particular case. Then, this corresponds to the identity matrix. Namely, dene for each pair of states 109 Figure 4.12: Single strand crossing the biangle s 1 ;s 2 2f1; 2; 3g the complex number Tr q (Id) s 2 s 1 := s 1 s 2 2C; (4.39) where s 1 s 2 denotes the Kronecker delta. So far, we have shown how to assign complex numbers to a handful of stated blackboard- framed oriented links in the biangle B with one or two components. Now, let (K;s) be an arbitrary stated blackboard-framed oriented link in the biangle having no kinks. By combing the linkK into a bridge position, as described in [5, §4] in the proof of Lemma 15, where in our case n = 3 we do in fact need to include additional local pictures containing crossings, then we can dene a complex number Tr q (K;s) 2C by a State Sum Formula, identical to that appearing in Equation (1) on p.1591 in [5, §4]. To dene the complex number Tr q (K;s) 2C for an arbitrary stated framed oriented linkK in the thickened biangleB[0; 1], rst isotope the link, by possibly introducing kinks, so that it has the blackboard framing. Let K 0 be the new stated blackboard-framed oriented link without kinks obtained by pulling tight the kinks in the un-framed oriented link underlying K. Then we have dened a complex number Tr q (K 0 ;s) 2C. Finally, dene Tr q (K;s) by modifying Tr q (K 0 ;s) 2C according to the un-kinking \skein relations" shown in Figures 4.13 and 4.14. 110 Note that the two positive (resp. negative) kinks shown are, in fact, locally isotopic. Figure 4.13: Skein relation for positive kinks Figure 4.14: Skein relation for negative kinks Proposition 4.9 (Reshetikhin-Turaev). Let B be a biangle. In the case n = 3, the trace function, prescribed above in this section, Tr q : ( isotopy classes of stated framed oriented links ) −!C; (4.40) sending a stated framed oriented link (K;s) in the thickened biangle B[0; 1] to the complex number Tr q (K;s) 2C, is well-dened. This construction also satises the State Sum Property, which is the obvious generaliza- tion to the case n = 3 of Property (1) of Proposition 13 in [5, §4]. Remark 4.10. The Reshetikhin-Turaev invariant for links in the biangle that we constructed is an a-typical normalization of the usual Reshetikhin-Turaev invariant, as can be seen in the 111 non-standard choice of the o-diagonal \duality" matrix appearing in Equation (4.22). This normalization has the advantage of being particularly symmetric in the current 3-dimensional setting, and turns out to be the unique such normalization yielding a well-dened quantum trace map. For a brief mention of this uniqueness property in the case n = 2, see the paragraph immediately preceding Proposition 26 in [5, §6]. 4.11 Denition of the SL 3 (C)-quantum trace map Our construction of the quantum trace map in the case n = 3 will follow exactly the same procedure explained in [5, §§3.4-6] for the case n = 2, where our Proposition 4.9 plays the role of Proposition 13 in [5, §4]. The essential piece that needs to be assembled is how Property (2)(a) of Theorem 11 in [5, §3.4] generalizes to the case n = 3. After this, the rest of the construction is identical. Generalizing Property (2)(a) to the case n = 3 is accomplished by using the quantum left and right matrices L ! and R ! , appearing in our Theorem 3.10. So let T be a triangle in the triangulation, and consider a single arc moving left or right across the triangle between two distinct boundary edges, as shown in Figures 4.15a and 4.15b. Dene two 3 3 matrices L ! and R ! with coecients in the Fock-Goncharov algebra FG ! SL 3 (C) (T) of the triangle, with generators labeled as in the two gures above, by L ! := 0 B B B B @ [D 1 3 L W 1 Z 1 XZ 2 W 2 ] [D 1 3 L W 1 Z 1 XW 2 ] + [D 1 3 L W 1 Z 1 W 2 ] [D 1 3 L W 1 Z 1 ] 0 [D 1 3 L Z 1 W 2 ] [D 1 3 L Z 1 ] 0 0 [D 1 3 L ] 1 C C C C A ; (4.41) where D 1 3 L inFG ! SL 3 (C) (T) is dened by D 1 3 L =W 1=3 1 Z 2=3 1 X 1=3 Z 1=3 2 W 2=3 2 ; (4.42) 112 (a) Left (b) Right Figure 4.15: Quantum left and right matrices and R ! := 0 B B B B @ [D 1 3 R W 2 Z 2 Z 1 W 1 ] 0 0 [D 1 3 R Z 2 Z 1 W 1 ] [D 1 3 R Z 2 W 1 ] 0 [D 1 3 R Z 1 W 1 ] [D 1 3 R W 1 ] + [D 1 3 R X 1 W 1 ] [D 1 3 R X 1 ] 1 C C C C A ; (4.43) where D 1 3 R inFG ! SL 3 (C) (T) is dened by D 1 3 R =W 1=3 2 Z 2=3 2 X 1=3 Z 1=3 1 W 2=3 1 : (4.44) Remark 4.11. We recall that the square brackets surrounding each monomial denotes the Weyl quantum ordering, dened in Equation (3.14), and depending on the quiver dis- played in Figure 3.1 dening the non-commutative structure in the Fock-Goncharov algebra FG ! SL 3 (C) (T). In particular, the ordering of generators (and their powers) within a square bracket are immaterial and so, for instance, any ordering of the generators in the denitions of D 1 3 L and D 1 3 R could have been chosen without changing the denitions of L ! and R ! . We recall that in Theorem 3.10 we saw that the quantum left and right matrices L ! and R ! are FG ! SL 3 (C) (T)-points of the quantum group SL q 3 . A puzzling empirical observation is 113 that this requires normalizing the quantum matrices by dividing out their determinants. In other words, if we tried to takeD L =D R = 1 in the denitions just above, then the quantum matrices not only would not be points of SL q 3 , but they would not be points of M q 3 . This can be seen, for instance, in the matrix L ! , which would have a 1 in the bottom right corner, and so would not be able to q-commute with the other entries of the matrix as required in order to have a point of M q 3 . Finally, generalizing Property (2)(a) of Theorem 11 in [5, §3.4] as desired, and recalling Figure 4.15, dene for each pair of states s 1 ;s 2 2f1; 2; 3g two elements in the SL 3 (C)-Fock- Goncharov algebraFG ! SL 3 (C) (T) of the triangle Tr q (Q left ) s 2 s 1 ; Tr q (Q right ) s 2 s 1 2FG ! SL 3 (C) (T); (4.45) by the matrix equations Tr q (Q left ) s 2 s 1 ! := L ! 2 M 3 FG ! SL 3 (C) (T) ! ; (4.46) Tr q (Q right ) s 2 s 1 ! := R ! 2 M 3 FG ! SL 3 (C) (T) ! : (4.47) This completes the denition of the quantum trace map Tr q in the case n = 3. To show it is well-dened, we have to check all the oriented versions of the local moves depicted in Figures 15-19 in [5, §5]. We did this by hand, using Mathematica (see §4.12 below). Lastly, the multiplicative property, Equation (4.19), for the quantum trace is proved exactly as in Lemma 19 and on p.1609 in [5, §§4, 6]. Note that the quantum trace Tr q K of a link K can be thought of as a tensor having dimension equal to the number of boundary points p i 2 @K, each associated to a state s i . If the statess i are segregated into two groupss i 1 ;:::;s i ` ands j 1 ;:::;s im , then the quantum trace of the link can be written as a matrix ((Tr q K ) s j 1 ;:::;s jm s i 1 ;:::;s i ` ) with coecients inFG ! SL 3 (C) (T). 114 Figure 4.16: One of the oriented versions of Move (II) 4.12 Computer check of local moves for n = 3 Here, and in Appendix 5.16, we will demonstrate the kinds of calculations that are needed in order to check that the quantum trace polynomial of Theorem 4.8 is independent of ambient isotopy of links K. We mentioned in the previous section that the two most dicult moves are the (now oriented) moves of type (II) and (IV) appearing in [5, §5] in their Figures 16 and 18. In Figure 4.16, we show a representative example of Move (II). According to the denition of the quantum trace (§4.11) as a State Sum Formula, the equality expressing Move (II) can be interpreted as an equality of 3 3 matrices (Tr q K ) s 2 s 1 ! := A 2 0 0 D 2 E 2 0 G 2 H 2 I 2 0 0 +q 1=3 0 q 4=3 +q 7=3 0 0 A 1 0 0 D 1 E 1 0 G 1 H 1 I 1 = q 1 3A 2 G 1 q 1 3A 2 H 1 q 1 3A 2 I 1 q 4 3E 2 D 1 +q 1 3D 2 G 1 q 4 3E 2 E 1 +q 1 3D 2 H 1 q 1 3D 2 I 1 q 7 3I 2 A 1 q 4 3H 2 D 1 +q 1 3G 2 G 1 q 4 3H 2 E 1 +q 1 3G 2 H 1 q 1 3G 2 I 1 ! = a 3 b 3 c 3 0 e 3 f 3 0 0 i 3 =: (Tr q K 0 ) s 2 s 1 ! 2 M 3 FG ! SL 3 (C) (T) ; (4.48) where we have used Figure 4.9a and Equations (4.22) and (4.25) for the middle matrix, and where, if we denote the left matrix of Equation (4.41) by L ! = L ! (W 1 ;Z 1 ;W 2 ;Z 2 ;X) and 115 Figure 4.17: One of the oriented versions of Move (IV) denote the right matrix of Equation (4.43) by R ! = R ! (W 1 ;Z 1 ;W 2 ;Z 2 ;X), then we dene A 2 0 0 D 2 E 2 0 G 2 H 2 I 2 := R ! (W 3 ;Z 3 ;W 1 ;Z 1 ;X); A 1 0 0 D 1 E 1 0 G 1 H 1 I 1 := R ! (W 2 ;Z 2 ;W 3 ;Z 3 ;X); a 3 b 3 c 3 0 e 3 f 3 0 0 i 3 := L ! (W 1 ;Z 1 ;W 2 ;Z 2 ;X) 2 M 3 FG ! SL 3 (C) (T) : (4.49) See Appendix 5.16, Section 3, for a computer check of Equation (4.48). In Figure 4.17, we show a representative example of Move (IV). According to the de- nition of the quantum trace (§4.11) as a State Sum Formula, the equality expressing Move (IV) can be interpreted as an equality of 9 9 matrices (c.f. Equation (4.21)) (Tr q K ) s 3 s 4 s 1 s 2 ! := 0 B B B B @ a 3 A 2 0 0 b 3 A 2 0 0 c 3 A 2 0 0 a 3 D 2 a 3 E 2 0 b 3 D 2 b 3 E 2 0 c 3 D 2 c 3 E 2 0 a 3 G 2 a 3 H 2 a 3 I 2 b 3 G 2 b 3 H 2 b 3 I 2 c 3 G 2 c 3 H 2 c 3 I 2 0 0 0 e 3 A 2 0 0 f 3 A 2 0 0 0 0 0 e 3 D 2 e 3 E 2 0 f 3 D 2 f 3 E 2 0 0 0 0 e 3 G 2 e 3 H 2 e 3 I 2 f 3 G 2 f 3 H 2 f 3 I 2 0 0 0 0 0 0 i 3 A 2 0 0 0 0 0 0 0 0 i 3 D 2 i 3 E 2 0 0 0 0 0 0 0 i 3 G 2 i 3 H 2 i 3 I 2 1 C C C C A =q +1=3 0 B B B B B B @ q 1 A 2 a 3 0 0 q 1 A 2 b 3 0 0 q 1 A 2 c 3 0 0 D 2 a 3 E 2 a 3 0 D 2 b 3 +(q 1 q)A 2 e 3 E 2 b 3 0 D 2 c 3 +(q 1 q)A 2 f 3 E 2 c 3 0 G 2 a 3 H 2 a 3 I 2 a 3 G 2 b 3 H 2 b 3 I 2 b 3 G 2 c 3 +(q 1 q)A 2 i 3 H 2 c 3 I 2 c 3 0 0 0 A 2 e 3 0 0 A 2 f 3 0 0 0 0 0 q 1 D 2 e 3 q 1 E 2 e 3 0 q 1 D 2 f 3 q 1 E 2 f 3 0 0 0 0 G 2 e 3 H 2 e 3 I 2 e 3 G 2 f 3 +(q 1 q)D 2 i 3 H 2 f 3 +(q 1 q)E 2 i 3 I 2 f 3 0 0 0 0 0 0 A 2 i 3 0 0 0 0 0 0 0 0 D 2 i 3 E 2 i 3 0 0 0 0 0 0 0 q 1 G 2 i 3 q 1 H 2 i 3 q 1 I 2 i 3 1 C C C C C C A =: (Tr q K 0 ) s 3 s 4 s 1 s 2 2 M 9 FG ! SL 3 (C) (T) ; (4.50) 116 where we have used Figure 4.10c and Equations (4.28) and (4.33) as part of the computation for the matrix on the right, and where, if we denote the left matrix of Equation (4.41) by L ! = L ! (W 1 ;Z 1 ;W 2 ;Z 2 ;X) and denote the right matrix of Equation (4.43) by R ! = R ! (W 1 ;Z 1 ;W 2 ;Z 2 ;X), then we dene, as above, A 2 0 0 D 2 E 2 0 G 2 H 2 I 2 := R ! (W 3 ;Z 3 ;W 1 ;Z 1 ;X); a 3 b 3 c 3 0 e 3 f 3 0 0 i 3 := L ! (W 1 ;Z 1 ;W 2 ;Z 2 ;X): (4.51) See Appendix 5.16, Section 4, for a computer check of Equation (4.48). 4.13 HOMFLYPT skein relation In [5], the quantum trace map is constructed as an algebra homomorphism from the SL 2 - skein algebraS q SL 2 (C) (S) to the quantum torusFG ! SL 2 (C) (S;). (Recall from the introduction §1 that a linkK determines a class [K] in the skein algebra.) Although we do not formulate our Theorem 4.8 in terms of the SL n -skein algebra S q SLn(C) (S), this is indeed a goal of our future work. The diculty is that the SL n -skein algebra has many relations, which are best understood in terms ofn-valent oriented graphs, or webs, embedded in the thickened surface S [0; 1]. We have taken some steps in this direction, namely in understanding quantum traces for webs, but do not currently have a precise statement. One important skein relation is the well-known HOMFLYPT relation from knot theory. For us, this appears in the form displayed in Figure 4.18. For instance, one can check from Figures 4.10a, 4.10b, 4.12, together with Equations (4.31), (4.32), (4.28), (4.39), that q 1=3 C q same q +1=3 (C q same ) 1 = (q 1 q)Id 9 2 M 9 (C): (4.52) 117 Figure 4.18: HOMFLYPT skein relation, in the case n = 3 4.14 Going to SL n (C) The proofs of the local moves, such as those demonstrated in the previous section, can be seen as equalities of matrices, which makes them easier to understand. In particular, it can be shown from the properties of the quantum group SL q n that our Theorem 3.10 is equivalent to the Move (III) shown in Figure 17 in [5, §5]. Also, using our strategy of decomposing the quantum left and right matrices that we used to prove our Theorem 3.10, the Move (I) shown in their Figure 15 is not dicult to prove. Clearly, Move (V) shown in their Figure 19 is trivial by our Figures 4.13 and 4.14. The dicult moves to prove are the Moves (IV) and (II) shown in their Figures 18 and 16. We are beginning to understand that Move (IV) can be deduced from the work in [14], and also that this move has to do with the properties of the braided tensor product SL q n SL q n of the dual quantum group SL q n with itself, see [47, 15]. Now, the strategy undertaken in [14] involves understanding Fock and Goncharov's matrices by counting paths in a graph. We are getting closer to understanding that this same path- counting strategy, combined with the classical geometric theory of snakes discussed in our Chapter 2, together should be enough to prove Move (II). Our restriction to the casen = 3 has been due to our inability to rigorously prove Moves (IV) and (II) for general n, but the strategy loosely discussed in the previous paragraph should very well be applicable in the general case. Moreover, our denition of the SL 3 (C)- quantum trace map is not, in its general form, particularly limited to the case n = 3. We know that the general SL n (C)-quantum left and right matrices L ! and R ! from our Theorem 3.10 should be used in place of Equations (4.41) and (4.43). And we know that the invariant 118 used in the biangles should be the classical SL n (C)-Reshetikhin-Turaev invariant, possibly with an a-typical normalization that is well-suited to the 3-dimensional setting, manifesting in the choice of the \duality" U-turn matrix appearing in Equation (4.22), in the R-matrices appearing in Equations (4.28) and (4.29), as well as in the skein relations for kinks appearing in Figures 4.13 and 4.14. 