University of Southern California Dissertations and Theses

On sutured construction of Liouville sectors-with-corners

(USC Thesis Other)

On sutured construction of Liouville sectors-with-corners

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content On sutured construction of Liouville sectors-with-corners
by
Jian Zhou
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MATHEMATICS)
December 2023
Copyright 2024 Jian Zhou

In memory of my grandfather, Huiyuan Liu.
ii

Acknowledgments
First and foremost, I would like to express my deepest gratitude to my advisor Prof. Sheel Ganatra
whose guidance, patience, and support have been truly indispensable. His extensive knowledge,
enthusiasm, and dedication have always been an inspiration to me, and I could not have reached
this point without his consistent encouragement throughout the journey. For all of this and more,
I am eternally indebted to Prof. Ganatra.
I am sincerely grateful to my committee members, Prof. Aravind Asok, Prof. Kris Pardo, and
oral committee members Prof. Harold Williams, Prof. Joseph Helfer and Prof. Gene Bickers.
Their generosity in dedicating time and in providing insightful feedback is greatly appreciated.
I extend my heartfelt thanks to the faculty members in the USC Mathematics Department for
their invaluable teachings and inspirations. My sincere appreciation also goes to staff members
Amy Yung and Susan Sath for their timely assistance and consistent efforts in streamlining every
procedure and process. In addition, I am profoundly grateful to Prof. Ian Biringer, Prof. Elisenda
Grigsby, and Prof. Joshua Greene at Boston College, my alma mater, for introducing me to the
magnificent world of geometry.
I feel truly fortunate to have had the opportunity to meet and converse with my fellow graduate students in the topology group and my academic siblings Debtanu Sen, Alec Sahakian, Sanat
Mulay, Yasin Uskuplu, Siyang Liu and Haosen Wu. Special thank you to Ying Tan, Linfeng Li,
iii

Haiping Yang, Jiajun Luo, and Yusheng Wu for their camaraderie and shared memories, especially
during the challenging times of the pandemic.
Lastly, I want to thank my parents Bangyou Zhou and Lihua Liu for their commitment to
providing me with the best education over the past years and the freedom to pursue my academic
aspirations. I am forever grateful for their support, sacrifices and the opportunities they have
afforded me.
iv

Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Symplectic and contact structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Convex hypersurfaces in contact manifolds . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Liouville sectors and sectors-with-corners . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 3: Constructing examples of Liouville sectors-with-corners from cotangent bundles 24
3.1 Cotangent bundles of manifolds-with-boundaries . . . . . . . . . . . . . . . . . . 24
3.2 Cotangent bundles of manifolds-with-corners . . . . . . . . . . . . . . . . . . . . 26
3.3 Complements of Legendrian conormals . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 4: Completion of sutured sectorial domains . . . . . . . . . . . . . . . . . . . . . 32
Chapter 5: Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.1 Fukaya categories of Liouville manifolds & their sheaf-theoretic models . . . . . . 49
5.1.1 Partially wrapped Fukaya categories . . . . . . . . . . . . . . . . . . . . . 49
5.1.2 Sheaf theoretic models of W(T
∗M,Λ) . . . . . . . . . . . . . . . . . . . . 52
5.2 Decomposing a knot into braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 Categorical link closure in Fukaya categories . . . . . . . . . . . . . . . . . . . . . 56
5.3.1 Categorical link closure as a link invariant . . . . . . . . . . . . . . . . . . 56
5.3.2 Perverse sheaves on n-punctured disk and the GMV action . . . . . . . . . 57
5.4 Comparing closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
v

List of Tables
4.1 Defining functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Hamiltonian vector fields and characteristic foliations . . . . . . . . . . . . . . . . 42
vi

List of Figures
1.1 An example of Liouville sector and stop . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Equator as dividing set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 XI is outward transverse on each boundary face. . . . . . . . . . . . . . . . . . . 21
4.1 A schematic diagram for A′ = A ∩ ∂∂X0. . . . . . . . . . . . . . . . . . . . . . . 38
5.1 A decomposition of knot into braids. . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 A quiver associated to Perv(D2
, {p1 , . . . , pn}) . . . . . . . . . . . . . . . . . . . . 58
vii

Abstract
Liouville sectors-with-corners and sectorial decompositions provide a local-to-global approach
to understand the geometry of noncompact sympletic manifolds. We provide new examples and
generalize two constructions from Ganatra-Pardon-Shende [9] to the case of Liouville sectorswith-corners. In particular, we define the notion of sutured sectorial domains and show that their
Liouville completions give rise to sectors-with-corners. Applying this sutured construction to
conormal lifts of knots and braids, we also discuss its connection to the study of knot invariants.
viii

Chapter 1
Introduction
Liouville manifolds are an important class of noncompact manifolds in symplectic geometry, encompassing basic examples such as cotangent bundles of closed manifolds and complex affine varieties. Liouville sectors, introduced and extensively studied by Ganatra-Pardon-Shende in a series
of recent papers [9, 10, 8], extend the concept of Liouville manifolds to manifolds-with-boundaries
and manifolds-with-corners. From a purely topological view, manifolds-with-boundaries and/or
corners are the building blocks for cutting and gluing manifolds, a technique that enables localto-global decompositions of manifolds into simpler, more standard components. The goal of this
thesis is to introduce new tools for finding Liouville sectors with (sectorial) corners as defined in
[10] which are in particular manifolds-with-corners.
Work of Ganatra-Pardon-Shende [10] has established a “sectorial descent" result, which shows
that given a “sectorial covering" of a Liouville manifold X by subsectors Xi
, an important Floer
theoretic invariant of X, the wrapped Fukaya category, can be obtained local-to-globally from
Xi and the overlaps Xi1 ∩ · · · ∩ Xik
. The most general version of sectorial descent allows for the
possibility of the boundaries of covering Liouville sectors to intersect along “sectorial corners,”
and many decompositions inevitably involve such corners, as illustrated in Corollary 1.0.4 below.
1

In view of this, coupled with the broad expectation that decompositions involving the simplest
standard pieces necessarily involve corners – similar to the way simplicial complexes triangulate
smooth manifolds – it is an important question to produce examples of such Liouville sectorswith-(sectorial)-corners and decompositions. However, to date Liouville sectors-with-corners
remain geometrically difficult to construct in practice.
In this thesis, we provide new examples and generalize two constructions from [9] to the
case of Liouville sectors-with-corners. Cotangent bundles of manifolds-with-boundaries are primary examples of Liouville sectors [9, Example 2.7]. Our first contribution is the formal proof
of a folklore result [10, Example 12.6] in Section 3.2, which generalizes those examples to allow
corners:
Proposition 1.0.1 (Proposition 3.2.1 below). If Q is a manifold-with-corners, then its cotangent
bundle T
∗Q is a Liouville sector-with-corners.
Our second and main result concerns a cornered generalization of a method of constructing
Liouville sectors from “sutured Liouville domains" [9, Definition 2.14]. These domains correspond
closely to Sylvan’s notion of a “stop" [19, Definition 2.3]. Broadly, a stop is a pair (X, f) where X
is a Liouville manifold and f is a closed subset (for example a Legendrian) at ∞ of X. It is easy
to construct examples of stops, e.g., given a Lefschetz fibration W : X → C take f = W−1
(∞).
The process of completing a sutured Liouville domain associated to a Liouville stop (the case f
is a Liouville hypersurface or admits a Liouville hypersurface thickening) is roughly equivalent
to removing from X a neighborhood at ∞ of the corresponding stop, and the outcome of this
process is a Liouville sector by [9, Lemma 2.13].
2

[0, 1] −∞ • [0, 1] • ∞
C
Figure 1.1: Let T
∗
[0, 1] be the cotangent bundle of an interval, which is a Liouville sector. Then
T
∗
[0, 1] is equivalent to the Liouville manifold C stopped at two points.
We define a notion of sutured sectorial domains (related constructions are discussed in [1])
and prove in Chapter 4:
Theorem 1.0.2 (Theorem 4.0.18). Completing a sutured sectorial domain yields a Liouville sectorwith-corners.
As a first application, we give a new example of sectors-with-corners built from cotangent
bundles of manifolds-with-boundary by removing the conormal of a submanifold touching the
boundary:
Corollary 1.0.3 (Proposition 3.3.2). If K ⊆ Q is a submanifold-with-boundary, with ∂K ⊆ ∂Q
and K ∩∂Q transversely, then (D
∗Q, D
∗ΛK) is a sutured sectorial domain. In particular, completing
(D
∗Q, D
∗ΛK) thus specifies a Liouville sector-with-corners corresponding to T
∗Q \ nbhd(ΛK).
This provides new examples of sectorial decompositions for which the main result of [10,
Theorem 1.35] now applies:
Corollary 1.0.4. If in addition H ⊆ Q is a hyperplane transverse to K that divides
Q = Q0 ∪H Q1, K = K0 ∪K∩H K1, with Ki ⊆ Qi
.
Then the completions of (D
∗Qi
, D
∗ΛKi
) give a sectorial covering of the completion of (D
∗Q, D
∗ΛK).
3

Corollary 1.0.4 is already interesting for the case of knots K ⊆ S
3 or R
3
. By decomposing a
knot into braids through cutting with hyperplanes, the resulting sectorial decompositions provide
a local-to-global approach for understanding the equivalence [2] [8] between two knot invariants
[18] [6]. See Section 5 for more discussion.
Outline of thesis.
In Chapter 2, we review necessary symplectic and contact background, revisiting the construction
[9, Lemma 2.13] of Liouville sectors from sutured Liouville domains. Then, in Chapter 3, we
present motivating examples of Liouville sectors and sectors-with-corners arising from cotangent
bundles, including the proof of Proposition 1.0.1 detailed in Section 3.2.
We introduce the notion of sutured sectorial domain (Definition 4.0.4), and prove our main
result Theorem 4.0.18 in Chapter 4 via a local model near corners (Lemma 4.0.12), a computation
of Hamiltonian vector fields (Lemma 4.0.25), and a smoothing lemma (Lemma 4.0.41).
Finally, in Chapter 5, we briefly recall the descent formula [10, Theorem 1.35] for (partially)
wrapped Fukaya categories, and provide in Section 5.2 new examples of sectorial decompositions
resulting from decomposing a knot into braids via hyperplane cuts. We then describe the adaptation of an algebraic tool called “categorical link closure" [3] to such sectorial decompositions,
and suggest possible future questions at the end of the thesis.
4

Chapter 2
Preliminaries
In this chapter we recall the geometries on symplectic and contact manifolds, including a key
product decomposition (2.2.19) in contact manifold near dividing set (Definition 2.2.3). We also
recall the sutured construction [9, Lemma 2.13] of Liouville sectors in Proposition 2.3.17.
For in-depth reviews on symplectic and contact geometries, see textbooks [4][16] and [11].
For the theory of convexity in Section 2.2, see Giroux’s papers [13][14] for his work on convex surfaces in dimension three, and Honda-Huang [15] for the general theory. For definitions,
properties and various applications of Liouville sectors, see [9][10][8].
2.1 Symplectic and contact structures
A symplectic structure on a vector space V is a nondegenerate skew-symmetric bilinear map
Ω : V × V → R. Here, nondegenerate means that if there is an v ∈ V such that Ω(v, u) = 0 for
all u ∈ V , then v = 0; in other words, the associated map V → V
∗ defined by Ω(v, ·) is injective
and hence a bijection. It also follows that V must be of even dimensions, by the standard basis for
skew-symmetric bilinear maps. The orthogonal complement of a subspace W ⊆ V is the linear
5

subspace W⊥ = {v ∈ V | Ω(v, w) = 0, ∀w ∈ W}. One says that W is isotropic if W ⊆ W⊥, is
coisotropic if W⊥ ⊆ W, and is Lagrangian if W = W⊥.
A symplectic manifold (M, ω) is a 2n-dimensional manifold, together with a de Rham 2-form
ω that is both closed and nondegenerate, meaning that dω = 0 where d is the exterior derivative,
and at every point q ∈ M the map ωq : TqM × TqM → R induces on each tangent space TqM a
symplectic structure.
Example 2.1.1. (Euclidean spaces) R
2n admits a symplectic form ω =
Pn
i=1 dxi ∧ dyi
.
Example 2.1.2. (Cotangent bundles) T
∗Q with local coordinates (q1, . . . , qn) in the base Q and
coordinates (p1, . . . , pn) in fiber admits a canonical symplectic form ωcan =
Pn
i=1 dqi ∧ dpi
. Given
a submanifold K ⊆ Q, the conormal bundle
N
∗K = {(q, p) ∈ T
∗Q| q ∈ K, p(v) = 0 ∀ v ∈ TqK} (2.1.3)
is an n-dimensional Lagrangian submanifold of (T
∗Q, ωcan).
Remarkably, the nondegeneracy of ωq provides a linear isomorphism TqM
∼→ T
∗
q M, which
sends a vector v to the covector ω(v, ·), varying smoothly in p. Hence on symplectic manifolds
there is a bijective correspondence between (here ιX denotes the interior product)