4.15 Proposed denition of the SL n -quantum trace map We have so far dened the SL 3 -quantum trace, and proved by computer calculation that it is well-dened, providing some examples in §4.12. This had a bit of an ad hoc feeling to it. However, we do have an evolving understanding of the quantum trace in terms of the theory of quantum groups, which we have neglected to discuss much in this thesis, the only explicit example of such a connection being Theorem 3.10. As another example, Proposition 4.9, which describes the well-known SL n -Reshetikhin- Turaev invariant in the case n = 3, uses the theory of quantum groups for its construction. For instance, the matrix appearing in Equation (4.28) is just the standard R-matrix for the dual quantum group SL q 3 , with the standard normalization q +1=3 appearing in front of the matrix. There is also an R-matrix for SL q n , where the normalization factor becomes q +1=n . The o-diagonal U-turn matrix with the strange exponents appearing in Equation 4.22 is a tad bit more mysterious at rst glance, given that the typical U-turn, or \duality", matrix in Reshetikhin and Turaev's theory is taken to be the identity matrix. This turns out, however, to also be connected to well-known quantum group theory. We mentioned in Remark 4.10 that this has to do with requiring that the U-turn matrix be \nicely symmetric" with respect to the three-dimensional setting. More specically, this \symmetric condition" for the U-turn matrix has to do with the ribbon element := +q 8=3 for SL q 3 , appearing (along with its inverse) in Figures 4.13 and 4.14 as the power ofq created when performing the kink-removing skein relations. Forn = 2, 119 the ribbon element is :=q 3=2 , and for general n the ribbon element is := (1) n1 q 1n 2 n 2C: (4.53) In particular, the ribbon element determines the kink-removing skein relations generally for SL q n . Next, dene the square root of the ribbon element to be p := +q 1n 2 2n 2C: (4.54) Then the denition of the (clockwise) U-turn matrix U q for general n is U q := p 0 B B B B B B B B B B B B B B @ q 1n 2 q 3n 2 . . . +q n5 2 q n3 2 +q n1 2 1 C C C C C C C C C C C C C C A 2 M n (C); (4.55) which indeed recovers Equation 4.22 in the case n = 3. Observe that the common ratio between adjacent entries in the matrix is equal toq. Notice also that when q = 1 this recovers the classical U-turn matrix of Proposition 2.29, see also Equation (4.11). Finally, to nish dening the quantum trace for general n, the quantum left and right matrices L ! and R ! appearing in Equations (4.41) and (4.43), with coecients in the SL n - Fock-Goncharov algebraFG ! SL 3 (C) (T), are replaced with their SL n counterparts, as discussed in the previous chapter, c.f. Theorem 3.10. Remark 4.12. Using these formulas, we also checked by computer the analogue of Figure 4.16 in the case n = 4, giving us condence that the above denition works for general n. However, we do not supply that calculation here. 120 Part III Classical aspects 121 Chapter 5 Tropical Fock-Goncharov coordinates for Kuperberg webs on surfaces The work discussed in this chapter is joint with Zhe Sun, and is based on the work of Xie [62], Goncharov and Shen [31], and Kuperberg [41]. In particular, most of the topological ideas appear, in one form or another, in Kuperberg's paper. For very related work, see [26, 46]. Other references on webs and their relation to \canonical" bases are [38, 22, 23]. Independently, Charlie Frohman and Adam Sikora [26] developed an identical topological description for webs as we did, and they dened coordinates for webs that are dierent but related to ours. We would like to profoundly thank Charlie for pointing us to Kuperberg's paper at the start, and for helping us with various technical details. We would also like to thank Dylan Allegretti for all of his help throughout this project. LetS be a punctured surface, as dened in§4.1. In contrast to Chapter 4, in this chapter we always assume that the punctured surface S has empty boundary. And we once again always assume that ideal triangulations never have any self-folded triangles. The integerN plays a more uid role, and should be interpreted according to the context. 122 5.1 Webs We now dene the primary objects of study in this chapter. Denition 5.1. A webW on the surfaceS is a boundary-less possibly-disconnected oriented tri-valent graph-with-curves that is embedded in S, satisfying (1) each vertex of W is either a source or sink, namely all the orientations either go in or out, respectively; (2) some components ofW may be closed oriented curves, in particular they do not contain any vertices. Figure 5.1: Web on the once punctured torus In Figure 5.1 we show a web on the once-punctured torus, having ve connected compo- nents, two of which are closed curves. The puncture is located at any of the four identied corners of the square. Denition 5.2. Two webs W 1 and W 2 on S are parallel-equivalent if W 1 can be taken to W 2 (as oriented graphs) by a sequence of moves of the following two types: (1) ambient isotopies of the surface S; 123 (2) a global parallel-move, exchanging oppositely-oriented boundaries of an \empty" em- bedded annulus AS, namely satisfying the property A\W =@A. Figure 5.2: Global parallel-move on the once punctured torus In Figure 5.2, we illustrate a global parallel-move. If two webs W 1 and W 2 are parallel- equivalent, then we say they belong to the same parallel-equivalence class of webs on S. Intuitively, we think of \parallel-equivalent" as meaning \homotopic" on the surface S. 5.2 Faces Let D denote the open unit disk inR 2 , and let D n denote the closed unit disk inR 2 with n distinct marked points M@D on its boundary, where n is an integer> 0. Denition 5.3. An n-face of a web W is a continuous function f W : D n ! S, called a face-map, such that (1) the open disk DD n is mapped homeomorphically byf W onto a componentf(D) W c S of the complement of the web W ; (2) each component of@D n M, also called a boundary component of the face, is mapped homeomorphically by f W onto a component of Wfvertices of Wg; (3) the marked points M@D n are mapped to vertices of the web W . 124 Figure 5.3: Web face We often identify the open disk D D n with its image under the mapping f W , and emphasize the integer n by writing D n to denote an n-face. In Figure 5.3 we show a typical n-face. In the example shown in the gure, n = 6. As another example, the web shown in Figure 5.1 has one 0-faceD 0 , two 2-facesD 2 , ve 4-faces D 4 , and one 6-face D 6 . An alternative name for a 4-face is a square. Note that we do not consider the marked closed diskD n nor its boundary to be oriented, and so it is only the un-oriented structure of the webW that is relevant for the denition of an n-face. Observe that when n > 0, then it is possible for the boundary @f W (D n ) W S of the image of an n-face to consist of fewer than n edges of the web. See Figure 5.1 for an example, where the unique 6-face consists of 5 edges of the web. Notice that non-face can have two adjacent boundary components in@D n M identied by the face-map f W , for in one case the oriented surface S would contain a M obius strip, and in the other case the web would have a boundary vertex, which is not allowed by the denition of webs. This impossibility is illustrated in Figure 5.4. Observe, by the property that each tri-valent vertex is a source or sink, that a web can never have n-faces for n an odd integer. Note that, in the cases n = 2 and n = 4, the boundary @f W (D n ) W S of the image of a 2-face D 2 or a 4-face D 4 always consists of exactly 2 or 4 edges, respectively, of the web W . Indeed, suppose otherwise. For the 2-face D 2 , this means there would be two 125 Figure 5.4: Prohibited face forming a \hat" adjacent boundary components of the face glued together, which is not possible by what we discussed above. And the only remaining case for the 4-face D 4 is when two opposite boundary components in@D 4 M are glued together. In one way a M obius strip is created, and in the other way the source-or-sink condition of the web is violated, and so we obtain a contradiction. This is illustrated in Figure 5.5. Figure 5.5: Prohibited 4-face 5.3 Essential webs General webs can be quite complicated. The following restriction drastically reduces this complexity. 126 Denition 5.4. A webW on the surfaceS is called essential if it does not have any 0-faces D 0 , 2-faces D 2 , or 4-faces D 4 . Note that if W is an essential web, and if W 0 is another web that is parallel-equivalent to W as dened in §5.1, then W 0 is also essential. We denote the set of parallel-equivalence classes of essential webs by W S . We sometimes abuse notation and write W 2 W S to indicate that W is an essential web. Remark 5.5. Kuperberg uses the terminology \non-elliptic" instead of \essential". 5.4 Knutson-Tao cone of an ideal triangulation Let N be a positive integer. Denition 5.6. A coneC is a submonoid ofZ N . Namely, it is a subsetCZ N that contains 0 and is closed under addition. A positive cone C is a cone that is contained inZ N >0 . Consider the punctured boundary-less connected oriented surfaceS, and let be an ideal triangulation of S. Dene the positive integerN to be twice the number of edgesES of plus the number of triangles TS of. For a xed triangulation ofS, Goncharov and Shen [31] essentially dened a positive coneC Z N >0 Z N using the ideas of Knutson-Tao [40]. We describe their construction now, and give a new (as far as we know) description of this cone, using ideas of Fock and Goncharov. Let TS be a triangle in the ideal triangulation of the surface S. We will associate to the triangle T a coneC T Z 7 . In §5.16, we will show that this is in fact a positive cone C T Z 7 >0 . We refer to this cone as the Knutson-Tao cone associated to the triangle T. We write an arbitrary integer point c2Z 7 as a 7-tuple c = (a 11 ;a 12 ;a 21 ;a 22 ;a 31 ;a 32 ;a) 2Z 7 : (5.1) 127 Following Fock and Goncharov, we imagine the seven integer coordinates of Z 7 as corre- sponding to seven dots attached to the triangle T, as depicted in Figure 5.6. - Figure 5.6: Seven tropical coordinates for a triangle and nine diamond inequalities, three of which are shown LetQ 3 Q denote the set of integer-thirds within the rational numbers. For any integer point c = (a 11 ;a 12 ;a 21 ;a 22 ;a 31 ;a 32 ;a) inZ 7 , dene a corresponding 9-tuple (D 11 ;D 12 ;D 13 ;D 21 ;D 22 ;D 23 ;D 31 ;D 32 ;D 33 ) 2 (Q 3 ) 9 ; (5.2) by the linear equations D 12 = (a +a 32 a 11 a 31 )=3; D 11 = (a 22 +a 31 a 0)=3; D 13 = (a 21 +aa 12 a 22 )=3; D 22 = (a +a 12 a 21 a 11 )=3; D 21 = (a 32 +a 11 a 0)=3; D 23 = (a 31 +aa 22 a 32 )=3; D 32 = (a +a 22 a 31 a 21 )=3; D 31 = (a 12 +a 21 a 0)=3; D 33 = (a 11 +aa 32 a 12 )=3: (5.3) 128 The integer-thirds D ij 2 Q 3 are called diamond numbers and are associated to nine diamonds embedded in the triangle T. These diamonds can be visualized in the discrete triangle 3 as shown in Figure 5.6. There, only three diamonds are displayed. The remaining six diamonds come from the 3-fold symmetry of the triangle T. The zeroes attached to the vertices of the triangle are not coordinates, but appear in the denition of the diamond numbers D ij . Denition 5.7. The Knutson-Tao coneC T Z 7 associated to the ideal triangleT is dened by C T = ( (a 11 ;a 12 ;a 21 ;a 22 ;a 31 ;a 32 ;a) 2Z 7 ; (D 11 ;D 12 ;D 13 ;D 21 ;D 22 ;D 23 ;D 31 ;D 32 ;D 33 ) 2Z 9 >0 (Q 3 ) 9 ) : (5.4) It is clear by linearity that this indeed denes a cone contained inZ 7 . Remark 5.8. Goncharov and Shen refer to the inequalitiesD ij 2Z >0 as the \Knutson-Tao rhombus inequalities". We prefer the terminology \diamond" to \rhombus". To see some examples of cone points c2 C T , view Figure 5.7. Formulaically, the eight cone points c = (a 11 ;a 12 ;a 21 ;a 22 ;a 31 ;a 32 ;a)2 Z 7 >0 appearing in the gure can be written down as c(R 1 ) = (0; 0; 1; 2; 2; 1; 1); c(L 1 ) = (0; 0; 2; 1; 1; 2; 2); c(R 2 ) = (2; 1; 0; 0; 1; 2; 1); c(L 2 ) = (1; 2; 0; 0; 2; 1; 2); c(R 3 ) = (1; 2; 2; 1; 0; 0; 1); c(L 3 ) = (2; 1; 1; 2; 0; 0; 2); c(T in ) = (2; 1; 2; 1; 2; 1; 3); c(T out ) = (1; 2; 1; 2; 1; 2; 3): (5.5) The eight 9-tuples D = (D 11 ;D 12 ;D 13 ;D 21 ;D 22 ;D 23 ;D 31 ;D 32 ;D 33 ) inZ 9 >0 (Q 3 ) 9 of dia- 129 mond numbers corresponding to these eight cone points can be computed as D(R 1 ) = (1; 0; 0; 0; 0; 0; 0; 0; 0); D(L 1 ) = (0; 1; 1; 0; 0; 0; 0; 0; 0); D(R 2 ) = (0; 0; 0; 1; 0; 0; 0; 0; 0); D(L 2 ) = (0; 0; 0; 0; 1; 1; 0; 0; 0); D(R 3 ) = (0; 0; 0; 0; 0; 0; 1; 0; 0); D(L 3 ) = (0; 0; 0; 0; 0; 0; 0; 1; 1); D(T in ) = (0; 0; 1; 0; 0; 1; 0; 0; 1); D(T out ) = (0; 1; 0; 0; 1; 0; 0; 1; 0): (5.6) Figure 5.7: Eight points in the Knutson-Tao cone for a triangle, four of which are shown We extend the denition of the Knutson-Tao cone from triangles to the whole triangula- tion in the obvious way. Let N be as dened in the introduction to current section, namely twice the number of edges ES of plus the number of triangles TS of . As we did in the case of a single triangle T, we think of each integer coordinate ofZ N as attached to 130 a point on the triangulation , either living on an edge or in the interior of a triangle. Each edge E or triangle T harbors two or one coordinate points, respectively. Each triangle T induces in a natural way a projection T :Z N −!Z 7 . Denition 5.9. The Knutson-Tao coneC associated to the ideal triangulation is dened to be the set of integer pointsc2Z N such that for each triangle T the 7-tuple T (c)2Z 7 is an element of the Knutson-Tao coneC T associated to T. It is clear from the denition that this indeed denes a cone contained inZ N , since each C T is a cone contained inZ 7 . 5.5 Goal: mapping from webs to the cone Let S be a punctured connected oriented surface equipped with an ideal triangulation , as described at the beginning of this chapter. Let W S be the set of essential webs W on the surface S, considered up to parallel-equivalence, as described in §5.3. And letC Z N be the Knutson-Tao cone associated to the ideal triangulation of the surface S. Here, the integerN is equal to twice the number of edgesE of the triangulation plus the number of triangles T of , as described in §5.4. We want to associate to each essential webW2W S a cone pointc = (W )2C that is well-dened on the parallel-equivalence class of the essential webW . Since the Knutson-Tao coneC depends on the triangulation , so too do the integer coordinates that we assign to the essential web. The strategy is to put an essential web W2W S into minimal position with respect to the ideal triangulation , and then to show that the possible minimal congurations can be reconstructed, up to isotopy, from the local information of how the web W = S T (W\T) looks over every triangle T of the ideal triangulation . 131 5.6 Webs-with-boundary With the goal of understanding webs locally, we now discuss webs in surfaces-with-boundary. For purposes of exposition, we will restrict to the case where the surface is an ideal polygon P k , which we will dene in a moment. Recall from §5.2, for k> 0 be a xed non-negative integer, that D k denotes the closed disk with k distinct marked points M @D k on its boundary. Let P k be a k-polygon. Namely,P k =D k M is obtained from the closed diskD k by removing thek marked points from its boundary. So whenk = 0, thenP 0 =D is just the closed disk, and whenk> 0, we think of the polygon P k as the closed disk with k punctures on its boundary. In this case, a component of @P k is an ideal arc, and we call P k an ideal polygon. The notion of a web W in a polygon P k is the same as in the boundary-less setting, recalling Denition 5.1, except now we let webs W have non-empty boundary @W 6= ? consisting of mono-valent vertices and satisfying @W = (@P k )\W . In particular, we allow for components of webs to be arcs with distinct endpoints, which are both mono-valent vertices, stretching from one boundary component of P k to another (possibly the same) boundary component ofP k . We require that websW intersectP k @P k , namely that webs do not consist of only a nite set of points on the boundary @P k of the polygon. For an example of a web in a polygon, see Figure 5.8. 5.7 External faces of webs-with-boundary For a non-negative integer n> 0, the notion of an n-face f W : D n ! P k of a web W in a polygon is the same as that for boundary-less surfacesS, except now we distinguish between internal and external faces. More precisely, the second item in Denition 5.3 is changed to read 2. each component of@D n M, also called a boundary component of the face, is mapped homeomorphically byf W onto a component ofWfvertices of Wg, or into the bound- 132 Figure 5.8: Web-with-boundary ary @P k of the polygon. Then, an internal n-face is one such that all the boundary components in @D n M are mapped to edges of the web, and an external n-face is one such that there exists at least one boundary component in @D n M that is mapped into the boundary @P k of the polygon. As usual, we often identify the open disk D with its image under the face-mapping f W , writing D int n or D ext n for emphasis. One dierence in the boundary setting is that external faces D ext n exist for n odd, which is not true for internal faces D int n . In Figure 5.9 we show a typical external n-face. In the example shown there, n = 7 and k = 3. Figure 5.9: External face 133 In Figure 5.10, from left to right, we show an external 2-face D ext 2 , also called a cap, an external 3-face D ext 3 , also called a fork, and two types of external 4-faces D ext 4 , which we refer to as type I, also called an H, and type II. It is not hard to see that these are the only possible examples of external 2-, 3-, and 4-faces, and also that external 0- and 1-faces do not exist. Figure 5.10: Cap, fork, H, and external 4-face of type II 5.8 Flat Riemannian metrics associated to webs-with- boundary LetW be a connected web in the closed diskP 0 =D. SinceW is connected, its complement in the open disk W c \DP 0 contains at most one annulus. If such an annulus exists, its closure contains the boundary@P 0 of the closed diskP 0 . Such an annulus exists if and only if the web W has empty boundary @W = (@P 0 )\W =?. Let W be a connected web in the closed disk P 0 = D such that @W6=?. Then every component of the complement ofW (in the open diskDP 0 ) is homeomorphic to an open disk, hence is either an internal or external face of the web. Consequently, the closed disk P 0 can be tiled by the dual graph of the web. More precisely, the vertices of the dual graph are the faces of the web, and the complement of the dual graph consists of triangles. An 134 example is shown in Figure 5.11. Figure 5.11: Tiling the disk with the dual graph of a web Such a tiling gives the closed disk P 0 =D the structure of a Flat Riemann surface with conical singularities and piecewise-geodesic boundary, by forcing each triangle in the tiling to be equilateral (all three angles are =3). By applying the Gauss-Bonnet theorem to this Riemann surface, we obtain the following result. Fact 5.10 (Gauss-Bonnet). Let W be a connected web with non-empty boundary @W6=? in the closed disk P 0 =D. Then 2 = X internal faces D int n 2 3 n ! + X external faces D ext n 3 (n 2) ! : (5.7) 5.9 Non-elliptic webs-with-boundary The notion of non-ellipticity provides a tremendous amount of regularity to a web, consid- ering the simplicity of the denition. Denition 5.11. A web W in a polygon P k is called non-elliptic if the web W has no internal 0-faces D int 0 , internal 2-faces D int 2 , or internal 4-faces D int 4 . 135 We recall that external 2-faces D ext 2 are also called caps, that external 2-faces D ext 3 are also called forks, and that external 4-faces D ext 4 come in two types, type I, also called H's, and type II, as shown in Figure 5.10 above. Lemma 5.12. Let W be a non-elliptic web in the closed disk P 0 = D such that W is connected, has non-empty boundary @W6=?, and has at least one tri-valent vertex. Then W has at least three forks and/or H's. Remark 5.13. It turns out, as a simple consequence of Lemma 5.12 together with the fact that webs are oriented having only sinks and sources, that non-elliptic webs W with empty boundary @W =? do not exist. Consequently, the hypotheses \@W6=?" in the lemma is super uous. Moreover, if we assume that W has no caps D ext 2 , then the hypothesis \W is connected" is also not necessary. Proof. Since @W6=?, we can give the closed disk P 0 = D the associated at Riemannian metric with singularities and apply the Gauss-Bonnet theorem, Fact 5.10. Since W is non- elliptic, the interior angle int (D int n ) 0 of every internal face D int n of W , which is equal to (=3)n, is not greater than zero, since n> 6. Notice also that the exterior angle ext (D ext n ) of an external face D ext n , which is equal to (=3) (n 2), is 0 if and only if n> 5. For external faces D ext n , we know that n> 2 since external 0- and 1-faces never exist. And since W is connected and contains at least one tri-valent vertex, it has no external 2-faces (else W would be an arc). So those external faces D ext n with a positive contribution satisfy n = 3; 4. The last part is because external 3-faces contribute 2=3 in the Gauss-Bonnet formula, and external 4-faces contribute =3. In Figure 5.12, we give examples showing how this lemma cannot be improved. 5.10 Essential and rung-less webs-with-boundary The concept of essential web in the boundary setting is more restrictive than in the boundary- less setting. 136 (a) Web with three forks (b) Web with six H's Figure 5.12: Small webs-with-boundary in the disk Denition 5.14. A web W in a polygon P k for k> 0 is called essential if the following two conditions are satised: (1) the web W is non-elliptic; (2) for any boundary component E @P k , and for any embedded compact arc with boundary on the edge E, then the geometric intersection number (W;E) of the web with the edgeE does not exceed the geometric intersection number(W;) of the web with the embedded arc . More concisely, (W;E)6(W;): (5.8) For a general picture, see Figure 5.13, and for some examples see Figure 5.14. We recall that another name for an external 2-face D ext 2 is a cap, another name for an external 3-faceD ext 3 is a fork, and another name for an external 4-faceD ext 4 of type I, recalling Figure 5.10, is an H. Note that essential webs cannot have any caps or forks. We frequently consider the operation of adding or removing an H from a web, as shown in Figure 5.15. In that example, the web on the left hand side of the gure is elliptic, and the web on the right hand side of the picture is non-elliptic but not essential. 137 Figure 5.13: Denition of an essential web The following claim is obvious from the denition. Fact 5.15. The following two statements hold: (1) removing an H preserves the property of being essential; (2) if an H is added to an essential web-with-boundary, such that the new web is still non-elliptic, then the new web is also essential. Given a polygon P k , we may consider its underlying closed disk D, equal to P k ifk = 0, and ifk> 0 such that the ideal polygonP k is obtained by removing nitely many punctures P from the boundary @D of the closed disk. Given any web W in the polygon P k , we may consider this web W as a web, denoted W say, in the underlying closed disk D. We sometimes refer to W as the induced web in D. Suppose W has an external face D ext W in D. Then it could be that the disk D ext W is not an external face of W viewed in the polygon P k . Indeed, this happens if and only if there is a puncture p2P lying on one of the boundary components of the external face D ext W . In this case, we say that the external face D ext W of the webW viewed in the closed disk D straddles the puncturep of the polygon P k . See the left 138 (a) Essential web (b) Non-elliptic web, not essential Figure 5.14: Examples and non-examples of essential webs hand side of Figure 5.14 for an example, where all three forks straddle a puncture, or the right hand side of the same gure, where two forks straddle a puncture. Notice that if W is a non-elliptic web in P k , then the induced web W in D is also non-elliptic. However, it could be that W is essential and W is not, by what we just said. SupposeW is a connected essential web inP 0 =D orP 1 , and letW be the induced web in the closed diskD. IfW does not have a tri-valent vertex, thenW is a single arc. SinceW is in P 0 or P 1 , it follows that W has a cap, violating that it is essential. So W in D has a tri-valent vertex. By Lemma 5.12 and Remark 5.13, W has at least three forks and/or H's. At most one of these three external faces of W can straddle a puncture of the polygon. SinceW is essential, it has no forks. SoW has an H. Removing this H yields a new connected essential web with strictly fewer tri-valent vertices. By repeating this process, we obtain a connected essential with no tri-valent vertices, a contradiction. We have proved the following claim. Fact 5.16. Essential webs in P 0 =D or P 1 do not exist. Denition 5.17. A web in the polygonP k fork> 0 is called rung-less if it does not contain an H. 139 Figure 5.15: Adding and removing an H from a web For an example, see Figure 5.16. Remark 5.18. Kuperberg uses the terminology \non-convex non-elliptic web in the k- clasped web space" for what we call an \essential web in the k-polygon", and he uses the terminology \core of a non-convex non-elliptic web in the k-clasped web space" for what we call a \rung-less essential web in the k-polygon". 