Vector fields on M

↔

1-forms on M

X 7→ ιXω = ω(X, ·).
(2.1.4)
6

Given a smooth function H ∈ C
∞(M) on a symplectic manifold (M, ω), we often refer to
such H as a Hamiltonian function, or simply a Hamiltonian; its differential dH gives a 1-form.
Under above duality (2.1.4), there is a unique vector field XH satisfying
ω(XH, −) = dH. (2.1.5)
By Cartan’s formula
LX = dιX + ιXd, (2.1.6)
the flow of XH preserves the symplectic form ω, as well as level sets of H:
LXH ω = d(ιXH ω) + ιXH
(dω) = d(dH) = 0; (2.1.7)
LXH H = ιXH
(dH) = ιXH
(ιXH ω) = 0. (2.1.8)
We call this vector field XH the Hamiltonian vector field of the Hamiltonian H. Let f, g ∈
C
∞(M), the Poisson bracket is defined by
{f, g} = ω(Xf , Xg). (2.1.9)
Example 2.1.10. (Classical mechanics) For Hamiltonians H on cotangent bundles (T
∗Q, ωcan)
from Example 2.2, solving ω(XH, −) = dH yields Hamiltonian vector fields of the form
XH =
Xn
i=1
∂H
∂pi
∂qi −
Xn
i=1
∂H
∂qi
∂pi
. (2.1.11)
7

In other words, the coefficients in XH are given by q˙i = ∂H/∂pi and −p˙i = ∂H/∂qi
, also known as
Hamiltonian equations.
Definition 2.1.12. An exact symplectic manifold (M, θ) is a manifold equipped with an 1-form
θ such that ω = dθ is symplectic. Such θ is called a Liouville form.
As before, an Liouville 1-form θ dualizes to a vector field Z defined by ιZω = θ (note that θ
determines both ω and Z). However, unlike Hamiltonian vector fields (2.1.7) which preserve the
symplectic form,
LZω = d(ιZω) + ιZ(dω) = dθ = ω, (2.1.13)
i.e., flowing under Z exponentially expandsthe symplectic area. On a symplectic manifold (M, ω),
a vector field satisfying the equation (2.1.13) is known as an Liouville vector field.
Example 2.1.14. (Cotangent bundles are exact symplectic) Let Q = (q1, . . . , qn) be a closed manifold and T
∗Q its cotangent bundle with local coordinates (q1, . . . , qn, p1 = dq1, . . . , pn = dqn).
The tautological 1-form θ =
P
i
pidqi on T
∗Q gives rise to ωcan = −dθ =
P
i
dqi ∧ dpi
, and its
associated Liouville vector field, determined by ιZ(dθ) = θ, is Z =
P
i
pi∂pi
.
It is worth noting that a closed manifold (compact without boundary) cannot possibly admit
a Liouville vector field, i.e, is not exact symplectic. For otherwise by (2.1.13) the flow of Z would
exponentially expand the 2-form ω, its exterior powers, and the volume form ω
2n while a closed
manifold has a finite volume. In next section, we will study certain classes of exact symplectic
manifolds [cf. Definition 2.3.3, 2.3.26] that are compact with boundaries and corners.
Definition 2.1.15. A contact manifold (Y, ξ) is a (2n + 1)-dimensional manifold, together with
a 2n-dimensional distribution ξ locally defined by some 1-form α, such that dα|ξ is symplectic.
8

Throughout the thesis, we assume ξ is coorientable (i.e., the quotient line bundle T Y/ξ is trivial),
so there exists a global defining contact form α with ξ = ker(α).
Definition 2.1.16. An embedded n-dimensional submanifold Λ
n ⊆ (Y
2n+1, ξ) is Legendrian if
its tangent bundle lies entirely in the contact distribution, that is, TpΛ ⊆ ξp for all p ∈ Λ.
Recall that a distribution on Y just means a vector subbundle of the tangent bundle T Y . So
each fiber ξp ⊆ TpY is a codimension one linear subspace of TpY , which can be described as the
kernel of some nonzero covector αp ∈ T
∗
p Y ,
ξp = ker αp, αp : TpY → R. (2.1.17)
The data of hyperplane ξp, as a kernel, stays invariant under multiplying scalars to αp, and so a
defining contact 1-form αp is then unique up to a scalar. Upon fixing a local defining form αp
near p, we obtain following 1-dimensional complement to ξp
ker dαp = {v ∈ TpY | dαp(v, w) = 0, ∀w ∈ TpY } ⊆ TpY, (2.1.18)
since dαp|ξp
is nondegenerate and (ker dαp) ∩ ξp = 0. With a global contact form α, the Reeb
vector field Rα characterized by



dα(Rα, −) = 0
α(Rα) = 1
(2.1.19)
is everywhere transverse to the contact distribution ξ – it consists of the “α-unit length vectors"
in the complementary subspace (2.1.18).
9

Example 2.1.20. (unit cotangent bundle) The unit cotangent bundle ST∗Q = ∂(D
∗Q) of a Riemannian manifold Q admits a canonical contact structure ξ = ker(α), where α = θ|ST ∗Q is the
restriction of the tautological 1-form θ in Example 2.1.14.
A contact vector field on a contact manifold (Y, ξ) is a vector field whose flow ϕt preserves
the contact distribution ξ for all t. In the presence of a global contact form α, a vector field V is
contact if
LV α = gα, for some function g : Y → R. (2.1.21)
Example 2.1.22. (Reeb is contact) LRα α = dιRα α + ιRα dα = d(α(Rα)) + dα(Rα, −) = 0.
Using a contact form, one can define a dynamics on a contact manifold similar to the duality
between vector fields and smooth functions on a symplectic manifold. More precisely, there is a
bijective correspondence between smooth functions I : Y → R and contact vector fields VI on
Y , via following relations



I = −α(VI )
LVIα = dI(Rα)α,
(2.1.23)
Note VI is indeed contact by taking g = dI(Rα) in (2.1.21). Apply Cartan’s formula,
LVIα = ιVI
dα + dιVIα = ιVI
dα − dI, (2.1.24)
the contact vector field of I can be obtained by solving
dα(VI , −) = dI + dI(Rα)α. (2.1.25)
10

2.2 Convex hypersurfaces in contact manifolds
Definition 2.2.1. A codimension one submanifold H ⊆ (Y, ξ) in contact manifold is called a convex hypersurface if there is a contact vector field defined near and transverse to H.
Example 2.2.2. (Unit sphere is convex) The vector field V = x∂x + y∂y + 2z∂z is a contact vector
field (2.1.21) on the contact manifold (R
3
, ξ) with ξ given as the kernel of α = dz +xdy −ydx, since
LV α = 2α. Note V is transverse to S
2
, hence the unit sphere S
2 ⊆ (R
3
, ξ) is a convex hypersurface.
Definition 2.2.3. Let H ⊆ (Y, ξ) be a convex hypersurface, and fix a choice of transverse contact
vector field V . The dividing set ΓH ⊆ H is the set of points where V is tangent to ξ, i.e.,
ΓH := {y ∈ H| α(Vy) = 0}. (2.2.4)
Example 2.2.5. (Equator as dividing set) Continue on Example 2.2.2, the dividing set on H = S
2 ⊆
(R
3
, ξ) is given by the zeros of α(V ) = 2z, which is the equator circle {z = 0}.
Figure 2.1: Equator as dividing set
Lemma 2.2.6. ΓH is nonempty.
11

Proof. Without loss of generality, we may assume V is outward pointing along the convex hypersurface H. The negative flow of V determines a neighborhood of H in Y
H × Ru≥0 ,→ Y, and ∂u := −V, (2.2.7)
on which the contact structure is the kernel of α = Idu + β, for function I = α(V ) and some
1-form β on H (see [7]). Suppose I is never zero on H. Then one can divide α by I and rewrite
the contact form as α
′ = du + β
′ with dβ′ > 0. So
Z
H
(dα′
)
n =
Z
H
(dβ′
)
n > 0. (2.2.8)
But by Stokes’ Theorem (for closed submanifold H with ∂H = ∅),
Z
H
(dα′
)
n =
Z
∂H=∅
α
′ ∧ (dα′
)
n−1 = 0, (2.2.9)
since (dα′
)
n
is exact with a primitive α
′ ∧ (dα′
)
n−1
.
Lemma 2.2.10. ΓH is an embedded submanifold of H, and moreover a codimension two contact
submanifold of Y .
Proof. The dividing set ΓH can characterized as the zero set of I = −α(V ) : H → R, and
we show that 0 is a regular value. Indeed, observe that I (with the negative sign) is the contact
Hamiltonian (2.1.23) associated to V , and by (2.1.25),
dI = dα(V, −) − dI(Rα)α. (2.2.11)
12

At a point y ∈ I
−1
(0), by definition Vy ∈ ker(α) = ξ and since dα|ξ is symplectic, there exists
W ∈ ξ (hence α(W) = 0) with dα(Vy, W) ̸= 0. Then
dI(W) = dα(Vy, W) − dI(Rα)α(W) = dα(Vy, W) ̸= 0, (2.2.12)
which shows that dI is a submersion at every y ∈ I
−1
(0). The contact distribution on ΓH is then
given naturally by ξ ∩ TΓH.
Lemma 2.2.13. Let H ⊆ (Y
2n+1, ξ) be a convex hypersurface intersecting a Legendrian Λ
n ⊆ Y
transversely at ΛH := H ∩ Λ. Moreover, suppose there exists a convex vector field V for H that is
tangent to Λ at ΛH. Then ΛH ⊆ ΓH, and ΛH is a Legendrian in ΓH.
Proof. If V is tangent to Λ at y ∈ ΛH, then Vy ∈ ξ since Λ is Legendrian (i.e., TΛ ⊆ ξ). Therefore
y ∈ ΓH by Definition 2.2.3; this shows ΛH ⊆ ΓH.
We have TΛH ⊆ TΛ ⊆ ξ since ΛH ,→ Λ (as a submanifold), and TΛH ⊆ TΓH since
ΛH ⊆ ΓH (by last paragraph). Then TΛH ⊆ ξ ∩ TΓH, contained in the contact distribution on
ΓH. By transversality, ΛH has codimension in Y
2n+1 equal to codim(H) + codim(Λ) = n + 2
and so dim(ΛH) = n − 1; whereas ΓH is a (2n − 1)-dimensional contact manifold by Lemma
2.2.10. Hence ΛH ⊆ ΓH is Legendrian.
Definition 2.2.14. The characteristic foliation ΣH of a hypersurface H ⊆ (Y, ξ) is the onedimensional foliation of H defined by
ΣH = ker dα|ξ∩T H. (2.2.15)
13

(Caution: this definition should not be confused with (2.3.5), the characteristic foliation of a hypersurface in symplectic manifold.)
Suppose H is convex. Fix a contact vector field V and let ΓH ⊆ H be the dividing set [Definition 2.2.3]. Then
Lemma 2.2.16. ΣH is transverse to ΓH.
Proof. By Lemma 2.2.10, ΓH ⊆ H is an embedded level set {I = 0}, so ΣH ∩ TΓH = 0 iff
ΣH ̸⊆ ker dI. By (2.2.11)
dI(ΣH) = dα(V, ΣH) − dI(R)α(ΣH) = dα(V, ΣH), (2.2.17)
where the term dI(R)α(ΣH) vanishes because ΣH ⊆ ξ ∩ T H and ξ = ker α. At points in ΓH,
V is transverse to H and tangent to ξ (hence ξ ∩ T H transversely at ΓH), on which the contact
hyperplane decomposes as
ξ = (ξ ∩ T H) ⊕ V. (2.2.18)
By definition, the characteristic foliation ΣH is the kernel of dα on this first summand, i.e., dα(ξ∩
T H, ΣH) = 0. Therefore dα(V, ΣH) must be nonzero, for otherwise ker(dα|ξ) contains ΣH but
ker(dα|ξ) is trivial since dα|ξ nondegenerate.
Thus, at the dividing set ΓH ⊆ H of a convex hypersurface H ⊆ (Y, ξ), on one hand there
is (by Definition 2.2.1) a contact vector field V transverse to H (also tangent to ξ at ΓH); on the
other hand (by previous lemma) the characteristic foliation ΣH ⊆ T H is transverse to ΓH inside
H, and flowing ΓH along the direction of ΣH moreover preserves the distribution ξ ∩ T H (see
the paragraph after (2.23) in [9]).
14

Denote this flow coordinate as t and together with (2.2.7), the neighborhood H × Ru≤0 ,→ Y
determined by the negative flow of V , we get local coordinates on contact manifold near dividing
set ΓH, given by
Y = ΓH × R|t|≤1 × Ru≤0, (2.2.19)
on which the contact form splits as
α = µ + tdu, (2.2.20)
where µ is a contact form on ΓH for the contact structure ξ ∩ TΓH.
2.3 Liouville sectors and sectors-with-corners
Definition 2.3.1. A Liouville domain is a compact exact symplectic manifold (X0, θ) with boundary a smooth hypersurface ∂X0 such that the Liouville vector field Z is transverse and strictly outward pointing along ∂X0.
The existence of a Liouville vector field transverse to ∂X0 shows that ∂X0 ⊆ X0 is a contacttype hypersurface, on which α := θ|∂X0 defines a contact form [16, Proposition 3.5.31]. Typical
examples of Liouville domains include the unit ball(D
2n
,
P
i
xidyi) and the disk cotangent bundle
(D
∗Q,P
i
pidqi) whose Liouville vector fields generate the radial expansion Z =
1
2
P
i
(xi∂xi +
yi∂yi) and fiberwise radial dilation Z =
P
i
pi∂pi towards infinity.
The condition that Z is outward transverse to ∂X0 also implies that near ∂X0 flowing by Z
gives an exact symplectic collar neighborhood ∂X0×(ϵ, 1]r which can be identified with the (negative half of) sympletization of the contact manifold ∂X0 with Liouville form r(θ|∂X0
) = rα.
15

Hence given a Liouville domain (X0, θ), there is a standard way of “completing" it to a Liouville manifold X, by attaching to X0 an infinite cylindrical end (i.e., the positive half of the
symplectization of ∂X0)
X := X0 ∪ (∂X0 × [1,∞)r), (2.3.2)
and extending θ to r(θ|∂X0
) = rα on the attached end. For Liouville manifold X, there is a welldefined notion of boundary at infinity ∂∞X, which is contactormorphic to ∂X0 under the flow
of complete Liouville vector field Z = r∂r.
Similarly, a Liouville manifold-with-boundary is an exact symplectic manifold-withboundary for which a neighborhood of infinity is given by the positive half of the symplectization
of a contact manifold-with-boundary ∂∞X.
Definition 2.3.3. [9, Definition 2.4] A Liouville sector X is a Liouville manifold-with-boundary
for which there exists a defining function I : ∂X → R such that
(1) I is linear at infinity, meaning that ZI = I;
(2) Hamiltonian vector field XI is transverse and outward pointing along ∂X.
Remark 2.3.4. Recall that the characteristic foliation of a hypersurface H in a symplectic manifold (X, ω) is the 1-dimensional distribution over H defined by
Cp = ker(ωp|TpH) = {v ∈ TpH| ωp(v, w) = 0 ∀w ∈ TpH}, p ∈ H. (2.3.5)
16

Let C denote the characteristic foliation on ∂X. Then by (2.1.5) and nondegeneracy of ω,
XI is transverse to ∂X ⇐⇒ T X = T ∂X ⊕ XI (2.3.6)
⇐⇒ dI|C = ω(XI , C) ̸= 0. (2.3.7)
If C is oriented in a way such that ω(N, C) > 0 for any outward pointing vector N, then condition
(2) of Definition 2.3.3 can be equivalently stated as dI|C > 0.
The contact boundary at infinity ∂∞X of a sector X has an“actual" boundary ∂∂∞X = ∂∞X∩
∂X, and it turns out that
Proposition 2.3.8. ∂∂∞X is a convex hypersurface (Definition 2.2.1) in ∂∞X.
Proof. We show that the Hamiltonian vector field XI of a defining function I : ∂X → R descends
to a contact vector field near ∂X ∩ ∂∞X. Choose an inclusion i : ∂∞X ,→ X by choosing a
neighborhood at ∞. This determines a choice of contact form on ∂∞X by α = i
∗
θ. By Cartan’s
formula
LXIα = di∗
θ(XI ) + (di∗
θ)(XI , −) (2.3.9)
= i
∗
(d(dθ(Z, XI )) + dI) (2.3.10)
= i
∗
(d(−dI(Z)) + dI) (2.3.11)
= i
∗
(d(−I) + dI) (2.3.12)
= 0, (2.3.13)
17

where the equality in (2.3.10) used ιZ(dθ) = θ and the equality in (2.3.12) used the condition that
I is linear at infinity. Hence by (2.1.21), XI is contact with g = 0. Together with the outwardpointing condition of XI , this proves the convexity of ∂∂∞X.
Just as Liouville manifolds (noncompact without boundary) are obtained from completing Liouville domains, it has been shown in [9] that Liouville sectors (noncompact symplectic manifolds
with boundaries) arise in a similar vein from a completion operation:
Definition 2.3.14. [9, Definition 2.14] A sutured Liouville domain (X0, F, θ) is a Liouville
domain (X0, θ) together with a codimension one submanifold-with-boundary F ⊆ ∂X0 such that
(F, θ|F ) is itself a Liouville domain. We call F a Liouville hypersurface in ∂X0.
Example 2.3.15. Every smooth closed Legendrian Λ ⊆ ∂X0 determines a sutured Liouville domain,
for the embedding Λ ⊆ ∂X0 can be thickened to a Liouville hypersurface embedding of F = D
∗Λ,
the disk cotangent bundle of Λ.
Given a sutured Liouville domain, the Reeb vector field R := ∂s on ∂X0 is transverse to F
since dθ|F is symplectic. Therefore, the flow along R in ∂X0 determines a codimension zero
neighborhood
A := nbhd(F) ∼= F × R|s|≤1 ⊆ ∂X0, (2.3.16)
and it follows that
Proposition 2.3.17. [9, Lemma 2.13] The completion of X0 away from A ⊆ ∂X0, namely,
X := Xc0 \ (A
◦ × Rr≥0) (2.3.18)
is a Liouville sector (after implicitly smoothing ∂X, which as constructed has corners).
18

Here Xb0 denotes the Liouville completion of X0, A◦ denotes the interior of A, and r denotes the
flow coordinate on the symplectization of ∂X0.
Proof of Proposition 2.3.17. Let θ be the Liouville form on X0. Then α = θ|∂X0 gives a contact
form on ∂X0, and we have α = θ|F +ds on the R-invariant neighborhood A = F ×R|s|≤1. Also,
recall the symplectic form on the completion Xc0 is given by
ω = d(rα) = rdα + dr ∧ α. (2.3.19)
Our goal is to write down defining function I : ∂X → R (cf. Definition 2.3.3) on the boundary
∂X, which by construction (2.3.18) equals precisely the boundary of A × Rr≥1. Now since
∂A = ∂(F × R|s|≤1) (2.3.20)
= F × {±1}s ∪ ∂F × R|s|≤1, (2.3.21)
the boundary ∂X = ∂(A × Rr≥1) consists of following faces:
∂X = A × {1}r
| {z }
H1
∪ F × {±1}s × Rr≥1
| {z }
H2
∪ ∂F × R|s|≤1 × Rr≥1
| {z }
H3
, (2.3.22)
We claim that
I = rs (2.3.23)
is a defining function on all H1, H2, and H3. By linearity, the Hamiltonian vector field XI dual
to dI = rds + sdr is
XI = rXs + sXr. (2.3.24)
19

We directly compute Xr = −∂s = −Rα since
ω(Xr, −) = (rdα + dr ∧ α)(−∂s, −)
= −rdα(Rα, −) − (dr ∧ α)(Rα, −)
= −r · 0 − dr(Rα)α + α(Rα)dr
= dr
and Xs = ∂r −
1
r
ZF (where ZF is the Liouville vector field on F) since
ω(Xs, −) = (rdα + dr ∧ α)(∂r −
1
r
ZF , −)
= r(dθ|F )(−
1
r
ZF , −) + dr ∧ α(∂r −
1
r
ZF , −)
= −(dθ|F )(ZF , −) + dr(∂r)α +
1
r
α(ZF )dr
= −θ|F + α +
1
r
(θ|F + ds)(ZF )dr
= ds +
1
r
(dθ|F )(ZF , ZF )dr
= ds.
Therefore, the Hamiltonian vector field is
XI = r∂r − s∂s − ZF , (2.3.25)
which is transverse and outward pointing along boundary faces
• H1 = A × {1}r by the component r∂r, which generates the Liouville flow on Xb0.
20

• H2 = F × {±1}s × Rr≥1 by the component −s∂s = ∓∂s. Note the negative sign in −∂s
gives exactly the required outward-pointingness, as it points into F × R|s|<1 × Rr>1 ⊆
Xc0 \ X, the removed complement (and hence “outside") of X in Xb0.
• H3 = ∂F × R|s|≤1 × Rr≥1 by the component −ZF (i.e., the negative Liouville flow on F),
which is transverse to ∂F, pointing into F, and so outward pointing.
Figure 2.2: XI is outward transverse on each boundary face.
After smoothing H1, H2, and H3 into a single boundary, the Hamiltonian vector field XI
remains outward pointing. (In other words, to show that X is a Liouville sector, it is enough to
21

check the existence of a same defining function I before smoothing; see [9, Remark 2.12].) A
quick computation shows that I is also cylindrical at infinity:
dI(Z) = ω(XI , Z) = ω(r∂r − s∂s − Zλ, r∂r)
= (rdθ|F + dr ∧ θF + dr ∧ ds)(−s∂s, r∂r)
= dr ∧ ds(−s∂s, r∂r)
= rs = I.
We conclude that X is a Liouville sector witnessed by I = rs : ∂X → R.
Our main result Theorem 4.0.18 generalizes the above sutured construction and produces:
Definition 2.3.26. [10, Definition 12.2 and 12.4] A Liouville sector-with-corners X is a Liouville
manifold-with-corners whose boundary faces ∂
1X, . . . , ∂
nX satisfy
(1) The characteristic foliations Ci of ∂
iX are ω-orthogonal over intersections:
ω(Ci
, Cj ) = 0. (2.3.27)
(2) There exists defining functions Ii
: ∂
iX → R linear at infinity such that
dIi
|Ci > 0, dIi
|Cj = 0, {Ii
, Ij} = 0, for i ̸= j. (2.3.28)
In terms of their Hamiltonian vector fields Xi
, equivalently,
ω(Xi
, Ci) > 0, ω(Xi
, Cj ) = 0, ω(Xi
, Xj ) = 0, for i ̸= j. (2.3.29)
22

Remark 2.3.30. The definition of “Liouville manifold-with-corners" directly generalizes that of “Liouville manifold-with-boundary" (see paragraph before Definition 2.3.3): it is simply an exact symplectic manifold-with-corners on which a neighborhood at infinity is identified with the positive
symplectization of a contact manifold-with-corners.
However, a “Liouville sector-with-corners" is more subtle than a “Liouville sector with naïve corners" (for instance, the space (2.3.18) before smoothing the faces H1, H2, H3 (2.3.22) indeed has
corners Hi ∩ Hj ). Besides the additional compatibility conditions between different defining functions (2.3.29), it is shown in [10, Lemma 12.1] that the orthogonal condition (2.3.27) is equivalent to
the intrinsic property that all corners ∂
iX ∩ ∂
jX are coisotropic; both conditions turn out [10] to be
necessary for the study of Floer theory on reasonable Liouville manifolds-with-corners.
Example 2.3.31. We will prove in Proposition 3.2.1 the folklore result that cotangent bundle of
compact manifold-with-corners (with mutually transverse boundaries) is a Liouville sector-withcorners.
23

Chapter 3
Constructing examples of Liouville sectors-with-corners
from cotangent bundles
The cotangent bundle T
∗Q equipped with its tautological 1-form θ provides one of the most
basic examples of Liouville manifold. In this chapter, we construct Liouville sectors and Liouville
sectors-with-corners from T
∗Q, as a warm-up for our main theorem in Chapter 4. Despite of its
particularity, cotangent bundles give rise to useful sectorial decompositions (by the sectors-withcorners constructed in Section 3.3) which we will study in Section 5.2.
3.1 Cotangent bundles of manifolds-with-boundaries
Proposition 3.1.1. [9, Example 2.7] Let Q be a compact manifold-with-boundary. Then X = T
∗Q
is a Liouville sector.
Proof. Here we elaborate on the claim in [9, Example 2.7], which states “any vector field on Q lifts
to a Hamiltonian vector field on T
∗Q, and the lift of a vector field transverse to ∂Q thus certifies
that T
∗Q is a Liouville sector," by explicitly writing down the process of “lifting" a vector field.
24