5.11 Ladders in ideal biangles Another name for an ideal 2-polygon P 2 is an ideal biangle. The boundary@P 2 of a biangle P 2 consists of two bi-innite ideal arcs E 1 andE 2 , called the boundary edges, each limiting to the two punctures on the boundary @D of the closed disk D underlying the biangle P 2 . See Figure 5.17. In this section, we want to characterize essential webs W in a biangle P 2 . So let W P 2 be a, possibly-disconnected, essential web, and let W be the induced non-elliptic web in the underlying closed diskD. SinceWP 2 is essential, it does not have any caps on the boundary edges E 1 and E 2 . However, the induced web W D may have caps. These caps are, in particular, arc components ofW which, taken individually, straddle 140 (a) Rung-less essential web (b) Essential web, not rung-less Figure 5.16: Examples and non-examples of rung-less webs Figure 5.17: Essential web in a biangle one of the two punctures of the biangleP 2 . LetC denote the union of such arc components. Forgetting C, we obtain a new non-elliptic web WC in D having no caps. Suppose that the induced web WC D has at least one tri-valent vertex. Then by Lemma 5.12 and Remark 5.13,WC has a total of at least three forks and/or H's. At most two of these three external faces can straddle the two punctures of the biangle P 2 , and so at least one of these external faces is an external face of the web W in the biangle P 2 . Since W is essential, W does not have any forks, and so W has an H in the biangle P 2 sitting on one of the two boundary edges E 1 or E 2 . Removing this H yields a new essential web with strictly fewer tri-valent vertices. We repeat this process until there are no tri-valent 141 vertices remaining. We call the resulting essential web W 0 P 2 . As an example, consider Figure 5.17. There, C consists of two arcs straddling a single puncture of the biangle. The induced web WC has two forks and one H. The two forks straddle distinct punctures, and the web W has an H. Since W 0 is non-elliptic without any tri-valent vertices, it consists of a disjoint union of arcs. SinceW 0 is essential, each arc component ofW 0 travels according to its orientation (we recall that webs are oriented) from one boundary edgeE 1 orE 2 of the biangleP 2 to the other boundary edge. We see then that the original webW is obtained fromW 0 by inserting horizontal \rungs" into the vertical arc components of W 0 , which are thought of as the sides of a \ladder". Here the notions of horizontal and vertical are dened with respect to the biangle. Denition 5.19. A ladder is a, possibly-disconnected, essential web in the biangle P 2 . We show an example in Figure 5.18. Notice that the web W 0 is not uniquely dened. Indeed, it depends on whether rungs are pushed out of the biangle through one boundary edge E 1 of the biangle P 2 or the other boundary edge E 2 . In the example shown, all of the H's are pushed out of the bottom boundary component. (a) Ladder with rungs re- moved, which is also a ladder (b) Ladder (c) Bird's-Eye View (BEV) multicurve Figure 5.18: Ladders in biangles 142 This discussion shows that, given an essential web W in the biangle P 2 , the number of in-ends (resp. out-ends) of the web W located at one boundary edge E 1 is equal to the number of out-ends (resp. in-ends) of the web located at the other boundary edge E 2 . On the right-most side of Figure 5.18, we show the Bird's-Eye View (or BEV) multicurve <W > of the essential webW . The BEV multicurve<W > is not a web in the biangle, but is an oriented multicurve, namely an embedding in the biangle of each component of a disjoint union of oriented arcs, such that the images intersect generically. The BEV multicurve is dened by shrinking each rung of the ladder to a point, hence the name, since if the web were looked at from high above, then the rungs would not be visible, only the multicurve. More precisely, the BEV multicurve is obtained by applying a sequence of local moves, each replacing an H with a \swap", as shown in Figure 5.19. It follows that the oriented arcs of the BEV multicurve travel monotonically across the biangle. The BEV multicurve < W > of an essential web W possesses as many oriented arcs as the geometric intersection number(W;E 1 ) =(W;E 2 ) of the web with either boundary edge E 1 orE 2 of the biangle P 2 . This intersection number is also equal to the number of in-ends of the web plus the number of out-ends on a single boundary edge. By construction, only oriented arcs moving in opposite directions, relative to the biangle, are allowed to cross. Figure 5.19: Local move dening the Bird's-Eye View (BEV) multicurve Moreover, two distinct oriented arcs in the BEV multicurve<W > do not intersect twice. Indeed, if they did, then by monotonicity they would form a possibly-empty embedded bigon B inside P 2 , as shown in Figure 5.20. The interior of this embedded bigon B might still 143 intersect the BEV multicurve <W >. Since oriented arcs moving in the same direction do not cross, any oriented arc entering the embedded bigonB would have to leave via the same side of the bigonB through which it entered. Consequently, there exists a, possibly-smaller, empty embedded bigon B 0 whose interior is disjoint from the BEV multicurve <W >. But then the original web W would have an internal square, violating that it is non-elliptic. (a) Internal squares in the web (b) Corresponding embedded bigons in the BEV multicurve Figure 5.20: Interpreting non-ellipticity from the standpoint of the BEV multicurve We gather that the essential web W in the biangle P 2 is completely determined, up to planar isotopy, by the ordering of the in-ends and out-ends of the web W on both boundary edgesE 1 andE 2 of the biangleP 2 . Specically, it is the unique ladder whose BEV multicurve < W > is obtained by connecting these oriented in- and out-ends by monotonic oriented arcs, such that arcs moving in the same direction do not cross, and such that two arcs do not cross twice. We say that we have applied the ladder construction. We summarize: Proposition 5.20. Consider the biangle P 2 with boundary edges E 1 and E 2 . Fix boundary conditions on E 1 and E 2 consisting of a sequence of oriented ends placed on each edge, such that the number of in-ends (resp. out-ends) on E 1 is equal to the number of out-ends 144 (resp. in-ends) on E 2 . Then, up to planar isotopy, there is a unique, possibly-disconnected, essential web W in the biangle P 2 matching this boundary data, and it is obtained by the ladder construction. Moreover, every essential web in the biangle is obtained in this way. 5.12 Honeycombs in ideal triangles Another name for an ideal 3-polygon P 3 is an ideal triangle. We now characterize the rung-less essential webs in ideal triangles P 3 , recalling Denition 5.17. Denition 5.21. For n > 0 a positive integer, by the out- (resp. in-) n-honeycomb H out n (resp. H in n ) in the ideal triangle P 3 , we mean the dual graph H n of the n-triangulation of the topological n-triangle ( (x;y;z)2R 3 >0 ; x +y +z =n ) ; (5.9) associated to the discrete n-triangle n = ( (a;b;c)2Z 3 >0 ; a +b +c =n ) ; (5.10) where the orientation of H n is such that all the arrows go out of (resp. into) the triangle. In Figure 5.21, we show the out-5-honeycomb H out 5 . It is easy to see that an-honeycombH n is a rung-less essential web in the ideal triangle. Proposition 5.22. In the ideal triangle P 3 , a connected rung-less essential web W with at least one tri-valent vertex is, up to planar isotopy, a n-honeycomb web H out n or H in n . Proof. We begin with an index argument. LetWD be the induced web in the closed disk D obtained by forgetting the punctures of the ideal triangle P 3 . By Lemma 5.12, W has at least three forks and/or H's. Since W is essential and rung-less, then W cannot have any forks or H's. Therefore, the induced webWD has exactly three forks and/or H's, each of 145 Figure 5.21: Honeycomb which straddles a puncture of the ideal triangleP 3 . Since these three external faces ofW are the only positive contributions in the Gauss-Bonnet theorem (Fact 5.10), they must all be forks. Moreover, since these three forks' contribution to Gauss-Bonnet is exactly 2, every other interior and exterior face must have zero contribution. Thus, each interior face has exactly six sides, and each external face, besides the three forks straddling the punctures, has exactly 5 sides. It remains to show that these restrictions force the web W to be a honeycomb H n . We prove this by induction onn, showing how to \grow" the honeycomb out from a corner. Given the geometric restrictions that we have derived, the rest of the proof is purely topological. Indeed, Figures 5.22, 5.23, and 5.24 show that the only obstructions to tiling the honey- comb cannot occur. Remark 5.23. Notice that Proposition 5.22 assumes the web W is connected. Since hon- eycombs attach to all three boundary edges of the ideal triangle, we gather that a rung-less essential web W in the ideal triangle that is not necessarily connected has at most one connected component that contains a tri-valent vertex. Consequently, a rung-less essential, 146 Figure 5.22: Proof of Proposition 5.22: 1 of 3 possibly-disconnected, web W in the ideal triangle consists of a, possibly-empty, oriented honeycomb H out n or H in n together with three nested families of disjoint oriented arcs associ- ated to each corner of the triangle. A typical example is shown in Figure 5.25. 5.13 Minimal position of a web on the surface We thank Charlie Frohman and Adam Sikora for sharing their understanding of the technical aspects required for this section. We return to the setting of§§5.1-5.3, namely we want to study boundary-less websW in boundary-less punctured surfaces S. Denition 5.24. We say that an essential web W 2 W S is in minimal position with re- spect to the triangulation , if W achieves the minimum geometric intersection number min W 0((W 0 ;E)) with every edge E of the triangulation T, where the minimum is taken with respect to all essential webs W 0 parallel-equivalent to the essential web W . A triangulation-move is a certain isotopy of the web on the surface that changes the web's position with respect to the triangulation. We are primarily interested in three types of triangulation-moves. The cap- and fork-moves are shown in Figures 5.26, and the H-moves are shown in 5.27. 147 Figure 5.23: Proof of Proposition 5.22: 2 of 3 It is not clear from the denition that minimal positions exist. The concern is that a position minimizing the global sum of the geometric intersection numbers over all the edges might not minimize the geometric intersection number on a particular edge. We now show that minimal positions exist, and in particular give an algorithm for putting a web into minimal position. Each step of the algorithm goes as follows. Either there exists a nite, possibly-empty, sequence of H-moves, after which a cap- or fork-move can be performed, or not. If so, apply either of these moves and repeat. If not, then we are done. The algorithm stops because cap- and fork-moves strictly decrease the global geometric intersection number of the web with the ideal triangulation, whereas H-moves do not change the global intersection number. For the following proposition, recall the denition of a global parallel-move between parallel-equivalent webs, from Denition 5.2. Proposition 5.25. First, let W 2 W S be an essential web. Then the essential web W 0 parallel-equivalent to W obtained by applying the algorithm, described above, is in minimal position with respect to the ideal triangulation . Second, if W 0 and W 00 are parallel-equivalent essential webs, both in minimal position 148 Figure 5.24: Proof of Proposition 5.22: 3 of 3 with respect to the ideal triangulation , then the web W 0 can be taken to the web W 00 by a sequence of moves of the following three types: (1) an H-move; (2) a global parallel-move; (3) an ambient isotopy of the surface S respecting the ideal triangulation . Proof. This is a routine application of the index arguments developed for and applied to webs-with-boundary in §§5.6-5.12, combined with standard topological arguments, such as those appearing in [16]. 