Let (q1, . . . , qn) be local coordinates on Q so that ∂Q is described by {q1 = 0}, and let
(p1, . . . , pn) be the coordinates on cotangent fiber. The boundary of X = T
∗Q is just the cotangent lift of ∂Q in X and in local coordinates
∂X = T
∗Q|∂Q = (0, q2, . . . , qn, p1, . . . , pn). (3.1.2)
Given a fixed vector field V ∈ X(Q), it induces at every point q ∈ Q a linear functional
⟨−, V ⟩ : T
∗
q Q → R, defined by applying a covector p ∈ T
∗
q Q to the vector V (q) ∈ TqQ. For
simplicity, we may assume our coordinates are chosen so that
V = ∂q1. (3.1.3)
is outward transverse along ∂Q = {q1 = 0} (by adjusting the local chart near ∂Q, if necessary).
We show that the pairing
I := ⟨−, V ⟩ : T
∗Q|∂Q → R, (3.1.4)
is a defining function (cf. Definition 2.3.3). At points (q, p) ∈ T
∗Q|∂Q, we have a vector V (q) =
∂q1 constant for all q, and a covector p = p1dq1 + · · · + pndqn ∈ T
∗
q Q. Therefore, in local
coordinates
I(q, p) = p(∂q1
) = p1, (3.1.5)
25

since dqi(∂qj ) = δij . Recall from Example 2.1.14 the tautological 1-form θ =
P
i
pidqi on T
∗Q,
which defines, by ωcan = −dθ and ιZ(dθ) = θ,
ωcan =
P
i
dqi ∧ dpi
, Z =
P
i
pi∂pi
. (3.1.6)
Then I = ⟨−, V ⟩ = p1 is a defining function, since
(1) Linear at infinity. By (3.1.5) (3.1.6), clearly
ZI = dI(Z) = dp1
P
i
pi∂pi

= p1 = I. (3.1.7)
(2) XI outward transverse along ∂X. For a generic Hamiltonian H on (T
∗Q, ωcan), the formula in Example 2.1.10 gives its dual vector field
XH =
Xn
i=1
∂H
∂pi
∂qi −
Xn
i=1
∂H
∂qi
∂pi
. (3.1.8)
In our case XI = ∂q1, which agrees with V on the zero section (i.e., the “hinted" Hamiltonian lift of V from Q to T
∗Q), and hence outward pointing along ∂X.
3.2 Cotangent bundles of manifolds-with-corners
Proposition 3.2.1. [10, Example 12.6] Let Q be a compact manifold-with-corners whose boundary
faces ∂
1Q, . . . , ∂
kQ are mutually transverse. Then X = T
∗Q is a Liouville-sector-with-corners.
2

Proof. We aim to generalize the approach in the proof for Proposition 3.1.1, and show that the
Hamiltonian lifts of a suitable collection of vector fields on Q satisfy the conditions for sectorwith-corners (cf. Definition 2.3.26).
By the transverse assumption, we may find local charts Q = (q1, . . . , qn) in which ∂
iQ =
{qi = 0} near the intersections. Similar to the manifold-with-boundary case in Proposition 3.1.1,
each boundary face ∂
iX ⊆ X corresponds to the restriction of T
∗Q to ∂
iQ
∂
iX := T
∗Q|∂
iQ, (3.2.2)
and is cut out by the equation {qi = 0} in the local coordinates X = (q1, . . . , qn, p1, . . . , pn).
To construct defining functions Ii
: ∂
iX → R, first choose vector fields Vi
(defined near
intersections) such that Vi
is transverse to ∂
iQ and tangent to ∂
jQ for all i ̸= j; these V
′
i
s are
guaranteed to exist since at point q ∈ ∂
iQ ∩∂
jQ by transversely TqQ = T ∂i
qQ ⊕ T ∂j
qQ. We may
simply take Vi = ∂qi by a suitable chart and define
Ii = ⟨−, Vi⟩ : T
∗Q|∂
iQ → R. (3.2.3)
Then similar to (3.1.5) in local coordinates
Ii(q, p) = p(Vi(q)) = (p1dq1 + · · · + pndqn)(∂qi) = pi
. (3.2.4)
Following (2.1.11), their associated Hamiltonian vector fields
Xi = ∂qi
(3.2.5)
27

are lifts of Vi = ∂qi to T
∗Q (in the sense that π∗XIi = Vi for projection π : T
∗Q → Q).
(1) Linear at infinity. By (3.2.4)
ZIi = dIi(Z) = dpi(
P
i
pidpi) = pi = Ii
. (3.2.6)
(2) ω(Ci
, C j ) = 0 . i
, Cj ) = 0. Since ∂
iX = {qi = 0},
T ∂iX = ker(dqi) = R⟨∂q1, . . . , ∂qn, ∂p1, . . . , ∂pn⟩/R⟨∂qi⟩, (3.2.7)
on which the characteristic foliation Ci with respect to ωcan =
P
i
dqi ∧ dpi
is
Ci = ker(ωcan|T ∂iX) = R⟨∂pi⟩. (3.2.8)
Hence ω(Ci
, Cj ) = 0, either by directly plugging into ωcan, or by noticing that all Ci
’s
are contained in the tangent space of fiber, thus Ci
’s are tangent to the vertical cotangent
bundle, i.e., a Lagrangian of X.
(3) ω(Xi
, C i) > 0.
ωcan(Xi
, Ci) = ωcan(∂qi
, ∂pi) = (dqi ∧ dpi)(∂qi
, ∂pi) = 1. (3.2.9)
(4) ω(Xi
, C j ) = 0 .
ωcan(Xi
, Cj ) = ωcan(∂qi
, ∂pj ) = 0 ∀ i ̸= j. (3.2.10)
28

(5) {Ii
, I j} = 0 . i
, Ij} = 0.
{Ii
, Ij} = ωcan(Xi
, Xj ) = ωcan(∂qi
, ∂qj ) = 0. (3.2.11)
We conclude that our choice of defining functions I
′
i
s satisfies the compatibility conditions (2.3.27)
(2.3.29) and thus X is a Liouville sector-with-corners.
3.3 Complements of Legendrian conormals
We include the following examples of Liouville sectors-with-corners as a corollary of our main
Theorem 4.0.18. Given a submanifold K ⊆ Q, consider its unit conormal bundle (assume Q is
Riemannian)
ΛK := N
∗K ∩ ST∗Q, (3.3.1)
i.e., the intersection of its (Lagrangian) conormal bundle (2.1.3) with the unit cotangent bundle
of Q (a contact manifold by Example 2.1.20).
Hence every K ⊆ Q specifies a Legendrian submanifold Λk ⊆ ST∗Q, meanwhile recall
ST∗Q = ∂(D
∗Q) is the contact boundary of Liouville domain D
∗Q whose completion gives
T
∗Q. If K in addition intersects the boundary of Q, then ΛK is a Legendrian-with-boundary,
and Liouville completing away from a Liouville hypersurface thickening nbhd(ΛK) ⊆ ST∗Q
produces a Liouville manifold-with-corners (since the complement in ST∗Q is now a contact
manifold-with-corners). We show that
29

Proposition 3.3.2. Let Q be a compact manifold-with-boundary, and K ⊆ Q a submanifold-withboundary such that ∂K ⊆ ∂Q and K ∩ ∂Q transversely. Then
X := T
∗Q \ (nbhd(ΛK) × Rr≥1) (3.3.3)
is a Liouville sector-with-corners. For an explicit definition of nbhd(ΛK), see (4.0.19).
Remark 3.3.4. This in a certain sense generalizes Example 2.3.15, i.e., every closed Legendrian Λ
in the contact boundary of a Liouville domain determines a sector; Proposition 3.3.2 shows that in
the special case of cotangent bundle, every Legendrian lift ΛK (of a submanifold K with boundary
touching the boundary of the manifold nicely) determines a sector-with-corners.
Proof of Proposition 3.3.2. Let V be a vector field on Q tangent to K near ∂K and outward pointing along ∂Q. Here we use the language in the proof of the forthcoming Theorem 4.0.18 and
show that the pair (D
∗Q, D
∗ΛK) is a sutured sectorial domain (Definition 4.0.4). Since D
∗ΛK ⊂
∂c(D
∗Q) is already a sectorial domain, it suffices to check the existence of a Hamiltonian vector
field on D
∗Q tangent to D
∗Λ near boundary.
As in Proposition 3.1.1, the vector field V gives rise to a defining function I = ⟨V, −⟩ on
∂s(D
∗Q) = D
∗Q|∂Q and its Hamiltonian lift XI on D
∗Q coincides with V on the base coordinates
and vanishes in the fiber directions. Therefore XI is tangent to ΛK (since the base coordinates
on ΛK = S(N∗K)|K equal K) and outward transverse at ∂ΛK = S(N∗K)|∂K. Hence near the
boundary we have
ΛK
∼= ∂ΛK × Ru≤0, (3.3.5)
where u is the collar coordinate given by XI .
30

By Lemma 2.2.13 and 2.3.8, ∂ΛK is a Legendrian in the dividing set Γ ⊆ ∂∂(D
∗Q). Then
we can thicken up ∂ΛK to D
∗
(∂Λk) ⊆ Γ, multiply another cotangent thickening D
∗
[0, 1], and
identify coordinates
D
∗
(∂ΛK) × D
∗
[0, 1] = D
∗
(∂ΛK) × R|t|≤1 × Ru≤0
∼= D
∗ΛK, (3.3.6)
where the isomorphism with D
∗ΛK follows from (2.2.19) the local model Γ × R|t|≤1 × Ru≤0
∼=
∂c(D
∗Q) near dividing set, and the fact that D
∗
(∂ΛK) ⊆ Γ and D
∗ΛK ⊆ ∂c(D
∗Q) are both
transverse to the Reeb vector field. Hence using this (3.3.6) local model of D
∗ΛK, we see that XI
is indeed tangent to D
∗ΛK near boundary (this is implicit in the choice ∂u := XI ).
31

Chapter 4
Completion of sutured sectorial domains
In this section, we prove our main result (Theorem 4.0.18) on constructing sectors-with-corners.
In light of Proposition 2.3.17 where sutured Liouville domains (Definition 2.3.14) give rise to
Liouville sectors, we begin by generalizing the definition of a sutured Liouville domain. The first
step is to generalize the notion of a Liouville domain:
Definition 4.0.1. A sectorial domain (X0, θ) is a compact exact symplectic manifold-withcorners, with boundary consisting of two boundary faces ∂cX0 and ∂sX0 such that
(1) (∂cX0 is contact): the Liouville vector field Z, associated to the 1-form θ, is pointing strictly
outwards along ∂cX0, and is tangent to ∂sX0 near ∂sX0 ∩ ∂cX0.
(2) (∂sX0 is sectorial): there exists a defining function I : ∂sX0 → R with ZI = I near ∂cX0 ∩
∂sX0, and its Hamiltonian vector field XI is transverse and outward pointing along ∂sX0.
We call ∂∂X0 := ∂cX0 ∩ ∂sX0 the corner of X0. Note that similar to Proposition 2.3.17, the corner
∂∂X0 is a convex hypersurface in the contact boundary ∂cX0, witnessed by the Hamiltonian vector
field XI which is contact near ∂∂X0.
32

As the name suggests, a sectorial domain is a Liouville domain with an additional non-contact
boundary resembling the boundary ∂X of a Liouville sector (thus termed “sectorial" compare
with Definition 2.3.3). Therefore roughly speaking, a sectorial domain is a “Liouville domainwith-corners," or a “compact Liouville sector.” A similar notion has appeared in literature; compare with the “sectorial Liouville subdomain" in [20].
Remark 4.0.2. The Liouville completion Xb0 = X0 ∪ (∂cX0 × Rr≥1) of a sectorial domain X0 is,
unsurprisingly, a Liouville sector. Indeed, given a defining function I : ∂sX0 → R, one can take
the defining function on ∂Xb0 = ∂sX0 ∪ (∂∂X0 × Rr≥1) to be (a smoothing of) I itself on the first
summand, and the linear extension r · I|∂∂X0
on the second summand (the sector boundary ∂Xc0 is
smooth if Z is tangent to ∂sX0 near ∂∂X0).
Example 4.0.3. The disk cotangent bundle D
∗Q of a compact manifold-with-boundary Q is a sectorial domain, and its completion T
∗Q is a Liouville sector by Section 3.1.
Definition 4.0.4. A sectorial hypersurface in a sectorial domain (X0, θ) is a compact codimension one submanifold-with-corners F ⊆ ∂cX0, such that (F, θ|F ) is exact symplectic and F has two
boundary faces ∂cF and ∂sF satisfying
(1) ∂cF is contact.
(2) ∂sF = F ∩ ∂sX0, with the intersection being transverse in ∂cX0.
(3) There exists a defining function I : ∂sX0 → R whose Hamiltonian vector field XI is tangent
to F near ∂sF.
33

Proposition 4.0.5. A sectorial hypersurface (F, θ|F ) is itself a sectorial domain.
Proof. It suffices to show that I|∂sF is a defining function. The Hamiltonian vector field XF
I
of
I|F is the symplectic orthogonal projection of XI to the tangent space of F. But by assumption
XI ∈ T F in a neighborhood of ∂sF, on which the Hamiltonian vector field XF
I would then equal
to XI , and so outward transverse when restricted to ∂sF (since XI is outward transverse to ∂sX0
and ∂sF ⊆ ∂sX0).
Let i : ∂sF ,→ ∂sX0 be the inclusion map, then I|∂sF = i
∗
I is also linear since
di∗
I(ZF ) = ωF (X
F
I
, ZF ) = −θ|F (X
F
I
) = −θ(XI )|F = dI(Z)|F = I|F = i
∗
I, (4.0.6)
with the third equality follows because XF
I = XI by tangency. This shows ∂sF is sectorial, and
F is indeed a sectorial domain.
A sutured sectorial domain (X0, F, θ) is then a pair of a sectorial domain (X0, θ) and a
sectorial hypersurface F. From now on, fix a defining function I : ∂sX0 → R; this specifies a
contact vector field V = XI transverse to the convex hypersurface ∂∂X0.
Lemma 4.0.7. Let (X0, F) be a sutured sectorial domain, and let Γ ⊆ ∂∂X0 be the dividing set
[Definition 2.2.3] with respect to V . Then
P := ∂sF ∩ Γ (4.0.8)
is a Liouville domain.
34

Proof. We first show that ∂sF ∩ Γ is nonempty if ∂sF is nonempty. Let I : ∂sX0 → R with
ZI = I near ∂cX0. Recall from Lemma 2.2.10 that the dividing set is defined as
Γ := (−α(XI ))−1
(0) (4.0.9)
for α = θ|∂cX0
. In X0 near ∂cX0, I is linear which means
I = dI(Z) = ω(XI , Z) = −ω(Z, XI ) = −θ(XI ), (4.0.10)
it suffices to know I|∂sF has zeros. By Proposition 4.0.5, I|∂sF is a defining function for the
sectorial domain F, and its extension r · I|∂sF induces a defining function on the completion of
F (which is a Liouville sector). Hence I|∂sF has zeros, since r · I|∂sF has zeros by [9, Lemma 2.5]
and r > 0.
It remains to show that P = (I|∂sF )
−1
(0) is an exact symplectic manifold with convex boundary. By the outward pointingness of XI on ∂sF (as shown in the previous proposition), dI is
nonzero on the characteristic foliation of ∂sF, and hence I is in particular a submersion. By
completing F and taking Ib : ∂dsF → R, [9, Lemma 2.5] shows that there is a diffeomorphism
∂dsF = R × Pb and moreover Pb is a Liouville manifold by [9, Definition 2.10] (choosing a section
of the projection ∂dsF → Pb defines a Liouville form on Pb which is well-defined up to adding df
for compactly supported f). When I = 0, dI(Z) = I = 0 so Z is tangent to the Ib= 0 locus, and
Pb = Ib−1
(0) is the completion of P = I
−1
(0) with respect to ZP = Z. Hence, it follows that ∂P
is convex as desired.
35

Recall from Section 2.2 the product decomposition Y ∼= Γ × R|t|≤1 × Ru≤0 (2.2.19) of contact
manifold Y near dividing set Γ ⊆ Y (a contact submanifold). Also recall that t := α(V ) and
∂u := V. Take Y = ∂cX0, previous lemma shows that P ⊆ Γ is symplectic and the flow of Reeb
vector field ∂s, being transverse to P, determines a Rs-invariant neighborhood P × R|s|≤1 ⊆ Γ
and hence a further decomposition of (2.2.19):
Corollary 4.0.11. Let (X0, F) be a sutured sectorial domain. There are local coordinates
∂cX0
∼= P × R|s|≤1 × R|t|≤1 × Ru≤0, (4.0.12)
on which
α = λ + ds + tdu (4.0.13)
for some Liouville 1-form λ on P.
Before stating the main result Theorem 4.0.18 for sutured sectorial domains (X0, F), as a
geometric preliminary, we shift our focus to study how the neighborhood A = F × R|s|≤1 ⊆
∂cX0 interacts and runs into the corner of X0 (note the corner completes to the “original" sector
boundary of Xc0). In the sutured construction (Proposition 2.3.17) of Liouville sectors one removes
cone(A◦
) from the completion Xc0, where A◦
stands for the interior of A = F × R|s|≤1 for F
a Liouville domain; however, when adapted to our cornered setting (where F has a sectorial
boundary), this interior A◦ becomes a bit more delicate.
In particular, by Definition 4.0.4, the suture hypersurface F ⊆ ∂cX0 runs into the corner of
X0 at
F ∩ ∂∂X0 = F ∩ ∂sX0 = ∂sF, (4.0.14)
36

hence its thickening A = F × R|s|≤1 ⊆ ∂cX0 (where s is the Reeb coordinate) intersects the
corner of X0 at the locus
A
′
:= A ∩ ∂∂X0 = ∂sF × R|s|≤1. (4.0.15)
At the first glance, such A′
(being contained in ∂∂X0) is not interior in the usual sense (i.e., not
open in ∂cX0). However, when considered as a subset of A ⊆ ∂cX0, with its manifold-withboundary topology:
Proposition 4.0.16. A′ ⊆ A◦
.
Proof. This is essentially similar to the fact that the interval {0} × [−1, 1] is not open in R
2 but
is open in the half disk {x
2 + y
2 ≤ 1, x ≥ 0} ⊆ R
2
. In local model ∂∂X0 × Ru≤0
∼= ∂cX0, the
ϵ-neighborhood Nϵ = ∂∂X0 × (−ϵ, 0]u is open in ∂cX0. Since A = F × R|s|≤1 already has the
u-coordinate in F (as ∂u = XI is tangent to F by Definition 4.0.4), we have
A ∩ Nϵ = (A ∩ ∂∂X0) × (−ϵ, 0]u = A
′ × (−ϵ, 0]u. (4.0.17)
Hence the set A′ × (−ϵ, 0]u is open in A with respect to the subspace topology of A ⊆ ∂cX0, and
then A′ ⊆ A′ × (−ϵ, 0]u ⊆ A implies that A′
is in the interior of A.
Theorem 4.0.18. Completion along the complement of a sutured sectorial domain determines a
Liouville sector-with-corners. Explicitly, given a sutured sectorial domain (X0, F, θ), let A = F ×
R|s|≤1 ⊆ ∂cX0 be the thickening of F by the Reeb vector field. Then
X := Xb0 \

A
◦ × Rr≥1

(4.0.19)
3

Figure 4.1: A schematic diagram for A′ = A ∩ ∂∂X0.
is a Liouville sector-with-corners (after implicitly smoothing the boundary strata to have two faces
in a manner described below.)
Here, Xc0 denotes the Liouville completion of X0 on which r is the flow coordinate, and A◦ denotes
the interior of A, which contains the locus A′ defined in (4.0.15).
Before giving the proof, we prepare ourselves with a general computation of Hamiltonian
vector fields on Xb0. Let α = θ|∂cX0 denote the contact form on ∂cX0. In local coordinates near
P ⊆ ∂∂X0 in Corollary 4.0.11, we obtain decomposition (4.0.13)
α = λ + ds + tdu, (4.0.20)
hence the sympletic form ω on Xb0 takes the form
ω = d(rα) = dr ∧ α + rdα (4.0.21)
= dr ∧ (λ + ds + tdu) + rd(λ + ds + tdu) (4.0.22)
= dr ∧ (λ + ds + tdu) + rdλ + rdt ∧ du. (4.0.23)
38

With respect to such ω, Hamiltonian vector fields XI are determined by following equation
dI = ιXI
(ω) = ιXI
(dr ∧ α + rdλ + rdt ∧ du)
= dr(XI )α − α(XI )dr + r

dλ(XI , −) + dt(XI )du − du(XI )dt
= dr(XI )(λ + ds + tdu) − (λ + ds + tdu)(XI )dr+
r

dλ(XI , −) + dt(XI )du − du(XI )dt
.
Collecting terms, we have
dI = rdλ(XI , −) +
















dr(XI )
tdr(XI ) + rdt(XI )
−rdu(XI )
−(λ + ds + tdu)(XI )
dr(XI )
















T 















ds
du
dt
dr
λ
















. (4.0.24)
Lemma 4.0.25. For I = s and I = r, their Hamiltonian vector fields are, respectively,
Xs = ∂r −
t
r
∂t −
1
r
Zλ; (4.0.26)
Xr = −∂s, (4.0.27)
where Zλ is the Liouville vector field associated to λ.
Proof. Take I = s. Since dλ(XI , −) is a 1-form on P = Γ ∩ F, and P is transverse to ∂s by
Lemma 4.0.7, the first term rdλ(XI , −) on the right hand side of (4.0.24) cannot contribute to

matching with dI = ds on the left. Thus dr(XI ) = 1. Similarly, dλ(XI , −) has no du, dt, or dr
terms, so the remaining coefficients in (4.0.24) immediately reduce to



rdt(XI ) = −t;
rdu(XI ) = 0;
(λ + ds + tdu)(XI ) = 0;
rdλ(XI , −) = −λ,
which admits a solution
Xs = ∂r −
t
r
∂t −
1
r
Zλ.
The last two equations hold because dλ(Zλ, −) = λ, and by uniqueness Xs is indeed the Hamiltonian dual of I = s.
The case for I = r is much simpler: all coefficients on the right hand side of (4.0.24) vanish
except that for dr and hence −(λ + ds + tdu)(XI ) = −α(XI ) = 1. Then
Xr = −∂s,
coinciding with the negative Reeb vector field on ∂cX.
Proof of Theorem 4.0.18. We wish show that
X := Xb0 \

(A
◦ ∪ A
′
) × Rr≥1

(4.0.28)
4

is a Liouville sector-with-sectorial-corners, that is, its boundary faces form a sectorial collection in
the sense of Definition 2.3.26 after partially smoothing into two faces. We begin by enumerating
the boundary strata of X:
• X already has a sector boundary H1 = ∂sX0 before completion.
• completing the corner ∂∂X0 (outside A∩∂∂X0) yields another stratum H2. This is separate
from H1 since the Liouville vector field ∂r is not necessarily tangent to ∂sX at the corner.
• Since A = F × R|s|≤1, removing the cone of A◦
from Xb0 yields following boundary strata
∂(A × Rr≥1) =(
H3
z }| {
A × {1}r) ∪ (∂A × Rr≥1)
=H3 ∪

∂(F × R|s|≤1) × Rr≥1

=H3 ∪

F × {±1}s × Rr≥1
| {z }
H4

∪

∂F × R|s|≤1 × Rr≥1

,
where the last summand, since ∂F = ∂sF ∪ ∂cF, is the union

∂cF × R|s|≤1 × Rr≥1
| {z }
H5

∪

∂sF × R|s|≤1 × Rr≥1
| {z }
A′×Rr≥1

.
Notice this second component is exactly the cone of A′ ⊆ A◦
(see Proposition 4.0.16), and
by construction it is not a boundary face of X (since cone(A) is removed).
We proceed to writing down defining functions on above boundary strata. Since (X0, F) is
a sutured sectorial domain, by Definition 4.0.4 there already exists a defining function, denoted
I : ∂sX0 = H1 → R, which is linear near the corner of X0 and restricts to a defining function on
∂sF. We claim tha

1. I1 = r · I|∂∂X0 gives defining functions on the “sector-type" boundaries H1, H2.
2. I2 = rs from (2.3.23) gives defining functions on the“contact-type" boundaries H3, H4, H5,
boundary component defining function
H1 = ∂sX0 rI0
H2 = (∂∂X0 \ (A ∩ ∂∂X0)) × Rr≥1 rI0
H3 = F × R|s|≤1 × {1}r rs
H4 = F × {±1}s × Rr≥1 rs
H5 = ∂cF × R|s|≤1 × Rr≥1 rs
Table 4.1: Defining functions
For their Hamiltonian vector fields, recall
XI1 = XI =: ∂u (4.0.29)
defines the transverse coordinate u near the corner of X0, and for I2 = rs, since d(rs) = rds +
sdr, by Lemma 4.0.25,
XI2 = rXs + sXr (4.0.30)
= r∂r − t∂t − s∂s − Zλ. (4.0.31)
boundary component Hamiltonian v.f. char. foliation
H1 = ∂sX0 ∂u R⟨∂t⟩
H2 = (∂∂X0 \ (A ∩ ∂∂X0)) × Rr≥1 ∂u R⟨∂t⟩
H3 = F × R|s|≤1 × {1}r r∂r − t∂t − s∂s − Zλ R⟨∂s⟩
H4 = F × {±1}s × Rr≥1 r∂r − t∂t − s∂s − Zλ R⟨∂r⟩
H5 = ∂cF × R|s|≤1 × Rr≥1 r∂r − t∂t − s∂s − Zλ char(∂cF)
Table 4.2: Hamiltonian vector fields and characteristic foliations
It is then straight forward to check the compatibility conditions (2.3.27)(2.3.29):
42