149 Figure 5.25: Disconnected rung-less essential web in the ideal triangle 5.14 Denition of the mapping LetW2W S be an essential web in the punctured boundary-less surfaceS, which is equipped with an ideal triangulation . We want to assign to W an element (W )2 C of the global Knutson-Tao cone C , such that if W and W 0 are parallel-equivalent essential webs, then (W ) = (W 0 ). Moreover, we wish for this to induce a one-to-one correspondence :W S !C between parallel-equivalence classes of essential webs and global cone points. We will need to replace the ideal triangulation with a split ideal triangulation b , meaning that we replace every edgeE of the triangulation with an ideal biangleB bounded by two edges E 1 and E 2 . An example is shown in Figure 5.28. By applying the algorithm of Proposition 5.25, we may put the web W into minimal position with respect to the non-split triangulation , while staying inside the parallel- equivalence class of W 2 W S . Let us do this, and call the resulting web W as well. By denition of minimal position, the restrictionW T ofW to an ideal triangleT in the non-split triangulation is essential, but it might not be rung-less, namely it might have H's. This is where the split ideal triangulation b comes in. We can remove an H from the 150 Figure 5.26: Cap- and fork-moves restricted webW T at the cost of isotoping it into an adjacent biangleB of b . Note that there is not necessarily a unique biangle in which to put the H, but dierent ways are connected by a simple modied H-move, as shown in Figure 5.29. Since removing an H preserves the property of being essential, we end up with a rung-less, possibly-disconnected, essential web in every triangle T of the split ideal triangulation b . Moreover, since the global web W is non-elliptic, by Fact 5.15 we also end up with essential webs W B in every biangle B of b . After all this is done, we say that the web W is in good position with respect to the split Figure 5.27: H-move 151 Figure 5.28: Split ideal triangulation ideal triangulation b . See Figures 5.31 and 5.32 for examples of good position. We gather by Propositions 5.20 and 5.22, as well as Remark 5.23, that in every biangle B the restricted web W B is a ladder, and in every triangle T the restricted web W T is a, possibly-empty, honeycomb H out n or H in n , where n2Z >0 , together with oriented arcs on the corners. We refer again to Figure 5.25. Figure 5.29: Modied H-move across a triangle between two biangles We recall now the Knutston-Tao coneC T for a single triangle T, thought of as a triangle in the split ideal triangulation b . As we did in §5.4, we associate seven integer coordinates to the triangle as depicted in Figure 5.6. In particular, we recall the eight cone pointsc(R 1 ), c(R 2 ), c(R 3 ), c(L 1 ), c(L 2 ), c(L 3 ), c(T out ), c(T in ) inC T dened in Equation (5.5). Remark 5.26. Notice that these integer coordinates are associated to the same seven dots drawn on the triangle as were the Fock-Goncharov coordinatesX i from Chapters 1-3, in the 152 casen = 3. However, in our current setting we think of these dots as supporting the tropical integerA-coordinates playing a dual role to that of the complexX-coordinates. We will associate the rst six c(R 1 );:::;c(L 3 ) of these eight cone points to six single oriented arcs located on the corner of the triangle, depending on (1) which corner the arc lies on, and (2) whether the arc goes clockwise or counter-clockwise around the vertex of the triangle. See the top two pictures of Figure 5.7 for two of these six arcs and their corresponding points c(R 1 );c(L 1 ) in the Knutson-Tao cone. Similarly, we will associate the two bottom pictures of that gure, dening the two cone points c(T out ) and c(T in ) of C T , to the n = 1 out- and in-1-honeycombs H out 1 and H in 1 in the triangle T, respectively, as illustrated in that gure. (Here, T stands for \tri-valent".) Moreover, we will associate the cone pointsnc(T out ) andnc(T in ) inC T to the out- and in-n-honeycombsH out n andH in n in the triangle T, respectively, for every integer n2Z >0 . In this way, simply by adding cone points, one for each connected component, we associate to each rung-less essential web W T in the triangle T a cone point c(W T ) inC T . It is not hard to see that the tropical coordinates assigned to the boundary edges E of the triangleT, which in particular are edges of the split ideal triangulation b , are completely determined by the total number of oriented in- and out-ends of the web lying on that edge. In particular, the tropical edge coordinates are preserved under the modied H-move, shown in Figure 5.29, since the eect of this move is only to swap two oppositely-oriented arcs on the same corner of the triangle. And trivially, that is by construction, the tropical coordinate assigned to the center point of the triangle (c.f. Figure 5.6) is unchanged if two parallel corner arcs are swapped. Since we will need it in just a moment, notice that the same argument shows that the global tropical coordinates are unchanged if we perform a global parallel- move, recalling Denition 5.2, to the whole web W , since this has the eect of swapping possibly many oppositely-oriented parallel corner arcs passing through many triangles. Moreover, since over the biangle B the restricted web W B is a ladder, the number of in- and out-ends on one boundary edge E 1 of the biangle is equal to the number of out- and 153 in-ends, respectively, on the other boundary edge E 2 . Consequently, if the boundary edges E 1 and E 2 are also boundaries of triangles T 1 and T 2 , then the tropical edge coordinates of the web W T 1 on the edge E 1 is equal to the corresponding tropical edge coordinates of the web W T 2 on the edge E 2 . This can be seen in Figures 5.31 and 5.32. Consequently, the disjoint union of cone points c(W T ) varying over all the triangles T in the split ideal triangulation b glue together to give a point c(W ) in the global Knutson-Tao coneC associated to the non-split ideal triangulation, recalling Denition 5.9. This is our denition of the mapping (W ). Given our previous work, it is not hard to see that the mapping is well-dened. Indeed, let W 0 2 W S be another essential web on the surface that is parallel-equivalent to W and has been put into minimal position with respect to the non-split ideal triangulation. Then, after further putting W 0 into good position with respect to the split ideal triangulation b , Proposition 5.25 immediately implies that the webW 0 is related to the webW by a sequence of modied H-moves, global parallel-moves, and trivial ambient isotopies preserving the split ideal triangulation b . As we have already discussed, all these moves preserve the tropical coordinates on every triangle T of b and therefore preserve the global tropical coordinates. Thus, the mapping :W S !C is well-dened. 5.15 Third theorem We state our main result, which is a precise version of Theorem 3 from the Introduction. Theorem 5.27 (with Zhe Sun). Let S be a boundary-less punctured surface equipped with an ideal triangulation that does not contain any self-folded triangles. Then the mapping :W S −!C ; (5.11) is a bijection from the set W S of parallel-equivalence classes of essential webs in S to the Knutson-Tao coneC associated to the ideal triangulation . 154 Sketch of proof. We proceed by constructing an explicit inverse function :C −!W S : (5.12) First, we analyze more carefully, for a single triangle T in the split ideal triangulation b , the Knutson-Tao coneC T . Specically, in §5.16, we will prove the following lemma: Lemma 5.28. An arbitrary cone point c2C T can be written uniquely in exactly one of the following three forms: c =n 1 c(R 1 ) +n 2 c(L 1 ) + +n 6 c(L 3 ); (n 1 ;n 2 ;:::;n 6 2Z >0 ); c =n 1 c(R 1 ) +n 2 c(L 1 ) + +n 6 c(L 3 ) +nc(T in ); (n 1 ;n 2 ;:::;n 6 2Z >0 ; n2Z >0 ); c =n 1 c(R 1 ) +n 2 c(L 1 ) + +n 6 c(L 3 ) +nc(T out ); (n 1 ;n 2 ;:::;n 6 2Z >0 ; n2Z >0 ); recalling the eight cone points c(R 1 ), c(R 2 ), c(R 3 ), c(L 1 ), c(L 2 ), c(L 3 ), c(T out ), c(T in ) in C T dened in Equation (5.5). Incidentally, this shows thatC T Z 7 >0 consists of positive integer points, as promised at the beginning of §5.4. Next, recalling Figure 5.25, the upshot is that the Knutson-Tao cone C T of the triangle does not parametrize all disconnected rung-less essential webs in the ideal triangle T, but does parametrize these webs up to local parallel moves, which swap oriented arcs on a corner, illustrated in Figure 5.30. Intuitively, the problem is that there is no natural way to order these oriented corner arcs. Now, if c2 C is a cone point in the global Knutson-Tao cone of the surface, then in particular the cone point c determines a family of local cone points c T , one for each triangle T in the split ideal triangulation b . Thus, we obtain a local picture overlaid on top of each triangle T, consisting of a, possibly-empty, honeycomb together with possibly a bunch of oriented corner arcs. We emphasize that this local triangle picture is well-dened only up to permutation of these corner arcs. We then make a choice of such local pictures, one for 155 Figure 5.30: Local parallel-move each triangle T, namely we x permutations of corner arcs locally over every triangle. Since the triangle cone points c T come from a single global cone point c, they are by denition compatible across the biangles B lying between triangles. That is, the number of in-ends (resp. out-ends) of a local triangle picture, corresponding to a local cone point c T 1 , at one boundary edge E 1 of the biangle B is equal to the number of out-ends (resp. in-ends) of the other local triangle picture, corresponding to a local cone point c T 2 , at the other boundary edgeE 2 of the biangle. We therefore may \combine" these two local triangle webs by gluing them together across the biangle B according to the ladder construction discussed in Proposition 5.20. We refer once again to Figures 5.31 and 5.32 for examples. The result is a global web on the surface. If we are lucky, then this web is essential, recalling Denition 5.4. However, the procedure may result in an elliptic web. In this case, it turns out that this un-desirable elliptic web only has internal squares, which can be removed in a controlled way, but not in a unique way, resulting in, a priori, dierent possible global essential webs. To summarize so far, given a global cone pointc2C we have a procedure for creating a global essential web on the surfaceS. However, this depends on two sets of choices. First, it depends on xing permutations of corner arcs on local triangle pictures. Second, it depends on removing internal squares (if they exist), after having glued together the local triangle pictures via the ladder construction. Call the resulting global essential web (c). The 156 following lemma is the most technical part of the proof of the theorem: Lemma 5.29. Given an initial global cone point c 2 C , the resulting choice of global essential web (c) on S is well-dened up to parallel-equivalence class, and so we have induced a well-dened mapping :C !W S . Proof. The topological content of this statement was independently proved in [26, §13, The- orem 22]. See Remark 5.30 below for a further discussion. This is a global statement. In other words, if two global essential webs are obtained in this way by making dierent choices, then just knowing what these webs look like over a local patch of the surface does not make it clear that the two global webs are indeed parallel- equivalent. For an example, compare Figures 5.31 and Figure 5.32, whose local triangle pictures only dier by a cyclic permutation of oriented corner arcs on the left hand triangle. In order to prove the second lemma, we describe an explicit algorithm taking one web to the other by a sequence of (1) modied H-moves (c.f. Figure 5.29), and (2) global parallel- moves (c.f. Figure 5.2). We do not discuss this proof here. Once the two above lemmas have been proved, it is more or less by construction that the well-dened mappings and are inverses of each other. Remark 5.30. In [26, §13], as the main step in the proof of Theorem 22 (their version of our Theorem 5.27), Frohman and Sikora prove precisely the same topological statement as our Lemma 5.29. Their proof is algebraic in nature, and crucially uses the non-trivial skein- theoretic result saying that the collection of parallel-equivalence classes of essential webs on the surface forms a linear basis for the SL 3 (C)-skein algebra (proved in [59, Theorem 9.5]). In contrast, our proof of Lemma 5.29 is purely topological-combinatorial in nature, and does not require any skein theory. In fact, we expect to be able to apply our Theorem 5.27, in combination with the SL 3 (C)-quantum trace map (c.f. §4), to give an alternative proof of Sikora and Westbury's theorem, using the same strategy as that appearing in [5, §8]. 