(1) Linear at infinity. Recall Z = r∂r on (Xb0, d(rα)), then
dI1(Z) = ω(XI1
, Z) = ω(∂u, r∂r) (4.0.32)
= (dr ∧ tdu)(∂u, r∂r) (4.0.33)
= −rt = rI0 = I1 (4.0.34)
dI2(Z) = ω(XI2
, Z) = ω(r∂r − t∂t − s∂s − Zλ, r∂r) (4.0.35)
= (dr ∧ ds)(−s∂s, r∂r) + (dr ∧ λ)(−Zλ, r∂r) (4.0.36)
= rs + 0 = I2 (4.0.37)
(2) ω(Ci
, C j ) = 0 . i
, Cj ) = 0. Note rdt∧du is the only 2-form in ω that involves dt, and therefore ω(∂t
, ∂s)
= ω(∂t
, ∂r) = 0. For H5, the characteristic foliation on the contact boundary ∂cF is directly
by its Reeb vector field, which is transverse to ∂u and hence ω-orthogonal to ∂t as well.
(3) ω(Xi
, C i) > 0. i
, Ci) > 0. Clearly, XI1 = ∂u is transverse and outward pointing along H1 and H2
following from the definition of ∂u. For XI2
, the component
• r∂r = ∂r is transverse to H3 = F × R|s|≤1 × {1}r,
• −s∂s = ∓∂s is transverse to H4 = F × {±1}s × Rr≥1,
• Zλ is transverse to H5 = ∂cF × R|s|≤1 × Rr≥1,
and are all outward pointing, similar to the discussion in Proposition 2.3.17.
(4) ω(Xi
, C j ) = 0 . It suffices to verify that XI1
is tangent to H3, H4, H5, and XI2
is tangent to
H1, H2. In local coordinates,
43

• ∂u is tangent to F ∼= P × R|t|≤1 × Ru≤0 near ∂sF, and therefore tangent to H3, H4,
and H5.
• −t∂t − s∂s − Zλ is tangent to the corner ∂∂X0
∼= P × R|s|≤1 × R|t|≤1 at r = 1, and
so XI2
is tangent to H1 and H2. One can also directly compute ω(XI2
, ∂t) = 0, as
rdt ∧ du is the only 2-form in ω containing dt but XI2
is transverse to ∂u.
(5) {Xi
, X j} = 0 . i
, Xj} = 0.
{XI1
, XI2 } = ω(XI1
, XI2
) = ω(∂u, r∂r − t∂t − s∂s − Zλ) (4.0.38)
= (dr ∧ tdu)(∂u, r∂r) + (rdt ∧ du)(∂u, −t∂t) (4.0.39)
= −rt + rt = 0. (4.0.40)
Finally, by applying Lemma 4.0.41 below twice, smoothing together the “sector type" boundaries
H1 ∪ H2 and “contact type" boundaries H3 ∪ H4 ∪ H5 yields a Liouville sector-with-corners.
Lemma 4.0.41. Let X be a Liouville sector with boundary faces H1, . . . , Hm, and suppose for some
1 < k < m,
(1) Hk+1, · · · , Hm is a sectorial collection (i.e., their characteristic foliations are ω-orthogonal
(2.3.27) and ∃ transverse Hamiltonian vector fields Xj satisfying (2.3.29) for k + 1 ≤ j ≤ m.)
(2) Further, ∃ a function I defined near H = H1 ∪ · · · ∪ Hk satisfying
ω(XI , Cj ) = ω(Xj
, Ci) = ω(Ci
, Cj ) = ω(XI , Xj ) = 0, for all i, j. (4.0.42)
44

Note that H1, . . . , Hk are not necessarily ω-orthogonal. Then after smoothing H′
i
s into a single
boundary
He = (H1 ∪ · · · ∪ Hk)
sm, (4.0.43)
X becomes a Liouville sector-with-sectorial-corners.
Proof. Here we follow [9, Remark 2.12] where the authors concern the smoothing of a Liouville
manifolds-with-corners into a Liouville manifold-with-boundary. In particular, a same defining
function I : Hi → R (separately with dI(Ci) > 0) before smoothing the corners (resulted from
intersecting Hi
’s) provides a defining function on the smoothed boundary; this is because that
the characteristic foliation at a point of the smoothed boundary He lies in the convex hull of the
characteristic foliations of nearby cornered boundaries Hi
:
CHe = s1Ci + · · · skCk, for si > 0,
Psi = 1. (4.0.44)
This takes care of ω(XI , CHe ) = dI(CHe ) > 0 since the positivity of dI is preserved under convex
combinations. Since I and XI stay the same after smoothing, ω(XI , Cj ) = ω(XI , Xj ) = 0 from
(4.0.42) remain unchanged. Hence for the sector-with-corners setting (2.3.27) (2.3.29), it remains
to only check that
ω(CHe , Cj ) = 0 and ω(CHe , Xj ) = 0. (4.0.45)
By (4.0.44) and linearity, (4.0.45) reduces to check ω(Ci
, Cj ) = ω(Ci
, Xj ) = 0 (4.0.42) on the
characteristic foliations Ci before smoothing.
45

Chapter 5
Applications
In this chapter we discuss the application of Liouville-sectors-with-corners and sutured sectorial domains to the study of knots K ⊆ R
3
. Recently, the partially wrapped Fukaya category
W(T
∗R
3
,ΛK), a Floer-theoretic invariant of Liouville sectors, has been used to prove the equivalence between two knot invariants which both arise from the Legendrian torus ΛK of a knot.
Recall that the “local data" of a knot is captured by braids, i.e., collections of twisted strands, and
gluing braids together recovers the original knot. Our main result shows that we can construct
a sector-with-corners that is equivalent – in the sense of having the same wrapped Fukaya category – to (T
∗R
3
,Λβi
) for each βi a braid, generalizing the construction of a sector equivalent
to (T
∗R
3
,ΛK) given in [10, Corollary 3.9] and reviewed later in Proposition 5.1.1. Moreover,
we show a decomposition of K into braids, as a result of slicing R
3 by hyperplanes, induces a
sectorial decomposition of (a sector equivalent to) (T
∗R
3
,ΛK) into (sectors-with-corners equivalent to) (T
∗R
3
,Λβi
). Hence, the descent result of [10, Theorem 1.35] for sectors-with-corners
then shows that “gluing" together (i.e., homotopy colimit of) the Fukaya categories W(T
∗R
3
,Λβi
)
recovers W(T
∗R
3
,ΛK).
46

In more details, the conormal bundle of a knot K ⊆ R
3 gives rise to a non-compact Lagrangian
submanifold LK
∼= S
1 × R2
in the cotangent bundle T
∗R
3 with its canonical symplectic structure, and by taking the “boundary at infinity" of LK, or equivalently intersecting with the unit
cotangent bundle (using a Riemannian metric), one obtains a Legendrian torus ΛK in the contact
manifold ∂∞(T
∗R
3
) ∼= S
∗R
3
. The Legendrian contact homology of the pair (S
∗R
3
,ΛK), also
known as the knot contact homology of K, is a topological knot invariant, strong enough to detect the unknot [17]. Recently, it has in fact been shown that the conormal torus ΛK “completely
knows" the original knot K: Shende studied the geometry of ΛK through the category of sheaves
with microsupport in ΛK, and proved that the Legendrian isotopy type of ΛK itself suffices to distinguish the smooth isotopy type of K [18]; and Ekholm-Ng-Shende enhanced the knot contact
homology into a complete knot invariant as well [6]. The relevant tools those authors used, the
category of sheaves with microsupport in ΛK and the knot contact homology, were subsequently
shown to be isomorphic [2] [8] through the intermediary of the partially wrapped Fukaya category (identified in the next paragraph). Our main result provides a possible method to refine the
comparison between approaches.
The partially wrapped Fukaya category is a symplectic invariant of certain pairs (M, f) of a
non-compact symplectic manifold M and a “stop at ∞" f ⊆ ∂∞M; equivalently it is an invariant
of the underlying Liouville sectors (which exists when f admits a Liouville thickening or “ribbon")
obtained by removing from M a neighborhood of f at ∞.
One may in particular consider W(T
∗R
3
,ΛK), the partially wrapped Fukaya category of T
∗R
3
“stopped" at the conormal torus ΛK. The equality of the knot contact homology of K and the
category of sheaves with singular support in ΛK, originally a conjecture of [19][5],is now a known
corollary of the results of [2] and [8] as follows: first, by appealing to the result in [2], one can
47

replace the knot contact homology with the equivalent partially wrapped Fukaya cohomology
in W(T
∗R
3
,ΛK), and then one can compare the latter category with its sheaf version using [8].
The main question we wish to explore is:
Question 1. Enhance the known approaches to comparing theories associated to knots, by showing
that the equivalence is compatible with local-to-global structures.
A knot can be realized by “closing" a braid (gluing its ends to a trivial braid), and a framework
for “local-to-globalizing" knot invariants through braids was introduced by Berest-EshmatovYeung in [3]. The authors developed a formalism, which named “categorical link closure"(5.3.2),
for extracting knot invariants from a given system of braid groups actions, one for each number of
strands, on a category. Roughly speaking, this procedure emulates (categorically) the topological
“gluing" involved in taking a braid closure; moreover it produces a knot invariant.
In the end of this chapter, we will outline an approach to Question 1: the basic idea is to
show that categorical link closure can be applied to “local" partially wrapped Fukaya categories
(of sectors-with-corners coming from braids) in a way that, by comparing to the local-to-global
homotopy pushout formula in [10], realizes the “global" category W(T
∗R
3
,ΛK) as the categorical
closure of a certain braid group action σ. By showing that such σ coincides with a suitable
combinatorical action (5.3.7) constructed on a corresponding (under [8]) “local sheaf category" in
[3], the equivalence in [8] would be local-to-globalized.
48

5.1 Fukaya categories of Liouville manifolds & their sheaftheoretic models
5.1.1 Partially wrapped Fukaya categories
Let M be a symplectic manifold. The ordinary Fukaya category of M has objects that are suitable
compact Lagrangian submanifolds of M, with morphism spaces given by the Lagrangian Floer
complexes. If M is a (possibly non-compact, properly embedded) Lagrangian L ⊆ M is called
conical if L is invariant under the flow of Z outside of a compact set. A choice of closed subset
f ⊆ ∂∞M is called a stop.
If M is a Liouville sector and f ⊆ ∂∞M a stop, the objects of the partially wrapped Fukaya
category W(M, f) are exact conical Lagrangian submanifolds of M disjoint from f, and morphisms are given by “wrapping" Lagrangians (under the geodesic flow of an Hamiltonian) in the
complement of f and ∂M.
In a historic view, partially wrapped Fukaya category generalizes other versions of noncompact Fukaya categories as special cases: the fully wrapped Fukaya category is the case f = ∅;
the Fukaya-Seidel category of a Lefschetz fibration W : M → C is the partially wrapped Fukaya
category for f = W−1
(−∞); and the infinitesimal Fukaya categories can be realized as a full
subcategory of W(M,Λ), by perturbing Lagrangians asymptotic to Λ at infinity away from the
stop f = Λ under the negative Reeb flow.
Let (X0, F0) be a sutured Liouville domain, and let X be its resulting Liouville sector by
Proposition 2.3.17; equivalently, one is given the data of a stopped Liouville manifold (Xb0, F0)
where Xb0 is the Liouville completion of X0 with stop f = F0. The authors in [10] showed that
49

Proposition 5.1.1 (Equivalent presentations of the same Fukaya category). [10, Corollary 3.9]
The inclusion of Liouville sectors induces a quasi-equivalence
W(X)
∼−→ W(Xb0, F0). (5.1.2)
Methods analogous to [10, Corollary 3.9] should establish:
Corollary 5.1.3 (Generalization of Proposition 5.1.1 to Liouville sector-with-corners). Replace
(X0, F) with a sutured sectorial domain (Definition 4.0.4), and replace X with the underlying Liouville sector-with-corners (by Theorem 4.0.18). Then the inclusion of Liouville sectors-with-corners
induces a quasi-equivalence
W(X)
∼−→ W(Xb0, F0). (5.1.4)
Among the many computation and generation results developed in [10], one of the main
results established was the descent formula, showing that partially wrapped Fukaya categories
have nice Van-Kampen type local-to-global properties:
Theorem 5.1.5 (Homotopy colimit formula). [10, Theorem 1.35] For any (Weinstein) sectorial
covering X1,. . . ,Xn of a Liouville sector X, where Xi
’s are Liouville sectors possibly with corners,
the induced functor
hocolim∅̸=I⊆{1,...,n}W
\
i∈I
Xi

∼−→ W(X) (5.1.6)
is a pre-triangulated equivalence. Here, hocolimit denotes the “homotopy colimit" of a diagram of
A∞ categories (see e.g. [10, Appendix A.4]).
A special case (also observed by Sylvan [19]) of the descent formula for a two element cover
is known as the homotopy pushout formula, and was stated using the language of stops:
50

Theorem 5.1.7 (Homotopy pushout formula). [10, Theorem 1.28] Let X, X1, X2 be Liouville
sectors such that X = X1 ∪ X2, and X1 ∩ X2 is a hypersurface inside X disjoint from ∂X. Write a
neighborhood of X1 ∩ X2 as F × T
∗
[0, 1] for some F ⊆ ∂∞X.
X1 X1 ∩ X2 X2
•
F
•
r1
•
r2
•
r2
Pick any stop r disjoint from ∂∞(X1 ∩ X2), and let ri
:= r ∩ (∂∞Xi)
◦
. Then following diagram of
A∞-categories
W(F)
W(X1,r1)
W(X2,r2)
W(X,r)
induces a Morita equivalence
hocolim
W(X2,r2) ← W(F) → W(X1,r1)
∼=
,→ W(X,r). (5.1.8)
(when X is Weinstein up to deformation, and the stops ri are mostly Legendrian, c.f. [10, Definition
1.6].)
Remark 5.1.9. Notice that in Theorem 5.1.7, the hypersurface F is “unstopped" in the sense that
∂∞(X1 ∩ X2) is required to be disjoint from the stop r ⊆ ∂∞X. This implies that F is a Liouville
5

manifold, as opposed to a Liouville sector in the more general homotopy colimit formula in Theorem
5.1.5.
Our main result shows that Theorem 5.1.7 remains true if ∂∞(X1∩X2) intersects the original
stop r in the following controlled way: for i = 1, 2, let X0,i be sectorial domain whose Liouville
completion is the sector Xi
; by a slight abuse of notation, let ri denote its contactomorphic image in the contact boundary of X0,i, and D
∗
ri denote the disk cotangent bundle of (the smooth
Legendrian locus of) ri
.
One would then require the pairs (X0,i, D
∗
ri) to be sutured sectorial domains, which reduce
to checking the existence of Hamiltonian vector fields tangent to D
∗
ri near boundaries. Indeed,
Theorem 4.0.18 then provides a model for the underlying sectors-with-corners for (X0,i, D
∗
ri),
and so one can use Corollary 5.1.3 and appeal the colimit formula. We will soon see an application
of this enhanced version of pushout formula in Section 5.2.
5.1.2 Sheaf theoretic models of W(T
∗M,Λ)
The main theorem [8, Theorem 1.1] we wish to appeal is the equivalence between the partially
wrapped category of T
∗M stopped at Λ, and the category of compact objects in the unbounded
derived category of sheaves on M with microsupport inside Λ, as stated below in Theorem 5.1.10.
First recall that given a stratification S of M, a sheaf is called S-constructible if it is locally
constant on each stratum, and we write ShS(M) for the category of unbounded S-constructible
sheaves. On the other hand, the microsupport of a sheaf is a conical subset in T
∗M, (roughtly
speaking are obstructions to propagating local sections of the sheaf) and for a fixed subset Λ in
S
∗M, we write ShΛ(M) for the category of unbounded constructible sheaves with microsupport
52

at infinity contained in Λ. If S is a Whitney stratification, then ShS(M) = ShN∗∞S(M), i.e. having
microsupport contained in N∗
∞S is equivalent to being S-constructible [8, Proposition 4.8]. It has
been established that:
Theorem 5.1.10. [8, Theorem 1.1] Let M be a real analytic manifold, and let Λ ⊆ S
∗M be a
subanalytic closed isotropic subset. There is a canonical equivalence of categories
Perf W(T
∗M,Λ)op = ShΛ(M)
c
(5.1.11)
which carries the linking disk at any smooth Legendrian point p ∈ Λ to a co-representative of the
microstalk functor at p, and carries the cotangent fiber at a point p ∈ M not in the image of Λ to a
co-representative of the stalk functor.
Here, Perf W(T
∗M,Λ) denotes the idempotent-completed pre-triangulated closure of the
partially wrapped Fukaya category of T
∗M stopped at Λ, and ShΛ(M)
c
is the category of compact
objects in ShΛ(M). Note an object X ∈ C is called compact if homC(X, −) commutes with
arbitrary direct sums.
Remark 5.1.12. One should have noticed that the category ShΛ(M), as defined previously, is not
the usual constructible sheaf category with finite cohomological stalks. Instead infinite rank sheaves
are allowed in ShΛ(M), and an compact object of ShΛ(M) may not be a finite-dimensional object.
For example [8, Corollary 6.1], let M be a smooth manifold and let stop Λ = ∅, then Sh∅(M)
c ∼=
Perf C−∗(ΩM), the perfect modules over chains on the based loop space of M.
Roughly speaking, allowing unbounded sheaves is meant to remedy the fact that morphism spaces
in wrapped Fukaya categories are oftentimes infinite-dimensional, even in the partially wrapped
53

setting. Consider for example the cotangent bundle T
∗S
1
stopped at one point. Then wrapping at
infinity is unstopped on the other end of the cylinder.
5.2 Decomposing a knot into braids
Let β ∈ Bn be an n-strand braid, and consider its link closure βˆ in unit three disk D3
. Let
U = {x ≤ ϵ} and V = {x ≥ −ϵ} be a cover of D3
, and denote
βleft := βˆ ∩ U, βright := βˆ ∩ V.
Figure 5.1: A decomposition of knot into braids.
U and V are compact manifolds with boundary, and so their cotangent bundles T
∗U and T
∗V
are Liouville sectors covering the Liouville sector T
∗D3
. Choose the stop r ⊆ ∂∞T
∗D3
to be the
conormal torus Λβˆ = N∗βˆ ∩ S
∗D3
.
Now, we are in exact need of a more refined homotopy pushout formula (see Remark 5.1.9) for
this cover T
∗U ∪ T
∗V = T
∗D3
, because the stop Λβ intersects with the overlap T
∗U ∩ T
∗V =
T
∗
(U ∩ V ) at infinity. However, note the resulting sub-stops ri are Legendrian lifts of braids
(submanifolds in U, V transverse to the boundary hyperplanes), which fall into the special case
54

of Proposition 3.3.2. Then by the paragraphs after Remark 5.1.9, the following diagram of partially
wrapped Fukaya categories
W

T
∗D3
,
`2n
i=1 N∗
∞Ii

W

T
∗D3
, N∗
∞βleft
W

T
∗D3
, N∗
∞βright
W

T
∗D3
,Λβˆ

gives a Morita equivalence
hocolim
W

T
∗D
3
, N∗
∞βleft) ← W
T
∗D
3
, ⨿
2n
i=1N
∗
∞Ii

→ W
T
∗D
3
, N∗
∞βright

∼=
,→ W
T
∗D
3
,Λβˆ

.
(5.2.1)
Since both βleft and βright are isomorphic to a disjoint union of n strands, the maps inside
above colimit can be viewed as:
W

T
∗D
3
, ⨿
n
i=1N
∗
∞Ii

← W
T
∗D
3
, ⨿
2n
i=1N
∗
∞Ii

→ W

T
∗D
3
, ⨿
n
i=1N
∗
∞Ii

(5.2.2)
In [10], it is also shown that there exists a fully faithful Künneth stablization functor
W(X, f) ,→ W(X, f) ⊗ W(C, {±∞})
:=W

X × C, (cX × {±∞}) ∪ (f × R)

,
In our case, this again gives Morita equivalences
W

T
∗D
2
, N∗
∞{p1, . . . , pn})
∼=
,→ W
T
∗D
3
, ⨿N
∗
∞Ii

which allow us to “dimensionally reduce" the data of (5.2.2) into
W

T
∗D
2
, N∗
∞{p1, . . . , pn}

← W
T
∗D
2
, N∗
∞{p1, . . . , p2n}

→ W
T
∗D
2
, N∗
∞{p1, . . . , pn}

(5.2.4)
5.3 Categorical link closure in Fukaya categories
5.3.1 Categorical link closure as a link invariant
The work of Berest-Eshmatov-Yeung [3] provided a new algebraic construction of the knot contact homology for L ⊆ R
3
, via defining, first in abstract setting, a link invariant called the categorical link closure L(A, σ) which takes as input an arbitrary braid groups actions generated by
Yang-Baxtor operators (A, σ). They showed that L(A, σ) recovers a combinatorical version of
knot contact homology when the input of braid group action is a certain Gelfand-MacPhersonVilonen (GMV) action [3, Section 5].
More precisely, let C be a category with finite colimits, and let A(n) denote the n-fold coproduct
of an object A ∈ C. Given a family of local and homogeneous braid group actions Bn → Aut A(n)
,
i.e., those generated by a single morphism σ : A ⨿ A → A ⨿ A representing the action of each
standard generator σi ∈ Bn (such pair of A and σ is called a Yang-Baxter operator), then
Definition 5.3.1. The categorical link closure [3, Definition 2.5] is the pushout in C:
L(A, σ)[β] := coeq
A
(n)
β
⇒
id
A
(n)

= colim
A
(n)
(β,id)
←−− A
(n) ⨿ A
(n)
(id,id)
−−→ A
(n)

, (5.3.2)

and is stable under the Markov moves. Hence L(A, σ)[β] is an invariant of the link closure βˆ of
β ∈ Bn [3, Theorem 2.7].
Example 5.3.3. A prototypical example of categorical link closure is that the classic link invariant
π1(R
3 \ βˆ) can be identified with L(F1, σ)[β], where F1 is the free group of rank 1 and the braid
acts on Fn via the Artin representation. Namely, it is well known that the braid group on n-strands
Bn can be identified with the mapping class group of (D \ {p1, . . . , pn}, ∂D) and therefore acts
naturally on π1(D \ {p1, . . . , pn}, ∂D) ∼= Fn. This braid group action is usually called the Artin
representation, and is given explicitly by
σi
:



xi
7→ xixi+1x
−1
i
xi+1 7→ xi
xj
7→ xj j ̸= i, i + 1
(5.3.4)
By the Artin-Birman Theorem, the fundamental group of the complement of a link βˆ ⊆ R
3 has
following representation, coinciding with L(F1, σ)[β]:
π1(R
3
\ βˆ) ∼=

x1, . . . , xn| β(x1) = x1, β(x2) = x2 . . . , β(xn) = xn

=: L(F1, σ)[β].
5.3.2 Perverse sheaves on n-punctured disk and the GMV action
Let Perv(D2
, {p1, . . . , pn}) be the category of perverse sheaves on a disk D2 with only possible
singularities at n points. Gelfand, MacPherson and Vilonen [12] have showed that Perv(D2
, {p1
57

, . . . , pn}) is equivalent to the category of finite-dimensional k-linear representations of the following quiver
Figure 5.2: A quiver associated to Perv(D2
, {p1 , . . . , pn})
such that Ti
:= e0 + aia
∗
i
are isomorphisms for all i. Equivalently, let
A
(n)
:= k⟨Q
(n)
⟩[T
−1
1
, . . . , T −1
n
] (5.3.5)
denote the path category of Q(n)
localized at the set of morphisms {T1, . . . , Tn}, then the category
of perverse sheaves is equivalent to the category of finite-dimensional modules over A(n)
:
Perv(D
2
, {p1, . . . , pn}) ∼= Mod A
(n)
. (5.3.6)
As in Example 5.3.3, the braid group Bn acts on the disk D2 with n marked points {p1, . . . , pn} as
the mapping class group, and this naturally induces an action on Perv(D2
, {p1, . . . , pn}). Under
the equivalence of (5.3.6), [12] showed that such of Bn on perverse sheaves corresponds to a strict
58

action on Mod A(n)
. This in turn, induces an action of Bn on the localized path category A(n)
,
denoted the GMV action in [3], and is given explicitly by the formulas
σi
:



ai
7→ Tiai+1
ai+1 7→ ai
aj
7→ aj j ̸= i, i + 1
a
∗
i
7→ a
∗
i+1T
−1
i
a
∗
i+1 7→ a
∗
i
a
∗
j
7→ a
∗
j
j ̸= i, i + 1
(5.3.7)
The k-category A(n)
, consider as an object in the category of small pointed k-categories Cat∗
k
,
is the coproduct (fusion product) of n copies of the k-category A = k⟨Q⟩[T
−1
], where k⟨Q⟩ is
the path category of the quiver
Q = 0 1
(5.3.8)
In the language of categorical link closure, Berest-Eshmatov-Yeung proved that:
Example 5.3.9. [3, Theorem 5.6] The GMV braid action on A(n) ∈ Cat∗
k
gives rise to a link invariant
L(A, σ), which associates an equivalence class of k-category L(A, σ)[β] for every braid β.
In particular, there exists a distinguished object in L(A, σ)[β] whose endormorphism algebra is
identified combinatorically with the fully noncommutative DGA computing the knot contact homology of βˆ.
59