157 Figure 5.31: Good position with respect to a split ideal triangulation (compare Figure 5.32) Figure 5.32: Dierent corner arc permutations for local triangle pictures yield dierent local ladder pictures (compare Figure 5.31), but not dierent global parallel-equivalence classes 5.16 Integer cones: proof of Lemma 5.28 LetCZ N be a cone (c.f. Denition 5.6). We assume the material of §5.4. Denition 5.31. We say a family of cone points c 1 ;:::;c k 2 C spans the cone C if every cone point c2C can be written as aZ >0 -linear combination of the cone points c 1 ;:::;c k . LetCZ N Q N be a cone, and let Q such that 02 . We say that c 1 ;:::;c k 2C are weakly independent over if, for all ! 1 ;:::;! k 2 , then ! 1 c 1 + +! k c k = 02Q N implies ! 1 = =! k = 0. 158 We say that c 1 ;:::;c k 2 C are strongly independent over if, for all ! 1 ;:::;! k and ! 0 1 ;:::;! 0 k in , then! 1 c 1 ++! k c k =! 0 1 c 1 ++! 0 k c k 2Q N implies! 1 =! 0 1 ;:::;! k =! 0 k . As a warm up, note that strongly independent over implies weakly independent over , since 02 . On the other hand, strongly independent overZ >0 is implied by (in fact, is equivalent to) weakly independent overZ, the latter which is equivalent to weakly indepen- dent overQ, which is the same as the usual notion of linear independence overQ. We say that the cone points c 1 ;:::;c k 2C form a weak basis of the coneC, if they span C and are weakly independent overZ >0 . Warning: A weak basis need not satisfy the familiar property that every element c2C can be written uniquely as c =n 1 c 1 + +n k c k for n 1 ;:::;n k 2Z >0 . Indeed, this requires that c 1 ;:::;c k be strongly independent overZ >0 . - Figure 5.33: (Recall of Figure 5.6) Seven tropical coordinates for a triangle and nine diamond inequalities, three of which are shown Recall the denition of the Knutson-Tao cone C T Z 7 associated to an ideal triangle T (c.f. Denition 5.7). We saw that there are eight \distinguished" cone points, displayed in Equation (5.5), namely c(R 1 );c(L 1 );c(R 2 );c(L 2 );c(R 3 );c(L 3 );c(T in );c(T out )2C T . Recall also the nine diamond numbers D ij =D ij (c) (i;j = 1; 2; 3), which are non-negative integers 159 by denition ofC T , associated to each of these eight cone points (c.f. Equation (5.6)). Note that these eight cone points c(R 1 );:::;c(T out ) are not strongly independent over Z >0 . For instance, c(L 1 ) +c(L 2 ) +c(L 3 ) = c(T in ) +c(T out )2C T . (The existence of such a relation is forced by rank considerations, as we are dealing with eight points inZ 7 .) Proposition 5.32. The cone points c(R 1 ), c(L 1 ), c(R 2 ), c(L 2 ), c(R 3 ), c(L 3 ), c(T in ), c(T out ) form a weak basis of the Knutson-Tao cone C T , namely, they span C T and are weakly inde- pendent overZ >0 . In particular,C T can be written in a better form (c.f. Denition 5.7) C T = ( (a 11 ;a 12 ;a 21 ;a 22 ;a 31 ;a 32 ;a) 2Z 7 >0 ; (D 11 ;D 12 ;D 13 ;D 21 ;D 22 ;D 23 ;D 31 ;D 32 ;D 33 ) 2Z 9 >0 (Q 3 ) 9 ) ; (5.13) meaning thatC T Z 7 >0 is actually a positive cone. Moreover,C T =C in T tC out T is the disjoint union of two subsets C in T := Span Z >0 c(R 1 );c(L 1 );c(R 2 );c(L 2 );c(R 3 );c(L 3 ) ! Z >0 c(T in ); C out T := Span Z >0 c(R 1 );c(L 1 );c(R 2 );c(L 2 );c(R 3 );c(L 3 ) ! Z >0 c(T out ); (5.14) the points c(R 1 );c(L 1 );c(R 2 );c(L 2 );c(R 3 );c(L 3 );c(T in ) are strongly independent overZ >0 ; the points c(R 1 );c(L 1 );c(R 2 );c(L 2 );c(R 3 );c(L 3 );c(T out ) are strongly independent overZ >0 . Corollary 5.33. Lemma 5.28 is an immediate consequence of the proposition. Proof. Let us warm up with the simple observation that, once we show the cone points c(R 1 );c(L 1 );c(R 2 );c(L 2 );c(R 3 );c(L 3 );c(T in );c(T out ) form a weak basis ofC T , then by deni- tion they spanC T , thus Equation (5.13) holds since these eight cone points are inZ 7 >0 . Now, we begin in earnest with some generalities. Let C 1 ;C 2 Z N be any two cones. Then we may talk about functions : C 2 ! C 1 which are Z >0 -linear dened in the usual 160 Figure 5.34: (Recall of Figure 5.7) Eight points in the Knutson-Tao cone for a triangle, four of which are shown way. Assume is a bijection. Then sends partitions ofC 2 to partitions ofC 1 . Moreover, assume extends to aQ-linear isomorphism e :Q N !Q N . Next, let c 1 ;:::;c k 2C 1 and c 0 1 ;:::;c 0 k 2C 2 be cone points such that (c 0 i ) = c i . Then: (I) if the c 0 i span C 2 , then the c i span C 1 , by the surjectivity andZ >0 -linearity of ; (II) if thec 0 i are weakly independent overZ >0 , then thec i are weakly independent overZ >0 , by the injectivity andZ >0 -linearity of ; and (III) if thec 0 i are strongly independent overZ >0 , then, as discussed in Denition 5.31, the c 0 i are independent overQ, hence the c i are independent overQ since e is aQ-linear isomorphism, hence the c i are strongly independent overZ >0 . By (I) and (II), if the c 0 i form a weak basis ofC 2 , then the c i form a weak basis ofC 1 . 161 We return to case of interest. PutC 1 =C T the Knutson-Tao cone of the triangle, and set c 1 =c(R 1 ); c 2 =c(L 1 ); c 3 =c(R 2 ); c 4 =c(L 2 ); c 5 =c(R 3 ); c 6 =c(L 3 ); c 7 =c(T in ); c 8 =c(T out ): (5.15) In light of the preceding generalities, we have reduced to proving the following result: Claim 5.34. There exists a cone C 2 Z 7 ; cone points c 0 1 ;:::;c 0 8 2 C 2 ; a partition C 2 = C >0 2 tC <0 2 ; a Z >0 -linear bijection : C 2 ! C T ; and, an extension e of to a Q-linear isomorphism e :Q 7 !Q 7 ; such that: We have (c 0 i ) =c i ; also (C >0 2 ) =C in T and (C <0 2 ) =C out T ; the cone pointsc 0 1 ;:::;c 0 8 form a weak basis of the coneC 2 ; the cone points c 0 1 ;:::;c 0 6 ;c 0 7 are strongly independent overZ >0 ; the cone points c 0 1 ;:::;c 0 6 ;c 0 8 are strongly independent overZ >0 . We prove the claim. DeneC 2 Z 7 by C 2 = ( (D 11 ;D 12 ;D 21 ;D 22 ;D 31 ;D 32 ;x)2Z 6 >0 Z; x6 min(D 12 ;D 22 ;D 32 ) ) : (5.16) We see thatC 2 is a cone. Put c 0 1 = (1; 0; 0; 0; 0; 0; 0); c 0 2 = (0; 1; 0; 0; 0; 0; 0); c 0 3 = (0; 0; 1; 0; 0; 0; 0); c 0 4 = (0; 0; 0; 1; 0; 0; 0); c 0 5 = (0; 0; 0; 0; 1; 0; 0); c 0 6 = (0; 0; 0; 0; 0; 1; 0); c 0 7 = (0; 0; 0; 0; 0; 0; 1); c 0 8 = (0; 1; 0; 1; 0; 1;1): (5.17) Check c 0 1 ;:::;c 0 8 are inC 2 . Dene C >0 2 :=C 2 \ (Z 6 >0 Z >0 ); C <0 2 :=C 2 \ (Z 6 >0 Z <0 ): (5.18) 162 Then C 2 =C >0 2 tC <0 2 is a partition. We will dene and e momentarily. Before that, we discuss spanning and independence. First, we show c 0 1 ;:::;c 0 8 spansC 2 . We see by Equation (5.17) that C >0 2 = Span Z >0 (c 0 1 ;:::;c 0 6 )Z >0 c 0 7 : (5.19) If c 0 2C <0 2 , then its last coordinate is x61. Since 16x6 min(D 12 ;D 22 ;D 32 ), we have c 0 = (D 11 ;x +D 0 12 ;D 21 ;x +D 0 22 ;D 31 ;x +D 0 32 ;x1); (5.20) for some D 11 ;D 0 12 ;D 21 ;D 0 22 ;D 31 ;D 0 32 2Z >0 andx2Z >0 . That is, c 0 =D 11 c 0 1 +D 0 12 c 0 2 +D 21 c 0 3 +D 0 22 c 0 4 +D 31 c 0 5 +D 0 32 c 0 6 + (x)c 0 8 : (5.21) Thus, C <0 2 = Span Z >0 (c 0 1 ;:::;c 0 6 )Z >0 c 0 8 ; (5.22) where the containment is by Equation (5.17). Second, we showc 0 1 ;:::;c 0 8 are weakly independent overZ >0 . Indeed, ifn 1 c 0 1 ++n 8 c 0 8 = 0, then n 1 =n 3 =n 5 = 0 and n 2 +n 8 ;n 4 +n 8 ;n 6 +n 8 ;n 7 n 8 = 0. Since n 2 ;n 4 ;n 6 ;n 8 are inZ >0 , it follows that n 2 = n 4 = n 6 = n 8 = 0, and so n 7 = n 8 = 0, as desired. Combined with the previous paragraph, we gather c 0 1 ;:::;c 0 8 form a weak basis ofC 2 . Third, we show c 0 1 ;:::;c 0 6 ;c 0 7 are strongly independent over Z >0 , which is equivalent to being independent overQ, which is clear. Fourth, it is similarly clear that c 0 1 ;:::;c 0 6 ;c 0 8 are independent overQ hence strongly independent overZ >0 . We now dene aZ >0 -linear bijection ': C T ! C 2 . Its inverse is the desiredZ >0 -linear 163 bijection :=' 1 :C 2 !C T . First, if c = (a 11 ;a 12 ;a 21 ;a 22 ;a 31 ;a 32 ;a) is inC T , then x := (a 11 a 12 +a 21 a 22 +a 31 a 32 )=3 =D 13 D 12 =D 23 D 22 =D 33 D 32 >D 12 ;D 22 ;D 32 ; (5.23) since the D ij 2Z >0 as c2C T . Thus, x> max(D 12 ;D 22 ;D 32 ) =min(D 12 ;D 22 ;D 32 ): (5.24) Therefore, we may dene a function ':C T !Z 6 >0 Z by '(c) = (D 11 ;D 12 ;D 21 ;D 22 ;D 31 ;D 32 ;x); (5.25) whose image is inC 2 (c.f. Equation (5.16)). We see ':C T !C 2 isZ >0 -linear. Check that '(c i ) =c 0 i . Since the c 0 i spanC 2 , it follows byZ >0 -linearity that' is surjective. Also, by Equations (5.19) and (5.22) andZ >0 -linearity, '(C in T ) =C >0 2 and '(C out T ) =C <0 2 (recall the denitions in Equation (5.14)). The formula for ' extends to dene aQ-linear isomorphism e ':Q 7 !Q 7 . Its inverse is the desiredQ-linear isomorphism e := e ' 1 :Q 7 !Q 7 . Indeed, the bijectivity of e ' follows by computing the values on the standard basis, giving the invertible matrix e '(e 1 ; e 2 ; e 3 ; e 4 ; e 5 ; e 6 ; e 7 ) = 1 3 0 B B B B B B B B B B B B B B B B B @ 0 0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 C C C C C C C C C C C C C C C C C A ; (5.26) 164 thinking of the e j 's as column vectors, and the matrix acting on the left. So e is dened. Since e ' is an injection, so is its restriction ': C T ! C 2 . And we argued above that ' : C T ! 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Higher laminations, webs andN = 2 line operators. 2013.url: https://arxiv.org/ abs/1304.2390. 171 Appendix A In the following Mathematica code, Sections 1-2 reproduce Equations (3.43) and (3.44), and Sections 3-5 verify that these two 2 2 sub-matrices ( a b c d ) of the quantum left and right matrices L ! and R ! , respectively, satisfy theq-commutation relations appearing in Equation (3.29), see also Equation (3.32). 172 �� (* SECTION 1: EXAMPLE: CLASSICAL LEFT 4x4 MATRIX *) S[Z1_, Z2_, Z3_] := {{Z1*Z2*Z3, 0, 0, 0}, {0, Z2*Z3, 0, 0}, {0, 0, Z3, 0}, {0, 0, 0, 1}}; L0={{1, 1, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}; L1[X_] := {{X, 0, 0, 0}, {0, 1, 1, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}; L2[X_] := {{X, 0, 0, 0}, {0, X, 0, 0}, {0, 0, 1, 1}, {0, 0, 0, 1}}; η = Z1^(-1/4)Z2^(-2/4)Z3^(-3/4)X1^(-1/4)X2^(-2/4)X3^(-1/4)Z1′^(-1/4)Z2′^(-2/4)Z3′^(-3/4); L = Expand[η*S[Z1, Z2, Z3].L0.L1[X1].L2[X2].L0.L1[X3].L0.S[Z1′, Z2′, Z3′]]; (* 2x2 sub-matrix a,b,c,d  *) L[[1, 3]] L[[1, 4]] L[[2, 3]] L[[2, 4]] �� Z1 3/4 Z2 Z3 1/4 Z3′ 1/4 X1 1/4 X2 X3 1/4 Z1′ 1/4 Z2 ′ + X2 Z1 3/4 Z2 Z3 1/4 Z3′ 1/4 X1 1/4 X3 1/4 Z1 ′ 1/4 Z2′ + X1 3/4 X2 Z1 3/4 Z2 Z3 1/4 Z3′ 1/4 X3 1/4 Z1′ 1/4 Z2′ �� Z1 3/4 Z2 Z3 1/4 X1 1/4 X2 X3 1/4 Z1′ 1/4 Z2 ′ Z3′ 3/4 �� Z2 Z3 1/4 Z3′ 1/4 X1 1/4 X2 X3 1/4 Z1 1/4 Z1′ 1/4 Z2 ′ + X2 Z2 Z3 1/4 Z3′ 1/4 X1 1/4 X3 1/4 Z1 1/4 Z1′ 1/4 Z2′ �� Z2 Z3 1/4 X1 1/4 X2 X3 1/4 Z1 1/4 Z1′ 1/4 Z2 ′ Z3′ 3/4 �� (* SECTION 2: EXAMPLE: CLASSICAL RIGHT 4x4 MATRIX *) (* Same definition for S[Z1,Z2,Z3] as in Section 1 *) R0 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 1, 1}}; R1[X_] := {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 1, 1, 0}, {0, 0, 0, X^(-1)}}; R2[X_] := {{1, 0, 0, 0}, {1, 1, 0, 0}, {0, 0, X^(-1), 0}, {0, 0, 0, X^(-1)}}; η = Z1^(-1/4)Z2^(-2/4)Z3^(-3/4)X2^(+1/4)X1^(+2/4)X3^(+1/4)Z1″^(-1/4)Z2″^(-2/4)Z3″^(-3/4); R = Expand[η*S[Z1, Z2, Z3].R0.R1[X2].R2[X1].R0.R1[X3].R0.S[Z1 ″, Z2″, Z3″]]; (* 2x2 sub-matrix a,b,c,d  *) R[[3, 1]] R[[3, 2]] R[[4, 1]] R[[4, 2]] �� X1 X2 1/4 X3 1/4 Z1″ 3/4 Z2″ Z3 1/4 Z3″ 1/4 Z1 1/4 Z2 �� X2 1/4 X3 1/4 Z2″ Z3 1/4 Z3″ 1/4 X1 Z1 1/4 Z1″ 1/4 Z2 + X1 X2 1/4 X3 1/4 Z2″ Z3 1/4 Z3 ″ 1/4 Z1 1/4 Z1″ 1/4 Z2 �� X1 X2 1/4 X3 1/4 Z1″ 3/4 Z2″ Z3″ 1/4 Z1 1/4 Z2 Z3 3/4 �� X3 1/4 Z2″ Z3″ 1/4 X1 X2 3/4 Z1 1/4 Z1″ 1/4 Z2 Z3 3/4 + X2 1/4 X3 1/4 Z2″ Z3″ 1/4 X1 Z1 1/4 Z1″ 1/4 Z2 Z3 3/4 + X1 X2 1/4 X3 1/4 Z2″ Z3″ 1/4 Z1 1/4 Z1″ 1/4 Z2 Z3 3/4 �� (* SECTION 3: DEFINITIONS NEEDED FOR QUANTUM CALCULATIONS *) (* number of coordinates *) n = 12; (* Poisson structure matrix *) (* Z3=1,Z2=2,Z1=3,Z3 ′=4,Z2′=5,Z1 ′=6,X1=7,X2=8,X3=9,Z3″=10,Z2 ″=11,Z1 ″=12 *) (* Encodes, e.g., Z3*Z2=q^(1)Z2*Z3, Z3*Z3′=q^(2)Z3′*Z3, Z2*Z3=q^(-1)Z3*Z2 *) P = {{0, 1, 0, 2, 0, 0, 0, -2, 0, 0, 0, 0}, {-1, 0, 1, 0, 0, 0, -2, 2, 0, 0, 0, 0}, {0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2}, {-2, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0}, {0, 0, 0, 1, 0, -1, 0, -2, 2, 0, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, -2, 2, 0, 0}, {0, 2, -2, 0, 0, 0, 0, -2, 2, 0, -2, 2}, {2, -2, 0, -2, 2, 0, 2, 0, -2, 0, 0, 0}, {0, 0, 0, 0, -2, 2, -2, 2, 0, -2, 2, 0}, {0, 0, 0, 0, 0, -2, 0, 0, 2, 0, -1, 0}, {0, 0, 0, 0, 0, 0, 2, 0, -2, 1, 0, -1}, {0, 0, 2, 0, 0, 0, -2, 0, 0, 0, 1, 0}}; (* i[]: For encoding a monic monomial in the n variables *) (* E.g., Z3^(1/2)*Z2^(-1/2)*Z1^(1/4)*Z3 is encoded by i[1,1/2],i[2,-1/2],i[3,1/4],i[1,1]  *) i[z__] := {z}[[1]]+ ⅈ{z}[[2]]; 2 �� thesis1.nb �� (* a[]: Weyl ordering [[.,.]] coefficient