5.4 Comparing closures
Inspired by the similarity between (5.2.4) and the categorical link closure (5.3.2), we begin by
noticing that
Proposition 5.4.1. The partially wrapped Fukaya categories W

T
∗D2
, N∗
∞{p1, . . . , pn}

is an nfold coproduct of W

T
∗D2
, N∗
∞{pt}

.
Thus, on the category C of partially wrapped Fukaya categories, one may possibly define a
Yang-Baxter style (5.3.2) braid group action, i.e., generated by a single morphism σ : A⨿A → A⨿
A and suitably extending to coproducts A(n)
, where A = W

T
∗D2
, N∗
∞{pt}

. It is then natural
to ask whether there is a braid group action whose induced categorical link closure operator on
partially wrapped Fukaya categories recovers the homotopy pushout formula, which is a known
local-to-global property in partially wrapped Fukaya categories:
Main Question. Construct an braid group action on W

T
∗D2 N∗{p1, . . . , pn}

generated by a
single morphism
σ : A ⨿ A → A ⨿ A, A = W

T
∗D
2
, N∗
∞{pt}

,
such that by taking categorical link closure with respect to σ, the outcome is
L(A, σ)[β] ∼= W(T
∗D
3
,Λβˆ), (5.4.2)
the partially wrapped Fukaya category of T
∗D3
stopped at the Legendrian torus Λβˆ for every braid
β ∈ Bn

Let us first comment on our approach and the significance of above question. On one hand,
thanks to the “dimensional reduction" by Künneth stabilization functor in (5.2.3), the braid groups
act on the zero section D2 by diffeomorphisms as in Example 5.3.3, which induces symplectomorphisms on cotangent bundle and gives rise to a natural action σ on W

T
∗D2
, N∗{p1, . . . , pn}

.
On the other hand, in light of the equivalence of categories Perf W(T
∗M,Λ)op ∼= ShΛ(M)
c
in
Theorem 5.1.10, we calculate (see Conjecture 5.4.4 below) that the sheaf models for Fukaya categories A(n) = W

T
∗D2
, N∗
∞{p1, . . . , pn}

are isomorphic to the path category of quivers Qn
in Figure 5.2, on which the GMV action (5.3.7) was originally defined. Consequently, one could
obtain another action τ on A(n) via pulling back the GMV action under the isomorphisms in
Theorem 5.1.10 and Conjecture 5.4.4 .
The above braid group actions σ and τ share the same geometric origin from Example 5.3.3,
and it would be interesting to compare their categorical link closures in partially wrapped Fukaya
categories. Our Main Conjecture states that (1) the actions σ and τ coincide; and (2) their closures
are equal and isomorphic to the partially wrapped Fukaya category stopped at the conormal torus.
In particular, the pathway from (1) to (2) requires understanding the computations of categories
with those functorial link closure operators, which might suggest a method to “local-to-globalize"
the equivalence in Theorem 5.1.10. Namely, by Theorem 5.1.10 and Main Conjecture (ii), the “local" Fukaya category A(n)
equipped with the geometric braid action σ is isomorphic to a category
of sheaves equipped with the GMV action; then passing to the closures yields an isomorphism
between the “global" categories, and the isomorphism is compatible with the diffeomorphisminduced braid group actions on both initial categories. Now we state:

Main Conjecture (Possible answer to Main Question). x
Such σ on W

T
∗D2
, N∗
∞{p1, . . . , pn}

arises geometrically from diffeomorphisms on disk, similar
to the way GMV action is defined on the category of perverse sheaves (paragraph preceding (5.3.7)).
More precisely,
(i) σ is induced by the action of Bn as the mapping class group on the zero section D2
, relative to
n marked points;
(ii) σ acts on the partially wrapped Fukaya category W

T
∗D2
, N∗
∞{p1, . . . , pn}

by the same
algebraic formulas as the GMV action (5.3.7) acts on the perverse sheaf category Perv(D2
,
{p1, . . . , pn}).
Our intuition for statement (ii) relies deeply on (5.1.11) the sheaf-theoretic model ShΛ(M)
c of
Fukaya categories, and perhaps most importantly, on the fact that ShΛ(M)
c
are compact objects
in the category of unbounded constructible sheaves on M. This is worth a bit of elaboration.
Choosing S to be the stratification D2 = {p1, . . . , pn}∪{complement}, Theorem 5.1.10 states
that
Perf W(T
∗D
2
, N∗
∞{p1, . . . , pn})
op = ShS(D
2
)
c
. (5.4.3)
The perverse sheaf category Perv(M, {p1, . . . , pn}) introduced in Section 3.2 is a full subcategory of Shb
S
(D2
), which is the “usual" constructible category of bounded sheaves. It is well-known
that Perv(M, {p1, . . . , pn}) generates Shb
S
(D2
), and by [8, Corollary 4.23], Shb
S
(D2
) can be identified as proper modules over ShS(D2
)
c
; meanwhile, by (5.3.6), Perv(M, {p1, . . . , pn}) can be
identified as finite modules over the path category A(n)
. Hence we expect:

Conjecture 5.4.4. As in (5.3.6), let A(n) = k⟨Q(n)
⟩[T
−1
1
, . . . , T −1
n
] be the path algebra of the quiver
localized at the collection of paths Ti = e0 + aia
∗
i
. Then
ShS(D
2
)
c ∼= Perf A
(n)
. (5.4.5)
As we attempted to explain in the preceding paragraph, an evidence behind the isomorphism
in (5.4.5) is that the proper modules over them are isomorphic, and a proof of Conjecture 5.4.4
could possibly be obtained by showing Mod C “determines" C similar to the case [8, Lemma
A.3]. Another somewhat ad-hoc way to show (5.4.5) is by explicitly locating the Lagrangians and
morphisms in W(T
∗D2
, N∗
∞{p1, . . . , pn}) that correspond to the vertices and edges in the quiver
Qn
.
At the time of writing, we expect that the vertices in Qn
are represented by a cotangent fiber
and n linking disks at N∗
∞{pi} (or possibly by their cones.) And we expect that a successful interpretation of the paths Ti = e0 + aia
∗
i
in (5.3.7) as morphisms between distinguished Lagrangians
in W(T
∗D2
, N∗
∞{p1, . . . , pn}) will reveal the geometric braid group action σ that we proposed
in Main Conjecture. Ideally, one could then use such a Floer-theoretic description of σ to show σ
63

generates the morphisms in diagram (5.2.2), which are defined [9] by pushing forward the inclusions of Liouville sectors; the notion of “forward stopped inclusion" in [10, Section 8.2] might help
to establish such equivalence. If true, performing categorical link closure with respect to σ would
indeed agree with gluing copies of Fukaya categories under the homotopy pushout formula, thus
proving the equivalence (5.4.2) in the Main Question.
Future directions of study
We close by listing two additional future directions of study.
1. One may study all the questions people study for compact Legendrians for the conormal
bundles of braids, which are non-compact Legendrians in T
∗R
3
. For example, Lagrangian
fillings of a compact Legendrian give rise to augmentations of the DGA associated to the
Legendrian; do non-compact Lagrangians fillings of Λβ also give rise to augmentations?
2. Can one extract invariants for braids (or tangles) themselves, by studying the partially
wrapped Fukaya category stopped at the Λβ?
64

Bibliography
[1] D. Alvarez-Gavela, Y. Eliashberg, and D. Nadler, Positive arborealization of polarized
weinstein manifolds, arXiv preprint arXiv:2011.08962, (2022).
[2] J. Asplund and T. Ekholm, Chekanov-eliashberg dg-algebras for singular legendrians, arXiv
preprint arXiv:2102.04858, (2021).
[3] Y. Berest, A. Eshmatov, and W.-K. Yeung, Perverse sheaves and knot contact homology,
Comptes Rendus. Mathématique, 355 (2017), pp. 378–399.
[4] A. Cannas da Silva, Lectures on Symplectic Geometry, Springer Berlin Heidelberg, 2008.
[5] T. Ekholm and Y. Lekili, Duality between lagrangian and legendrian invariants, arXiv
preprint arXiv:1701.01284, (2017).
[6] T. Ekholm, L. Ng, and V. Shende, A complete knot invariant from contact homology, Inventiones mathematicae, 211 (2018), pp. 1149–1200.
[7] J. B. Etnyre, Introductory lectures on contact geometry, arXiv preprint math/0111118, (2001).
[8] S. Ganatra, J. Pardon, and V. Shende, Microlocal morse theory of wrapped fukaya categories, arXiv preprint arXiv:1809.08807, (2018).
[9] , Covariantly functorial wrapped floer theory on liouville sectors, Publications mathématiques de l’IHÉS, 131 (2020), pp. 73–200.
[10] , Sectorial descent for wrapped fukaya categories, Journal of the American Mathematical
Society, (2023).
[11] H. Geiges, An introduction to contact topology, vol. 109, Cambridge University Press, 2008.
[12] S. Gelfand, R. MacPherson, and K. Vilonen, Perverse sheaves and quivers, (1996).
[13] E. Giroux, Topologie de contact en dimension 3 (autour des travaux de yakov eliashberg),
Séminaire Bourbaki, 1992 (1993), p. 93.
[14] , Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, arXiv
preprint math/9908178, (1999).
[15] K. Honda and Y. Huang, Convex hypersurface theory in contact topology, arXiv preprint
arXiv:1907.06025, (2019).
65

[16] D. McDuff and D. Salamon, Introduction to symplectic topology, vol. 27, Oxford University
Press, 2017.
[17] L. Ng, Framed knot contact homology, arXiv preprint math/0407071, (2004).
[18] V. Shende, The conormal torus is a complete knot invariant, in Forum of Mathematics, Pi,
vol. 7, Cambridge University Press, 2019, p. e6.
[19] Z. Sylvan, On partially wrapped fukaya categories, Journal of Topology, 12 (2019), pp. 372–
441.
[20] , Orlov and viterbo functors in partially wrapped fukaya categories, arXiv preprint
arXiv:1908.02317, (2019).
66

Abstract (if available)

Abstract Liouville sectors-with-corners and sectorial decompositions provide a local-to-global approach to understand the geometry of noncompact sympletic manifolds. We provide new examples and generalize two constructions from Ganatra-Pardon-Shende to the case of Liouville sectors-with-corners.
In particular, we define the notion of sutured sectorial domains and show that their Liouville completions give rise to sectors-with-corners. Applying this sutured construction to conormal lifts of knots and braids, we also discuss its connection to the study of knot invariants.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Cornered Calabi-Yau categories in open-string mirror symmetry

PDF

Some stable splittings in motivic homotopy theory

PDF

On the index of elliptic operators

PDF

The Z/pZ Gysin sequence in symplectic cohomology

PDF

On the Chow-Witt rings of toric surfaces

PDF

On foliations of higher dimensional symplectic manifolds and symplectic mapping class group relations

PDF

Tight contact structures on the solid torus

PDF

Picard group actions on constructible sheaves via contact isotopies

PDF

Computations for bivariant cycle cohomology

PDF

Algebraic plane curves: from multiplicities and intersection numbers to Bézout's theorem

PDF

Inequitable access to primary care in ethnic minority populations

PDF

Resolving black hole horizons: with superstrata

PDF

Motivic cohomology of BG₂

PDF

Construction of orthogonal functions in Hilbert space

PDF

Essays on effects of reputation and judicial transparency

PDF

Contact surgery, open books, and symplectic cobordisms

PDF

Motivation, persistence, and achievement in East Asian languages learning: an expectancy-value theory perspective

PDF

Hot electron driven photocatalysis and plasmonic sensing using metallic gratings

PDF

Middle school counselors’ perceptions of their role supporting LGBTQ+ youths’ belonging

PDF

Koszul-perverse sheaves and the A⁽¹⁾₂ dual canonical basis

Asset Metadata

Creator Zhou, Jian (author)

Core Title On sutured construction of Liouville sectors-with-corners

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Degree Conferral Date 2023-12

Publication Date 01/06/2024

Defense Date 01/03/2024

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

Tag conormal torus,Fukaya category,Liouville sector,OAI-PMH Harvest,symplectic geometry

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ganatra, Sheel (committee chair), Asok, Aravind (committee member), Pardo, Kris (committee member)

Creator Email nickj2357@gmail.com,zhou018@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113803848

Unique identifier UC113803848

Identifier etd-ZhouJian-12594.pdf (filename)

Legacy Identifier etd-ZhouJian-12594

Document Type Dissertation

Format theses (aat)

Rights Zhou, Jian

Internet Media Type application/pdf

Type texts

Source 20240116-usctheses-batch-1119 (batch), 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 author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

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

Repository Email cisadmin@lib.usc.edu