for a monomial *) (* E.g., since [[Z3*Z2]] = q^(-1/2)Z3*Z2, then a i[1,1],i[2,1] =q^(-1/2) *) a[w__] := tempw = Flatten[w]; temp = SumSum-(1/2)*Im tempw iImtempwjP Retempw i, Retempw j , j, i+1, Length[tempw], i, 1, Length[tempw]; q^temp; (* Test: *) (* ai[1,1],i[2,1] *) (* b[]: Coefficient resulting from re-ordering a monomial according to the order 1,2,...,n above *) (* E.g., since Z2*Z3=q^(-1)Z3*Z2, then b i[2,1],i[1,1]=q^(-1), but bi[1,1],i[2,1] =1 *) b[w__] := temp=0; tempw = Flatten[w]; Do  DoIfRetempw Length[tempw]-i1+1  ⩵n-i3+1, Do IfRetempw i2 ≥n-i3+1, , temp = temp+1*ImtempwLength[tempw]-i1+1 Imtempwi2PRetempwLength[tempw]-i1+1 , Retempwi2, i2, Length[tempw]-i1+2, Length[tempw] , i1, Length[tempw], i3, n; q^temp; (* Test: *) (* bi[2,1],i[1,1] *) (* bi[1,1],i[2,1] *) (* bi[2,1],i[1,1],i[2,1] *) (* c[]: Prints a possibly re-ordered monomial as if the variables commuted *) (* E.g., prints either Z3*Z2 or Z2*Z3 when Z2*Z3 is entered, based on Mathematica's whims *) c[w__]:= temp = ConstantArray[0, n]; tempw = Flatten[w]; Do  tempRetempwi = temp Re tempw i +Imtempwi, i, Length[tempw] ; (* the following depends on the ordering of the n variables *) (* Z3=1,Z2=2,Z1=3,Z3 ′=4,Z2 ′=5,Z1′=6,X1=7,X2=8,X3=9,Z3″=10,Z2″=11,Z1″=12 *) Z3^temp[[1]]*Z2^temp[[2]]*Z1^temp[[3]]* Z3′^temp[[4]]*Z2′^temp[[5]] *Z1 ′^temp[[6]]*X1^temp[[7]]*X2^temp[[8]]* X3^temp[[9]]*Z3″^temp[[10]]*Z2″^temp[[11]]*Z1″^temp[[12]] ; (* Test: *) (* ci[2,1],i[1,1] *) (* ci[1,1],i[2,1] *) (* ci[2,1],i[1,1],i[2,1] *) (* f[m1,m2]: Computes [[m_1]]*[[m_2]] = q^r m, where m should be viewed as if in the order 1,2,...,n, but, as for c[], Mathematica may not present the monomial in the proper order, so, for instance, if the output is q^(-1) Z_2*Z_3, the correct output is actually q^(-1) Z_3*Z_2 *) f[x_, y_] := a[x]*a[y]*b[{x, y}]*c[{x, y}]; (* Test: *) (* fi[2,1],i[1,1]  *) (* fi[1,1],i[2,1]  *) (* fi[2,1],i[1,1],i[2,1] *) thesis1.nb �� 3 �� (* SECTION 4: CHECKING COMMUTATION RELATIONS FOR QUANTIZED 2x2 SUB-MATRIX OF LEFT MATRIX, C.F. SECTION 1 *) (* Encoding 2x2 sub-matrix, e.g. a := [[a_1]]+[[a_2]]+[[a_3]] *) (* Recall *) (* Z3=1,Z2=2,Z1=3,Z3 ′=4,Z2′=5,Z1 ′=6,X1=7,X2=8,X3=9,Z3″=10,Z2 ″=11,Z1 ″=12 *) a1 = i[1, 1/4], i[2, 2/4], i[3, 3/4], i[4, 1/4], i[5, -2/4], i[6, -1/4], i[7, -1/4], i[8, -2/4], i[9, -1/4], i[10, 0], i[11, 0], i[12, 0] ; a2 = i[1, 1/4], i[2, 2/4], i[3, 3/4], i[4, 1/4], i[5, -2/4], i[6, -1/4], i[7, -1/4], i[8, 2/4], i[9, -1/4], i[10, 0], i[11, 0], i[12, 0] ; a3 = i[1, 1/4], i[2, 2/4], i[3, 3/4], i[4, 1/4], i[5, -2/4], i[6, -1/4], i[7, 3/4], i[8, 2/4], i[9, -1/4], i[10, 0], i[11, 0], i[12, 0] ; b1 = i[1, 1/4], i[2, 2/4], i[3, 3/4], i[4, -3/4], i[5, -2/4], i[6, -1/4], i[7, -1/4], i[8, -2/4], i[9, -1/4], i[10, 0], i[11, 0], i[12, 0]; c1 = i[1, 1/4], i[2, 2/4], i[3, -1/4], i[4, 1/4], i[5, -2/4], i[6, -1/4], i[7, -1/4], i[8, -2/4], i[9, -1/4], i[10, 0], i[11, 0], i[12, 0]; c2 = i[1, 1/4], i[2, 2/4], i[3, -1/4], i[4, 1/4], i[5, -2/4], i[6, -1/4], i[7, -1/4], i[8, 2/4], i[9, -1/4], i[10, 0], i[11, 0], i[12, 0] ; d1 = i[1, 1/4], i[2, 2/4], i[3, -1/4], i[4, -3/4], i[5, -2/4], i[6, -1/4], i[7, -1/4], i[8, -2/4], i[9, -1/4], i[10, 0], i[11, 0], i[12, 0]; (* Checking relations *) (* da-ad = (q-q^(-1))bc *) Expandf d1, a1 +fd1, a2+fd1, a3- fa1, d1+f a2, d1+fa3, d1 -(q-q^(-1))* fb1, c1+fb1, c2 (* bc = cb *) Expandf b1, c1+fb1, c2 - fc1, b1+fc2, b1 (* ca = qac *) Expandf[c1, a1]+f[c1, a2]+f[c1, a3]+f[c2, a1]+f[c2, a2]+f[c2, a3] -q* f[a1, c1]+f[a2, c1]+f[a3, c1]+f[a1, c2]+f[a2, c2]+f[a3, c2]  (* dc = acd *) Expandf d1, c1+fd1, c2 -q* f c1, d1+fc2, d1   (* ba = qab *) Expandf b1, a1+fb1, a2+fb1, a3 -q* f a1, b1+fa2, b1 +fa3, b1  (* db = qdb *) Expandf d1, b1 -q* f b1, d1 ��0 ��0 ��0 ��0 ��0 ��0 4 �� thesis1.nb �� (* SECTION 5: CHECKING COMMUTATION RELATIONS FOR QUANTIZED 2x2 SUB-MATRIX OF RIGHT MATRIX, C.F. SECTION 2 *) (* Encoding 2x2 sub-matrix, e.g. d := d_1+ d_2+ d_3  *) (* Recall *) (* Z3=1,Z2=2,Z1=3,Z3 ′=4,Z2′=5,Z1 ′=6,X1=7,X2=8,X3=9,Z3″=10,Z2 ″=11,Z1 ″=12 *) a1 = i[1, 1/4], i[2, -1/2], i[3, -1/4], i[4, 0], i[5, 0], i[6, 0], i[8, 1/4], i[7, 1/2], i[9, 1/4], i[10, 1/4], i[11, 1/2], i[12, 3/4]; b1 = i[1, 1/4], i[2, -1/2], i[3, -1/4], i[4, 0], i[5, 0], i[6, 0], i[8, 1/4], i[7, -1/2], i[9, 1/4], i[10, 1/4], i[11, 1/2], i[12, -1/4]; b2 = i[1, 1/4], i[2, -1/2], i[3, -1/4], i[4, 0], i[5, 0], i[6, 0], i[8, 1/4], i[7, 1/2], i[9, 1/4], i[10, 1/4], i[11, 1/2], i[12, -1/4] ; c1 = i[1, -3/4], i[2, -1/2], i[3, -1/4], i[4, 0], i[5, 0], i[6, 0], i[8, 1/4], i[7, 1/2], i[9, 1/4], i[10, 1/4], i[11, 1/2], i[12, 3/4]; d1 = i[1, -3/4], i[2, -1/2], i[3, -1/4], i[4, 0], i[5, 0], i[6, 0], i[8, -3/4], i[7, -1/2], i[9, 1/4], i[10, 1/4], i[11, 1/2], i[12, -1/4]; d2 = i[1, -3/4], i[2, -1/2], i[3, -1/4], i[4, 0], i[5, 0], i[6, 0], i[8, 1/4], i[7, -1/2], i[9, 1/4], i[10, 1/4], i[11, 1/2], i[12, -1/4]; d3 = i[1, -3/4], i[2, -1/2], i[3, -1/4], i[4, 0], i[5, 0], i[6, 0], i[8, 1/4], i[7, 1/2], i[9, 1/4], i[10, 1/4], i[11, 1/2], i[12, -1/4] ; (* da-ad = (q-q^(-1))bc *) Expandf d1, a1+fd2, a1+fd3, a1- fa1, d1+f a1, d2+fa1, d3 -(q-q^(-1))* fb1, c1+fb2, c1 (* bc = cb *) Expandf b1, c1+fb2, c1 - fc1, b1+fc1, b2 (* ca = qac *) Expandf[c1, a1] -q* f[a1, c1] (* dc = acd *) Expandf d1, c1+fd2, c1+fd3, c1 -q* f c1, d1+fc1, d2 +fc1, d3 (* ba = qab *) Expandf b1, a1+fb2, a1 -q* f a1, b1+fa1, b2  (* db = qdb *) Expandf d1, b1+fd2, b1+fd3, b1+fd1, b2+f d2, b2 +fd3, b2 -q* f b1, d1+fb1, d2 +fb1, d3+fb2, d1+f b2, d2 +fb2, d3 ��0 ��0 ��0 ��0 ��0 ��0 thesis1.nb �� 5 �� Appendix B In the following Mathematica code, Section 3 veries Equation (4.48), and Section 4 veries Equation (4.50). Refer to Figures 4.15, 4.16, and 4.17. 178 �� (* SECTION 1: DEFINITIONS NEEDED FOR QUANTUM CALCULATIONS *) (* SEE APPENDIX A, SECTION 3, FOR ADDITIONAL NOTES *) (* number of coordinates *) n = 7; (* Poisson structure matrix *) (* W1=1,Z1=2,W2=3,Z2=4,W3=5,Z3=6,X=7, going around in order clockwise *) P = {{0, -1, 0, 0, 0, -2, 2}, {1, 0, 2, 0, 0, 0, -2}, {0, -2, 0, -1, 0, 0, 2}, {0, 0, 1, 0, 2, 0, -2}, {0, 0, 0, -2, 0, -1, 2}, {2, 0, 0, 0, 1, 0, -2}, {-2, 2, -2, 2, -2, 2, 0}}; i[z__] := {z}[[1]]+ ⅈ{z}[[2]]; a[w__] := tempw = Flatten[w]; temp = SumSum-(1/2)*Im tempw iImtempwjP Retempw i, Retempw j , j, i+1, Length[tempw], i, 1, Length[tempw]; q^temp; b[ w__] := temp= 0; tempw = Flatten[w]; Do  DoIfRetempw Length[tempw]-i1+1  ⩵n-i3+1, Do IfRetempw i2 ≥n-i3+1, , temp = temp+1*ImtempwLength[tempw]-i1+1 Imtempwi2PRetempwLength[tempw]-i1+1 , Retempwi2, i2, Length[tempw]-i1+2, Length[tempw] , i1, Length[tempw], i3, n; q^temp; c[w__]:= temp = ConstantArray[0, n]; tempw = Flatten[w]; Do  tempRetempwi = temp Re tempw i +Imtempwi, i, Length[tempw] ; (* W1=1,Z1=2,W2=3,Z2=4,W3=5,Z3=6,X=7 *) W1^temp[[1]]*Z1^temp[[2]]*W2^temp[[3]]* Z2^temp[[4]]*W3^temp[[5]] *Z3^temp[[6]]*X^temp[[7]] ; f[x_, y_] := a[x]*a[y]*b[{x, y}]*c[{x, y}]; (* g[m], like f[m1,m2] but for only one monomial m *) g[x_] := a[x]*b[x]*c[x]; �� (* SECTION 2: ENCODING THE THREE LEFT MATRICES (LOWER CASE LETTERS) AND THREE RIGHT MATRICES (CAPITAL LETTERS) *) (* W1=1,Z1=2,W2=3,Z2=4,W3=5,Z3=6,X=7 *) (* going from 1-edge to 2-edge *) a3 = i[1, 2/3], i[2, 1/3], i[3, 1/3], i[4, 2/3], i[7, 2/3] ; b31 = i[1, 2/3], i[2, 1/3], i[3, 1/3], i[4, -1/3], i[7, -1/3]; b32 = i[1, 2/3], i[2, 1/3], i[3, 1/3], i[4, -1/3], i[7, 2/3] ; c3 = i[1, 2/3], i[2, 1/3], i[3, -2/3], i[4, -1/3], i[7, -1/3]; e3 = i[1, -1/3], i[2, 1/3], i[3, 1/3], i[4, -1/3], i[7, -1/3]; f3 = i[1, -1/3], i[2, 1/3], i[3, -2/3], i[4, -1/3], i[7, -1/3] ; i3 = i[1, -1/3], i[2, -2/3], i[3, -2/3], i[4, -1/3], i[7, -1/3]; A3 = i[1, 1/3], i[2, 2/3], i[3, 2/3], i[4, 1/3], i[7, 1/3] ; D3 = i[1, 1/3], i[2, 2/3], i[3, -1/3], i[4, 1/3], i[7, 1/3]; E3 = i[1, 1/3], i[2, -1/3], i[3, -1/3], i[4, 1/3], i[7, 1/3] ; G3 = i[1, 1/3], i[2, 2/3], i[3, -1/3], i[4, -2/3], i[7, 1/3] ; H31 = i[1, 1/3], i[2, -1/3], i[3, -1/3], i[4, -2/3], i[7, -2/3]; H32 = i[1, 1/3], i[2, -1/3], i[3, -1/3], i[4, -2/3], i[7, 1/3] ; I3 = i[1, -2/3], i[2, -1/3], i[3, -1/3], i[4, -2/3], i[7, -2/3]; (* going from 2-edge to 3-edge *) a1 = i[3, 2/3], i[4, 1/3], i[5, 1/3], i[6, 2/3], i[7, 2/3] ; b11 = i[3, 2/3], i[4, 1/3], i[5, 1/3], i[6, -1/3], i[7, -1/3]; b12 = i[3, 2/3], i[4, 1/3], i[5, 1/3], i[6, -1/3], i[7, 2/3] ; c1 = i[3, 2/3], i[4, 1/3], i[5, -2/3], i[6, -1/3], i[7, -1/3]; e1 = i[3, -1/3], i[4, 1/3], i[5, 1/3], i[6, -1/3], i[7, -1/3]; f1 = i[3, -1/3], i[4, 1/3], i[5, -2/3], i[6, -1/3], i[7, -1/3] ; i1 = i[3, -1/3], i[4, -2/3], i[5, -2/3], i[6, -1/3], i[7, -1/3]; A1 = i[3, 1/3], i[4, 2/3], i[5, 2/3], i[6, 1/3], i[7, 1/3] ; D1 = i[3, 1/3], i[4, 2/3], i[5, -1/3], i[6, 1/3], i[7, 1/3]; E1 = i[3, 1/3], i[4, -1/3], i[5, -1/3], i[6, 1/3], i[7, 1/3] ; G1 = i[3, 1/3], i[4, 2/3], i[5, -1/3], i[6, -2/3], i[7, 1/3] ; H11 = i[3, 1/3], i[4, -1/3], i[5, -1/3], i[6, -2/3], i[7, -2/3]; H12 = i[3, 1/3], i[4, -1/3], i[5, -1/3], i[6, -2/3], i[7, 1/3] ; I1 = i[3, -2/3], i[4, -1/3], i[5, -1/3], i[6, -2/3], i[7, -2/3]; (* going from 3-edge to 1-edge *) a2 = i[5, 2/3], i[6, 1/3], i[1, 1/3], i[2, 2/3], i[7, 2/3] ; b21 = i[5, 2/3], i[6, 1/3], i[1, 1/3], i[2, -1/3], i[7, -1/3]; b22 = i[5, 2/3], i[6, 1/3], i[1, 1/3], i[2, -1/3], i[7, 2/3] ; c2 = i[5, 2/3], i[6, 1/3], i[1, -2/3], i[2, -1/3], i[7, -1/3]; e2 = i[5, -1/3], i[6, 1/3], i[1, 1/3], i[2, -1/3], i[7, -1/3]; f2 = i[5, -1/3], i[6, 1/3], i[1, -2/3], i[2, -1/3], i[7, -1/3] ; i2 = i[5, -1/3], i[6, -2/3], i[1, -2/3], i[2, -1/3], i[7, -1/3]; A2 = i[5, 1/3], i[6, 2/3], i[1, 2/3], i[2, 1/3], i[7, 1/3] ; D2 = i[5, 1/3], i[6, 2/3], i[1, -1/3], i[2, 1/3], i[7, 1/3]; E2 = i[5, 1/3], i[6, -1/3], i[1, -1/3], i[2, 1/3], i[7, 1/3] ; G2 = i[5, 1/3], i[6, 2/3], i[1, -1/3], i[2, -2/3], i[7, 1/3] ; H21 = i[5, 1/3], i[6, -1/3], i[1, -1/3], i[2, -2/3], i[7, -2/3]; H22 = i[5, 1/3], i[6, -1/3], i[1, -1/3], i[2, -2/3], i[7, 1/3] ; I2 = i[5, -2/3], i[6, -1/3], i[1, -1/3], i[2, -2/3], i[7, -2/3]; 2 �� thesis2.nb �� (* SECTION 3: CHECK OF MOVE (II) EXAMPLE *) (* 11 *) Expand (g[a3]) - q^(-1/3)*f[A2, G1] (* 12 *) Expand gb31 +gb32 - q^(-1/3)* f[A2, H11]+f[A2, H12]  (* 13 *) Expand (g[c3]) - q^(-1/3)*f[A2, I1] (* 21 *) Expand (0) - -q^(-4/3)*f[E2, D1] +q^(-1/3)*f[D2, G1] (* 22 *) Expand (g[e3]) - -q^(-4/3)*f[E2, E1] +q^(-1/3)* f[D2, H11]+f[D2, H12]  (* 23 *) Expand gf3 - q^(-1/3)*f[D2, I1] (* 31 *) Expand (0) - q^(-7/3)*f[I2, A1] -q^(-4/3)* f[H21, D1]+f[H22, D1] +q^(-1/3)*f[G2, G1] (* 32 *) Expand (0) - -q^(-4/3)* f[H21, E1]+f[H22, E1] +q^(-1/3)* f[G2, H11]+f[G2, H12]  (* 33 *) Expand gi3 - q^(-1/3)*f[G2, I1] ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��(* SECTION 4: CHECK OF MOVE (IV) EXAMPLE *) (* 11/11 *) Expand f[a3, A2] -q^(1/3)* q^(-1)*f[A2, a3] (* 12/11 *) Expand f[a3, D2] thesis2.nb �� 3 �� -q^(1/3)* f[D2, a3] (* 13/11 *) Expand f[a3, G2] -q^(1/3)* f[G2, a3] (* 12/12 *) Expand f[a3, E2] -q^(1/3)* f[E2, a3] (* 13/12 *) Expand f[a3, H21]+f[a3, H22] -q^(1/3)* f[H21, a3]+f[H22, a3] (* 13/13 *) Expand f[a3, I2] -q^(1/3)* f[I2, a3] (* 11/21 *) Expand fb31, A2+fb32, A2 -q^(1/3)* q^(-1)* fA2, b31 +fA2, b32 (* 12/21 *) Expand fb31, D2+fb32, D2 -q^(1/3)* fD2, b31 +fD2, b32 +(q^(-1)-q)*f[A2, e3] (* 13/21 *) Expand fb31, G2+fb32, G2 -q^(1/3)* fG2, b31 +fG2, b32  (* 21/21 *) Expand f[e3, A2] -q^(1/3)* f[A2, e3] (* 22/21 *) Expand f[e3, D2] -q^(1/3)* q^(-1)*f[D2, e3] (* 23/21 *) Expand f[e3, G2] -q^(1/3)* f[G2, e3] (* 12/22 *) Expand fb31, E2+fb32, E2 -q^(1/3)* fE2, b31 +fE2, b32  (* 13/22 *) Expand fb31, H21 +fb32, H21+fb31, H22+fb32, H22  -q^(1/3)* fH21, b31+fH22, b31+fH21, b32 +fH22, b32 (* 22/22 *) Expand f[e3, E2] -q^(1/3)* q^(-1)*f[E2, e3] (* 23/22 *) Expand f[e3, H21]+f[e3, H22] -q^(1/3)* f[H21, e3]+f[H22, e3] (* 13/23 *) Expand fb31, I2+fb32, I2 -q^(1/3)* fI2, b31 +fI2, b32  (* 23/23 *) Expand f[e3, I2] -q^(1/3)* f[I2, e3] (* 11/31 *) Expand f[c3, A2] -q^(1/3)* q^(-1)*f[A2, c3] (* 12/31 *) Expand f[c3, D2] -q^(1/3)* f[D2, c3]+(q^(-1)-q)*fA2, f3 (* 13/31 *) Expand  f[c3, G2]  4 �� thesis2.nb �� -q^(1/3)* f[G2, c3]+(q^(-1)-q)*fA2, i3 (* 21/31 *) Expand ff3, A2 -q^(1/3)* fA2, f3 (* 22/31 *) Expand ff3, D2 -q^(1/3)* q^(-1)*fD2, f3  (* 23/31 *) Expand ff3, G2 -q^(1/3)* fG2, f3+(q^(-1)-q)*fD2, i3 (* 31/31 *) Expand fi3, A2 -q^(1/3)* fA2, i3 (* 32/31 *) Expand fi3, D2 -q^(1/3)* fD2, i3 (* 33/31 *) Expand fi3, G2 -q^(1/3)* q^(-1)*fG2, i3  (* 12/32 *) Expand f[c3, E2] -q^(1/3)* f[E2, c3] (* 13/32 *) Expand f[c3, H21]+f[c3, H22] -q^(1/3)* f[H21, c3]+f[H22, c3] (* 22/32 *) Expand ff3, E2 -q^(1/3)* q^(-1)*fE2, f3  (* 23/32 *) Expand ff3, H21 +ff3, H22  -q^(1/3)* fH21, f3+fH22, f3+(q^(-1)-q)*fE2, i3 (* 32/32 *) Expand fi3, E2 -q^(1/3)* fE2, i3 (* 33/32 *) Expand fi3, H21 +fi3, H22  -q^(1/3)* q^(-1)* fH21, i3+fH22, i3  (* 13/33 *) Expand f[c3, I2] -q^(1/3)* f[I2, c3] (* 23/33 *) Expand ff3, I2 -q^(1/3)* fI2, f3 (* 33/33 *) Expand fi3, I2 -q^(1/3)* q^(-1)*fI2, i3  ��0 ��0 ��0 ��0 ��0 thesis2.nb �� 5 �� 0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 ��0 6 �� thesis2.nb ��

Abstract (if available)

Abstract The thesis is divided into two parts. The first part discusses the quantum world. We generalize Bonahon and Wong's SL₂(ℂ)-quantum trace map for finite-type surfaces to the case of SL₃(ℂ), and then we propose a definition for an SLₙ(ℂ)-version of the quantum trace map. Along the way, we establish a “building block” result that we expect to be an important ingredient in the construction of the SLₙ(ℂ)-quantum trace map. In particular, we relate Fock and Goncharov's geometric theory of SLₙ(ℂ) to the algebraic theory of the quantum group U_q(slₙ). ❧ The second part studies the classical setting. We show that the Sikora-Westbury linear basis for the algebra of functions on the SL₃(ℂ)-character variety, namely those algebraic functions that correspond to planar essential webs on the surface, can be indexed by an integer cone defined by Knutson-Tao diamond inequalities. The theory of Fock and Goncharov furnishes the correspondence between webs and their integer coordinates inside the cone.

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Asset Metadata

Creator Douglas, Daniel Charles (author)

Core Title Classical and quantum traces coming from SLₙ(ℂ) and U_q(slₙ)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 07/14/2020

Defense Date 05/14/2020

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cone,Fock and Goncharov,higher Teichmüller theory,knot,low dimensional topology,OAI-PMH Harvest,quantum groups,quantum torus,quantum trace map,quiver,representation theory,skein algebra,surfaces,trace,tropical,webs

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Bonahon, Francis (committee chair), Kalia, Rajiv (committee member), Lauda, Aaron (committee member)

Creator Email dcdougla@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-331638

Unique identifier UC11663220

Identifier etd-DouglasDan-8681.pdf (filename),usctheses-c89-331638 (legacy record id)

Legacy Identifier etd-DouglasDan-8681.pdf

Dmrecord 331638

Document Type Dissertation

Rights Douglas, Daniel Charles

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA