# A Lagrangian filling for every cluster seed

Roger Casals  University of California Davis, Dept. of Mathematics, USA
<EMAIL_ADDRESS>and  Honghao Gao  Tsinghua University, Yau
Mathematical Sciences Center, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

We show that each cluster seed in the augmentation variety is inhabited by an
embedded exact Lagrangian filling. This resolves the matter of surjectivity of
the map from Lagrangian fillings to cluster seeds. The main new technique to
produce these Lagrangian fillings is the construction and study of a quiver
with potential associated to curve configurations. We prove that its
deformation space is trivial and show how to use it to manipulate Lagrangian
fillings with $\mathbb{L}$-compressing systems via Lagrangian disk surgeries.

###### 2010 Mathematics Subject Classification:

Primary: 53D12. Secondary: 57K33, 13F60.

##  1\. Introduction

We show that each cluster seed is inhabited by an embedded exact Lagrangian
filling. Heretofore, the surjectivity of the map from Lagrangian fillings to
cluster seeds remained open for essentially all braids. The argument is based
on applying Lagrangian disk surgeries to an initial Lagrangian filling with an
$\mathbb{L}$-compressing system. We are able to avoid the appearance of
immersed curves, a known technical issue of this problem, by introducing and
studying a new object: a quiver with potential for each such
$\mathbb{L}$-compressing system. The manuscript first establishes the key
properties of these new quivers with potentials, including their rigidity and
invariance. We then show how to use these properties to construct a Lagrangian
filling for every cluster seed.

Scientific context. Let $\beta$ be a positive braid word. Consider the
Legendrian link $\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ obtained as
the Legendrian lift of the rainbow closure of $\beta$, as defined in [CN22,
Section 2.2] or cf. Section 4.1. Here $\xi_{st}=\ker\\{dz-ydx\\}$ where
$(x,y,z)\in\mathbb{R}^{3}$ are Cartesian coordinates.

Consider its augmentation variety $X(\Lambda_{\beta},T)$, where
$T\subset\Lambda_{\beta}$ is a set of marked points, one per component. See
Section 4.7 or [CN22, Section 5.1] and references therein for more details on
such augmentation varieties. This affine algebraic variety
$X(\Lambda_{\beta},T)$ is smooth and isomorphic to the open (double) Bott-
Samelson variety with pair of braids $(\beta,\mbox{e})$, where $e$ is the
identity braid, up to trivial $\mathbb{C}^{\times}$-factors. It is known that
its ring of regular functions $\mathbb{C}[X(\Lambda_{\beta},T)]$ is a cluster
algebra. See [SW21, GSW20a] for both these facts or cf. Section 4.7 below. An
alternative, intrinsically symplectic construction of such cluster structures
is provided in [CW] via the microlocal theory of sheaves.

A salient property of this cluster algebra structure constructed on
$\mathbb{C}[X(\Lambda_{\beta},T)]$ is that, in all known cases, an oriented
embedded exact Lagrangian filling $L$ of
$\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$, embedded in the
symplectization $(\mathbb{R}^{4},\lambda_{st})$ of
$(\mathbb{R}^{3},\xi_{st})$, gives a cluster seed in
$\mathbb{C}[X(\Lambda_{\beta},T)]$. Specifically, the filling $L$ gives an
open toric chart in $X(\Lambda_{\beta},T)$ and there is a choice of
$\mathbb{L}$-compressing system $\Gamma$ for $L$ that endows this toric chart
with cluster $\mathcal{A}$-coordinates, thus defining a seed
$\mathfrak{c}(L,\Gamma)$ in $\mathbb{C}[X(\Lambda_{\beta},T)]$. The
description of the cluster seed is explicit for Lagrangian fillings $L$
associated to pinching sequences of $\beta$, see [CN22] and [GSW20a], and
[CZ22a] provides a diagrammatic calculus to describe more general cluster
seeds.

Note that these embedded exact Lagrangians $L$ for $\Lambda_{\beta}$ must
typically be surfaces of higher genus. In addition, if $L$ and $L^{\prime}$
are compactly supported Hamiltonian isotopic, then the toric charts in
$X(\Lambda_{\beta},T)$ associated to these seeds must be equal. Again, see
[EHK16, GSW20a] or [CW]. Such invariance property has been successfully used
to distinguish Lagrangian fillings, cf. [CG22, CZ22a, GSW20b].

The Question. Let $\operatorname{Lag}(\Lambda_{\beta})$ be the set of
Hamiltonian isotopy classes of embedded exact Lagrangian fillings of
$\Lambda_{\beta}$ in the symplectization of $(\mathbb{R}^{3},\xi_{st})$. A
Hamiltonian isotopy class is given by the equivalence relation $L\sim
L^{\prime}$ iff there exists a compactly supported Hamiltonian diffeomorphism
$\varphi\in\mbox{Ham}^{c}(\mathbb{R}^{4},\lambda_{st})$ such that
$\varphi(L)=L^{\prime}$. By [EHK16, Section 3.5], cf. also [CN22, Theorem
3.6], there exists a map
$\mathfrak{C}^{\circ}:\operatorname{Lag}(\Lambda_{\beta})\longrightarrow\mbox{Toric}(X(\Lambda_{\beta},T))$,
where $\mbox{Toric}(X(\Lambda_{\beta},T))$ is the set of open unparametrized
algebraic toric charts $(\mathbb{C}^{\times})^{d}\subset X(\Lambda_{\beta},T)$
in the affine variety $X(\Lambda_{\beta},T)$, where
$d=\dim_{\mathbb{C}}X(\Lambda_{\beta},T)$. An $\mathbb{L}$-compressing system
$\Gamma$ for a Lagrangian filling $L\in\operatorname{Lag}(\Lambda_{\beta})$
endows the toric chart $\mathfrak{C}^{\circ}(L)$ with toric coordinates
$A(\Gamma)$, e.g. see [CW, Section 4.6]. Let
$\operatorname{Lag}^{c}(\Lambda_{\beta})$ be the set of pairs $(L,\Gamma)$
consisting of a Lagrangian filling in
$L\in\operatorname{Lag}(\Lambda_{\beta})$ and an $\mathbb{L}$-compressing
system $\Gamma$ for $L$, up to Hamiltonian isotopy, such that $A(\Gamma)$ are
cluster coordinates for a cluster seed of the cluster algebra structure in
$\mathbb{C}[X(\Lambda_{\beta},T)]$ above. Finally, let
$\operatorname{Seed}(X(\Lambda_{\beta},T))$ be the set of cluster seeds in
$\mathbb{C}[X(\Lambda_{\beta},T)]$. In summary, at its coarsest level, the
constructions cited above, sending a Lagrangian filling with an
$\mathbb{L}$-compressing system $(L,\Gamma)$ to the cluster seed
$\mathfrak{c}(L,\Gamma):=(\mathfrak{C}^{\circ}(L),A(\Gamma))$, yields a map of
sets:

$\mathfrak{C}:\operatorname{Lag}^{c}(\Lambda_{\beta})\longrightarrow\operatorname{Seed}(X(\Lambda_{\beta},T)),\quad(L,\Gamma)\mapsto\mathfrak{C}(L,\Gamma):=\mathfrak{c}(L,\Gamma).$

In our view, the surjectivity and injectivity of this map $\mathfrak{C}$ is a
central open problem in low-dimensional contact and symplectic topology. It
lies at the core of the study of Legendrian knots in
$(\mathbb{R}^{3},\xi_{st})$ and, more generally, the symplectic topology of
Weinstein 4-manifolds. In fact, just understanding the exact cardinality of
any fiber of $\mathfrak{C}$ would be remarkable. Note that [CKQ22, Prop. 5.3]
implies that the forgetful map
$\iota:\operatorname{Seed}(X(\Lambda_{\beta},T))\longrightarrow\mbox{Toric}(X(\Lambda_{\beta},T))$
sending a cluster seed to its underlying toric chart is injective. Thus,
surjectivity of $\mathfrak{C}$ would imply that $\mathfrak{C}^{\circ}$
surjects onto $\iota(\operatorname{Seed}(X(\Lambda_{\beta},T)))$.
The state of affairs is as follows:

  * -

Injectivity of $\mathfrak{C}$ is a generalization of the nearby Lagrangian
conjecture for surfaces with boundary to a statement about embedded exact
Lagrangians in Weinstein neighborhoods of Lagrangian skeleta. At core, it
states that any embedded exact Lagrangian in a neighborhood of the arboreal
Lagrangian skeleton consisting of $L$ and Lagrangian disks attached to it is
either Hamiltonian isotopic to $L$ or to those Lagrangians obtained from it by
Lagrangian surgeries along the disks in the skeleton. By [EP96],
$\mathfrak{C}$ is injective if $\Lambda_{\beta}$ is the max-tb Legendrian
unknot. Injectivity of $\mathfrak{C}$ remains open for any other Legendrian
link $\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$.

  * -

Surjectivity of $\mathfrak{C}$ is a reconstruction statement, from an
algebraic invariant back to actual 4-dimensional symplectic topology. Indeed,
it inputs algebraic data, provided by the ring of functions
$\mathbb{C}[X(\Lambda_{\beta},T)]$ and a seed for its cluster structure, and
should output symplectic topological data, an embedded exact Lagrangian
filling of $\Lambda_{\beta}$ and an $\mathbb{L}$-compressing system.

By the finite type classification of cluster algebras, established in [FZ03],
$\operatorname{Seed}(X(\Lambda_{\beta},T))$ is known to be a finite set only
for a few exceptional cases, i.e. the ADE cases. For those ADE cases,
$\mathfrak{C}$ can be verified to be surjective by direct computation: this
has recently been established in [Hug22, ABL21b] by using weaves [CZ22a]. See
also [ABL21a] for the affine ADE cases where
$\operatorname{Seed}(X(\Lambda_{\beta},T))$ is still finite, up to the natural
tame quotient. These finite type and affine type are rare: essentially all
braids $\beta$ have $\operatorname{Seed}(X(\Lambda_{\beta},T))$ be an infinite
set, cf. [GSW20b, Section 4] or [Cas22, Section 5]. Confer [CG22, CN22, CW,
STWZ19, STW16] and references therein for partial results on $\mathfrak{C}$
and [Cas22, Section 5] for further discussions. Surjectivity of $\mathfrak{C}$
remains open for any other Legendrian link
$\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$.

There has been significant activity in recent times in the study of Lagrangian
fillings, see e.g. [ABL21b, ABL21a, CS22, CG22, CN22, CL22, CGG+22, CW, CZ22a,
EHK16, GSW20a, GSW20b, Gol22, Hug22, Hug23, STWZ19]. These results use
techniques from and connecting to the microlocal theory of sheaves, Floer
theory and cluster algebras among others. That said, all fall objectively
short of showing anything close to the surjectivity of $\mathfrak{C}$. This
manuscript introduces a genuinely new technique: the definition, study and use
of a quiver with potential $(Q,W)$ to construct Lagrangian fillings.

The Main Result. The goal of this manuscript is to establish the surjectivity
of the map $\mathfrak{C}$. In order to state the main result, which is a
stronger version of surjectivity, we introduce the following concepts. Let
$L\subset(\mathbb{D}^{4},\lambda_{st})$ be an exact oriented Lagrangian
filling, $\gamma\subset L$ an embedded oriented curve and $\Lambda_{\gamma}$
its Legendrian lift to the ideal contact boundary $\partial{\mathcal{O}}{\it
p}(L)$ of a convex open neighborhood ${\mathcal{O}}{\it p}(L)$ of $L$ in
$\mathbb{D}^{4}$. Note that ${\mathcal{O}}{\it p}(L)$ is symplectomorphic to a
convex neighborhood of the zero section in $(T^{*}L,\lambda_{st})$. By
definition, $\gamma$ is said to be $\mathbb{L}$-compressible if there exists a
properly embedded Lagrangian 2-disk
$D\subset(T^{*}\mathbb{R}^{2}\setminus{\mathcal{O}}{\it p}(L))$ such that
$\partial\overline{D}\cap\partial{\mathcal{O}}{\it
p}(L)=\Lambda_{\gamma}\subset\mathbb{R}^{4}$ and the union of
$\overline{D}\cup\nu_{\gamma}$ is a smooth Lagrangian disk, where
$\nu_{\gamma}\subset{\mathcal{O}}{\it p}(L)$ is the Lagrangian conormal cone
of $\gamma$.

A collection $\Gamma=\\{\gamma_{1},\ldots,\gamma_{b}\\}$ of oriented embedded
curves in $L$, with a choice of $\mathbb{L}$-compressing disks
$\mathscr{D}(\Gamma)=\\{D_{1},\ldots,D_{b}\\}$, one for each curve, is said to
be an $\mathbb{L}$-compressing system for $L$ if $D_{i}\cap D_{j}=\emptyset$
for all $i,j\in[b]$ and the homology classes of the curves in $\Gamma$ form a
basis of $H_{1}(L;\mathbb{Z})$. Two $\mathbb{L}$-compressing systems
$\Gamma,\Gamma^{\prime}$ for $L$ are said to be equivalent if there exists a
sequence of triple point moves and bigon moves, i.e. Reidemeister IIIs and
non-dangerous tangencies, that applied to the curves in $\Gamma$ lead to the
curves in $\Gamma^{\prime}$. See Section 2 for further discussions and cf.
Figure 2 and Figure 8 for such moves. Given a Lagrangian filling $L$ with an
$\mathbb{L}$-compressing system $\Gamma$ and a disk $D\in\mathscr{D}(\Gamma)$,
Lagrangian disk surgery produces an embedded exact Lagrangian filling
$\mu_{D}(L)$. For details on Lagrangian disk surgeries, see Section 4 below or
cf. [Pol91, Yau17].

It is not always the case that $\mu_{D}(L)$ inherits an
$\mathbb{L}$-compressing system from $(L,\Gamma)$: curves in $\Gamma$ might
become immersed under Lagrangian disk surgery. Therefore, the set of curves
$\mu_{D}(\Gamma)$ obtained from $\Gamma$ after Lagrangian disk surgery on $D$
might not be an $\mathbb{L}$-compressing system. This is a well-known problem
in this approach, cf. Section 4. The main technical achievement of this
manuscript is to show that this can be corrected, showing that certain
$\mathbb{L}$-compressing systems persist under arbitrary sequences of
Lagrangian disk surgeries. The quiver with potential that we introduce and
study is a crucial ingredient for implementing this correction.

The core result of the manuscript is the following theorem, which implies the
surjectivity of $\mathfrak{C}$ for all positive braids $\beta$. The entire
article is devoted to developing a new technique that proves this result.

######  Theorem 1.1.

Let $\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ be the Legendrian link
associated to a positive braid word $\beta$, and $T\subset\Lambda_{\beta}$ a
set of marked points with one marked point per component.

Then there exists an embedded exact Lagrangian filling
$L\subset(\mathbb{R}^{4},\lambda_{st})$ of $\Lambda_{\beta}$ and an
$\mathbb{L}$-compressing system $\Gamma$ for $L$ such that the following
holds:

  * (i)

If $\mu_{v_{\ell}}\ldots\mu_{v_{1}}$ is any sequence of mutations, where
$v_{1},\ldots,v_{\ell}$ are mutable vertices of the quiver
$Q(\mathfrak{c}(L,\Gamma))$ associated to the cluster seed
$\mathfrak{c}(L,\Gamma)$ of $L$ in $\mathbb{C}[X(\Lambda_{\beta},T)]$, then
there exists a sequence of embedded exact Lagrangian fillings $L_{k}$ of
$\Lambda_{\beta}$, each equipped with an $\mathbb{L}$-compressing system
$\Gamma_{k}$, with associated cluster seeds

$\mathfrak{c}(L_{k},\Gamma_{k})=\mu_{v_{k}}\ldots\mu_{v_{1}}(\mathfrak{c}(L,\Gamma))$

in $\mathbb{C}[X(\Lambda_{\beta},T)]$, for all $k\in[\ell]$.

  * (ii)

Each $\mathbb{L}$-compressing system $\Gamma_{k}$ for $L_{k}$ is such that
Lagrangian disk surgery on $L_{k}$ along any Lagrangian disk in
$\mathscr{D}(\Gamma_{k})$ yields an $\mathbb{L}$-compressing system. In
addition, $\Gamma_{k+1}$ is equivalent to such an $\mathbb{L}$-compressing
system via a sequence of triple point moves and local bigon moves. $\Box$

Theorem 1.1 is a reconstruction result that we prove by first introducing a
new object: a quiver with potential $(Q(\mathcal{C}),W(\mathcal{C}))$
associated to a curve configuration $\mathcal{C}$. Such a curve configuration
can be extracted from the Legendrian link $\Lambda_{\beta}$ and its initial
Lagrangian filling $L$. See [DWZ10] for general algebraic results on quiver
with potentials.111The word potential has different meanings in the
literature. Here a quiver with potential is meant in the sense of [DWZ10]. It
is unrelated to the potentials from [PT], for instance, which count
holomorphic disks bounded by Lagrangians and thus vanish for exact
Lagrangians. Our results will show that the right-equivalence class of this
particular quiver with potential $(Q(\mathcal{C}),W(\mathcal{C}))$ associated
to $\Lambda_{\beta}$ is rigid, it is invariant under triple point moves and
bigon moves, and changes according to QP-mutation under Lagrangian disk
surgeries. A different choice of potential $W$ for the quiver
$Q(\mathcal{C})$, not tailored to our problem and the specific geometry of
embedded polygons for curve configurations $\mathcal{C}$, would be of no
use.222Similarly, previously constructed potentials in the context of CY3
categories or triangulated surfaces, e.g. [Gon17, Section 5.1], [Smi15] or
[LF09], cannot be used either, unless proven to faithfully count embedded
polygons and be invariant under the necessary 4D moves.

This quiver with potential encodes an arboreal skeleton for the Weinstein
relative pair $(\mathbb{C}^{2},\Lambda_{\beta})$, cf. [Eli18, Section 2] or
[Cas22, Section 2]. Intuitively, proving its rigidity is showing that its
right-equivalence class is invariant under infinitesimal deformations.
Technically, we show that its deformation space, the trace space of the
Jacobian algebra modulo the ground ring, is trivial. Thanks to the
constructions in Sections 2 and 3, we are able to use this rigidity in Section
4 to show that we can geometrically realize any arbitrary sequence of
algebraic mutations in the cluster algebra $\mathbb{C}[X(\Lambda_{\beta},T)]$
by a sequence of Lagrangian disk surgeries on that arboreal skeleton for
$(\mathbb{C}^{2},\Lambda_{\beta})$. Near the entirety of the manuscript is
devoted to the construction, study and use of this new quiver with potential.

######  Remark 1.2.

Each embedded exact Lagrangian filling $L$ in Theorem 1.1 gives a closed
embedded exact Lagrangian surface $\overline{L}$ in $W(\Lambda_{\beta})$, the
Weinstein 4-fold obtained by attaching a Weinstein 2-handle to each component
of $\Lambda_{\beta}$. If the cluster seeds of $L,L^{\prime}$ are different,
then $\overline{L}$ and $\overline{L^{\prime}}$ are not Hamiltonian isotopic
in $W(\Lambda_{\beta})$, only possibly Lagrangian isotopic, cf. [CN22, Section
7]. Theorem 1.1 shows that each of these (typically infinitely many) different
closed exact Lagrangians in $W(\Lambda_{\beta})$ is embedded in a different
closed arboreal Lagrangian skeleton, cf. [Cas22, Section 2.4]. Note also that
for each $(L_{k},\Gamma_{k})$, the object
$\mathscr{L}_{k}:=C_{1}\oplus\ldots\oplus C_{\pi_{0}(\Lambda_{\beta})}\oplus
T^{*}_{p_{1}}D_{1}\oplus\ldots\oplus T^{*}_{p_{b}}D_{b}$ is a compact
generator of the wrapped Fukaya category of $W(\Lambda_{\beta})$, where
$p_{i}\in D_{i}$ are interior points, $T^{*}_{p_{i}}D_{i}$ the local cotangent
fibers and $C_{j}$ are the co-cores of the Weinstein 2-handles. Therefore,
Theorem 1.1 geometrically constructs a compact generator for each vertex of
the cluster exchange graph of the cluster algebra
$\mathbb{C}[X(\Lambda_{\beta},T)]$. In particular, when the dg-algebras
$\mbox{End}(\mathscr{L}_{k})$ are non-positively graded, Theorem 1.1
geometrically constructs bounded $t$-structures for the wrapped Fukaya
category. $\Box$

The map is surjective. The construction of the cluster algebra structure for
$\mathbb{C}[X(\Lambda_{\beta},T)]$ has the following property, relating
Theorem 1.1 to the map $\mathfrak{C}$; cf. [CW, GSW20a] or Section 4.7 below.
Given a Lagrangian filling $L$ of $\Lambda_{\beta}$ and an
$\mathbb{L}$-compressing system $\Gamma$, the cluster variables in
$\mathfrak{c}(L,\Gamma)$ are indexed by the curves in $\Gamma$ and the arrows
in the quiver of $\mathfrak{c}(L,\Gamma)$ record geometric intersections of
curves in $\Gamma$. The cluster variables are described by a microlocal
parallel transport, cf. [CW, Section 4]. In addition, the cluster seed
$\mu_{v}(\mathfrak{c}(L,\Gamma))$ obtained by algebraically mutating at a
vertex $v=\gamma$ of the quiver, indexed by some $\gamma\in\Gamma$, is
precisely $\mathfrak{c}(\mu_{D_{\gamma}}(L,\Gamma))$, where $D_{\gamma}$ is
the $\mathbb{L}$-compressing disk associated to the curve $\gamma\in\Gamma$.
Therefore, Theorem 1.1 implies the desired surjectivity:

######  Corollary 1.3.

Let $\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ be the Legendrian link
associated to a positive braid word $\beta$, $T\subset\Lambda_{\beta}$ a set
of marked points, one per component, and $X(\Lambda_{\beta},T)$ its
augmentation variety. Then

$\mathfrak{C}:\operatorname{Lag}^{c}(\Lambda_{\beta})\longrightarrow\operatorname{Seed}(X(\Lambda_{\beta},T))$

is surjective, i.e. each cluster seed is induced by an embedded exact
Lagrangian filling endowed with an $\mathbb{L}$-compressing system.

We conjecture that the map $\mathfrak{C}$ is injective. Corollary 1.3 and a
proof that $\mathfrak{C}$ is injective would settle the core symplectic
aspects of the classification of Hamiltonian isotopy classes of Lagrangian
fillings for $\Lambda_{\beta}$, proving that they are equivalent to studying a
class of cluster algebras, an algebraic matter.

Acknowledgements. We are grateful to Mikhail Gorsky for his thorough reading
of the manuscript and his useful comments, and to Daping Weng and James Hughes
for helpful remarks. We also thank Joel Hass for kindly explaining his
algorithm with P. Scott to us. R. Casals is supported by the NSF CAREER
DMS-1942363, a Sloan Research Fellowship of the Alfred P and a UC Davis
College of L&S Dean’s Fellowship. Sloan Foundation. H. Gao is supported by a
Tsinghua start-up grant and a Tsinghua Du-Shi research grant.

Structure of the manuscript. The study and use of our quiver with potential
(QP) is as follows:

  * (i)

Section 2 introduces and develops the new concept: the quiver with potential
$(Q(\mathcal{C}),W(\mathcal{C}))$ associated to a curve configuration
$\mathcal{C}$. Proposition 2.5 and Lemma 2.7 show that the right equivalence
class of $(Q(\mathcal{C}),W(\mathcal{C}))$ is invariant under triple point
moves and local bigon moves. Then Lemma 2.21 and Proposition 2.22 show that
the QP associated to a reduction of the $\gamma$-exchange on $\mathcal{C}$
yields the QP-mutation of $(Q(\mathcal{C}),W(\mathcal{C}))$ at the
corresponding vertex.

  * (ii)

Section 3 introduces the curve configurations $\mathcal{C}(\mathbb{G})$
associated to plabic fences $\mathbb{G}$. Proposition 3.12 shows that the
curve QP associated to $\mathcal{C}(\mathbb{G})$ is rigid.

  * (iii)

Section 4 constructs an initial Lagrangian filling with an
$\mathbb{L}$-compressible system whose associated curve configuration is of
the form $\mathcal{C}(\mathbb{G})$ for a plabic fence $\mathbb{G}$. It then
develops a few necessary technical results until Section 4.8, where we show
how to use the rigidity of the QP for $\mathcal{C}(\mathbb{G})$ to prove
Theorem 1.1. It also includes Theorem 4.10, a variant of Theorem 1.1.

Notation. We denote by $[n]$ the set $\\{1,\ldots,n\\}$. The group of
compactly supported diffeomorphisms of a smooth manifold $\Sigma$ is denoted
by $\mbox{Diff}^{c}(\Sigma)$. In this article, a quiver will refer to a
multidigraph with no loops, but possibly with 2-cycles. The set of vertices of
$Q$ is denoted by $Q_{0}$ and its set of arrows by $Q_{1}$. We often
abbreviate quiver with potential to QP, as in [DWZ10]. For specificity, we use
the ground ring $R=\mathbb{C}$ for the (complete) path algebra of the quiver.

##  2\. The curve quiver with potential

Let $\Sigma$ be an oriented surface and
$\mathcal{C}=\\{\gamma_{1},\ldots,\gamma_{b}\\}$, $b\in\mathbb{N}$, a
collection of embedded oriented closed connected curves
$\gamma_{i}\subset\Sigma$, $i\in[b]$, whose pairwise intersections are all
transverse. Two such collections $\mathcal{C}$ and $\mathcal{C}^{\prime}$ will
be considered equal if there exist a diffeomorphism
$\varphi\in\mbox{Diff}^{c}(\Sigma)$ such that
$\varphi(\mathcal{C})=\mathcal{C}^{\prime}$. In particular, two such
collections $\mathcal{C}$ and $\mathcal{C}^{\prime}$ related by compactly
supported isotopies are considered to be equal. We assume that the only
intersection points of curves in $\mathcal{C}$ are double intersection points,
i.e. exactly two curves $\gamma_{i},\gamma_{j}\in\mathcal{C}$ intersect
transversely at a given intersection point.

######  Definition 2.1.

Let $\Sigma$ be an oriented surface and
$\mathcal{C}=\\{\gamma_{1},\ldots,\gamma_{b}\\}$, $b\in\mathbb{N}$, a
collection of embedded oriented closed connected curves
$\gamma_{i}\subset\Sigma$, $i\in[b]$. By definition, $\mathcal{C}$ is said to
be a curve configuration, or simply a configuration, if the classes
$[\gamma_{1}],\ldots,[\gamma_{b}]$ in $H_{1}(\Sigma;\mathbb{Z})$ form a basis.
$\Box$

In particular, for a curve configuration
$\mathcal{C}=\\{\gamma_{1},\ldots,\gamma_{b}\\}$ in $\Sigma$, we must have
$b=b_{1}(\Sigma)$. The collections of curves $\mathcal{C}$ needed to prove
Theorem 1.1 will be configurations, i.e. consist of $b_{1}(\Sigma)$ curves
whose homology classes span $H_{1}(\Sigma;\mathbb{Z})$. Throughout this
manuscript, from now onward, we assume all collections of curves $\mathcal{C}$
that we use satisfy such hypothesis, i.e. they are curve configurations.

###  2.1. The QP $(Q(\mathcal{C}),W(\mathcal{C}))$ associated to $\mathcal{C}$

Let $\mathcal{C}$ be a curve configuration. Consider the integers

$A_{ij}(\mathcal{C}):=|p\in\gamma_{i}\cap\gamma_{j}:\mbox{sign}(p)\mbox{ is
positive}|\in\mathbb{N},$

where $\mbox{sign}(p)=\pm$ is the sign of the intersection point $p$.

######  Definition 2.2.

Let $\mathcal{C}$ be a curve configuration. The quiver $Q(\mathcal{C})$ is
defined to have vertex set
$Q(\mathcal{C})_{0}=\\{\gamma_{1},\ldots,\gamma_{b}\\}$ and the arrow set
$Q(\mathcal{C})_{1}$ is given by the condition that the number of arrows from
$\gamma_{i}$ to $\gamma_{j}$ is $A_{ij}(\mathcal{C})$, $i,j\in[b]$. $\Box$

The quiver $Q(\mathcal{C})$ is referred to as the curve quiver of
$\mathcal{C}$. Since the curves in $\mathcal{C}$ are embedded, the quiver
$Q(\mathcal{C})$ has no loops; it might nevertheless have 2-cycles. The arrows
from the vertex associated to $\gamma_{i}$ to that of $\gamma_{j}$ are in
natural bijection with the positive intersection points
$p\in\gamma_{i}\cap\gamma_{j}$ between $\gamma_{i}$ and $\gamma_{j}$: we
indistinctly identify arrows in $Q(\mathcal{C})$ and such intersection points
$p\in\Sigma$. Let us now discuss the preliminaries to introduce the potential
$W(\mathcal{C})$.

By definition, an $\ell$-gon $P$ bounded by $\mathcal{C}$, $\ell\in\mathbb{N}$
and $\ell\geq 2$, is a closed contractible subset $P\subset\Sigma$ with a PL-
smooth boundary $\partial P$ and embedded interior such that:

  * (i)

There are $\ell$ connected components in $\partial P\setminus V$, where
$V\subset\partial P$ is the set of non-smooth points of $\partial P$, i.e. $V$
is the set of vertices of $P$. That is, there are $\ell$ sides to $P$.

  * (ii)

For each smooth connected component $\partial P_{j}\subset(\partial P\setminus
V)$, $j\in[\ell]$, there exists a $\gamma_{i_{j}}$ such that $\partial
P_{j}\subset\gamma_{i_{j}}$ and the orientations either coincide for all
$j\in[\ell]$ or they are opposite for all $j\in[\ell]$. That is, each oriented
side of $P$ is an oriented subspace of a curve in $\mathcal{C}$ or each
oriented side of $P$ is an oriented subspace of a curve in
$\overline{\mathcal{C}}$. Here
$\overline{\mathcal{C}}=\\{-\gamma_{1},\ldots,-\gamma_{b}\\}$ denotes the same
configuration of curves as in $\mathcal{C}$ where we have switched the
orientation of each curve $\gamma\in\mathcal{C}$.

  * (iii)

At a small enough neighborhood $U\subset\Sigma$ of a vertex $v\in V$, which
locally is given by the intersection of two curves
$\gamma_{i},\gamma_{j}\in\mathcal{C}$, the intersection
$P\cap(U\setminus(U\cap(\gamma_{i}\cup\gamma_{j})))$ is a unique quadrant.
That is, vertices of an $\ell$-gon only use one of the quadrants; in a
combinatorial sense, $\ell$-gons bounded by $\mathcal{C}$ are convex.

An $\ell$-gon $P$ bounded by $\mathcal{C}$ determines a cyclically ordered set
of vertices. Conversely, it is uniquely determined by its cyclically ordered
set of vertices and its orientation, clockwise or counter-clockwise. Since the
vertices of $P$ must be intersection points between the curves in
$\mathcal{C}$, which bijectively correspond to arrows in $Q(\mathcal{C})$,
such $P$ is uniquely determined by a (cyclic) word of composable arrows
starting and ending at the same vertex, along with its orientation. In other
words, by a signed monomial in $\operatorname{HH}_{0}(Q(\mathcal{C}))$, the
trace space of the path algebra $\mathbb{C}\langle Q(\mathcal{C})\rangle$ of
$Q(\mathcal{C})$. This correspondence is written $P=v_{1}\dots v_{\ell}$ where
$v_{1},\ldots,v_{\ell}$ are the vertices of $P$ read according to the order
induced by the orientation of $\partial P$.

######  Remark 2.3.

Note that the connected components of
$\Sigma\setminus(\gamma_{1}\cup\ldots\cup\gamma_{b})$ might or might not be
$\ell$-gons bounded by $\mathcal{C}$, due to the orientations of the curves in
$\mathcal{C}$. Also, typically polygons bounded by $\mathcal{C}$ are not
connected components of $\Sigma\setminus(\gamma_{1}\cup\ldots\cup\gamma_{b})$
as they might have curves in $\mathcal{C}$ cross through them. $\Box$

Let $\Gamma_{\ell}^{+}$, resp. $\Gamma_{\ell}^{-}$, be the set of $\ell$-gons
$P$ bounded by $\mathcal{C}$ where the orientation of $\partial P$ coincides
with, resp. it is opposite to, the orientations of $\Sigma$.

For each intersection point of two curves
$\gamma_{i},\gamma_{j}\in\mathcal{C}$ that represents an arrow from
$\gamma_{i}$ to $\gamma_{j}$ in $Q(\mathcal{C})$, or vice-versa, we decorate
(shade) two consecutive quadrants as follows. If the tangent vectors of
$\gamma_{i}$ and $\gamma_{j}$, in this order, are an oriented basis of the
tangent space at the intersection point, then we shade the two quadrants in
the side of $\gamma_{i}$ where $\gamma_{j}$ points outwards from the
intersection point; see Figure 1, where the shading of the quadrants is
depicted. If that basis gives the reverse orientation, we shade the two
quadrants in the side of $\gamma_{j}$ where $\gamma_{i}$ points outwards from
the intersection point.

Figure 1. The shading of two quadrants at an intersection point. The two
consecutive quadrants that are shaded are the first and the second quadrants,
where we take the oriented basis as the two axes.

Given a polygon $P=v_{1}\ldots v_{\ell}$ bounded by $\mathcal{C}$, each vertex
$v_{i}$ of $P$ is assigned the sign $\sigma(v_{i};v_{1}\ldots v_{\ell})=1$ if
$v_{i}$ uses a non-shaded quadrant, and the sign $\sigma(v_{i};v_{1}\ldots
v_{\ell})=-1$ if $v_{i}$ uses a shaded quadrant. By definition, the vertex
sign $\sigma(v_{1}\ldots v_{\ell})$ of the polygon $P$ is

$\sigma(v_{1}\ldots v_{\ell})=\prod_{i=1}^{\ell}\sigma(v_{i};v_{1}\ldots
v_{\ell}).$

######  Definition 2.4.

The potential $W(\mathcal{C})\in\operatorname{HH}_{0}(Q(\mathcal{C}))$ of
$Q(\mathcal{C})$ is defined by

$W(\mathcal{C})=\sum_{v_{1}\dots
v_{\ell}\in\Gamma_{\ell}^{+}}\sigma(v_{1}\dots v_{\ell})\cdot v_{\ell}\dots
v_{1}\quad-\sum_{w_{1}\dots w_{\ell}\in\Gamma_{\ell}^{-}}\sigma(w_{1}\dots
w_{\ell})\cdot w_{1}\dots w_{\ell},$

where the sums run over all possible $\ell\in\mathbb{N}$, $\ell\geq 2$, and
all possible elements of $\Gamma_{\ell}^{\pm}$. The pair
$(Q(\mathcal{C}),W(\mathcal{C}))$ is referred to as the curve quiver with
potential of $\mathcal{C}$. We often abbreviate and refer to such a pair as a
curve QP or a cQP. $\Box$

In Definition 2.4 we always write the vertices on the boundary left to right
as read counter-clockwise; in this manner the monomial is an actual cycle in
the quiver $Q(\mathcal{C})$. We often consider QPs up to right-equivalence,
i.e. up to automorphisms of the path algebra; see [DWZ10, Definition 4.2] for
details.

###  2.2. Properties of curve QPs under planar moves

A configuration of curves $\mathcal{C}$ can be modified by smooth isotopies of
$\Sigma$. The combinatorics of $\mathcal{C}$, including the intersection
pattern and polygons bounded by $\mathcal{C}$, do not change under such
isotopies. We can modify $\mathcal{C}$ more significantly by choosing one
curve $\gamma\in\mathcal{C}$ and smoothly isotope it to another curve
$\gamma^{\prime}\subset\Sigma$. The new configuration
$\mathcal{C}^{\prime}:=(\mathcal{C}\cup\\{\gamma^{\prime}\\})\setminus\\{\gamma\\}$
has different combinatorics than that of $\mathcal{C}$. In this article, we
will consider two configurations $\mathcal{C}$ and $\mathcal{C}^{\prime}$
equivalent if they can be connected by a sequence of triple moves and bigon
moves; moves that we now introduce.

####  2.2.1. Behavior under triple point moves.

Let $\gamma_{1},\gamma_{2},\gamma_{3}\in\mathcal{C}$. The two moves in Figure
2 will be referred to as triple point moves. In general, triple moves will
refer to any local move that is smoothly isotopic to either of the two moves
in Figure 2, possibly after switching orientations of arrows. The two moves in
the figure capture all triple moves, up to rotational symmetries. A triple
move applied to a configuration $\mathcal{C}$ is a local operation: there
exists a neighborhood $U\subset\Sigma$ such that $\mathcal{C}\cap U$ is as in
Figure 2 (left), the new configuration $\mathcal{C}^{\prime}$ coincides with
$\mathcal{C}$ outside of $U$, and $\mathcal{C}^{\prime}\cap U$ is as in Figure
2 (right). That is, this is a change in a configuration $\mathcal{C}$ which is
compactly supported, as the boundary conditions in the local model in Figure 2
coincide before and after the move.

Figure 2. The two triple point moves.

Let us study how the curve QP $(Q(\mathcal{C}),W(\mathcal{C}))$ behaves under
such triple point moves applied to $\mathcal{C}$.

######  Proposition 2.5.

Let $(Q(\mathcal{C}),W(\mathcal{C}))$ be a curve QP associated to
$\mathcal{C}$. Then $(Q(\mathcal{C}),W(\mathcal{C}))$ is invariant under
triple point moves, up to right-equivalence.

###### Proof.

There are two cases to consider for a triple point move, depending on the
orientations, see Figure 2 for the two cases. First, we consider the local
model in Figure 3 (left) and denote by $\gamma_{i}^{in}$, resp.
$\gamma_{i}^{out}$, the tail, resp. the head, of the segment of $\gamma_{i}$
in the local model; the head is where the arrow is drawn.

Figure 3. (Left) Local model for Case I before triple move. (Right) Local
quiver.

Let $R=(r_{ij})$ be the matrix such that $r_{ij}$ equals the sums of monomials
on $p_{lk}$ in $W(\mathcal{C})$ that are used by regions that intersect the
local model entering at $\gamma_{l}^{in}$ and exiting at $\gamma_{k}^{out}$,
from either side. Then the matrix $R$ reads

$R=\begin{pmatrix}0&p_{21}+p_{23}p_{31}&p_{31}\\\ p_{21}&0&p_{23}\\\
p_{31}&p_{23}&0\end{pmatrix}.$

Indeed, the second entry in the first row corresponds to regions that enter in
the upper right through $\gamma_{1}^{in}$ and exist through $\gamma_{2}^{out}$
on the upper left. There are two such regions: using $p_{21}$ or using
$p_{23}p_{31}$. Since both regions are oriented clockwise, there is an overall
positive sign, and since none of the quadrants being used are shaded, the sign
remains positive. The third entry of the first row is similar. The first entry
$p_{21}$ and the third entry $p_{23}$ on the second row are actually
$p_{21}=-(-p_{21})$ and $p_{23}=-(-p_{23})$: both of the regions they record
are oriented counter-clockwise, which gives an overall minus sign, and in
addition both of them use a unique quadrant which happens to be shaded. This
introduces another minus sign and therefore the entry has two minus signs,
thus a positive coefficient in the end. The first entry on the third row is
also $p_{31}=-(-p_{31})$, in that sense, whereas the second entry on the third
row has two positive signs directly. From now onwards we will compute with
both types of signs in mind (orientation sign and shaded quadrants signs),
without further specifying when a positive sign is actually an even number of
negative signs. The local quiver, recording the intersections that occur only
in the local model, is depicted in Figure 3 (right).

After the triple point move we have the local model in Figure 4 (left) and its
corresponding local quiver in Figure 4 (right). Note that the quivers in
Figure 3 (right) and Figure 4 (right), before and after, are identical: thus
it is clear that $Q(\mathcal{C})$ is invariant under this particular triple
point move.

Figure 4. (Left) Local model for Case I after triple move. (Right) Local
quiver.

The matrix of regions $R^{\prime}$ after the triple move is computed as with
$R$ above. It reads:

$R^{\prime}=\begin{pmatrix}0&q_{21}&q_{31}\\\ q_{21}-q_{23}q_{31}&0&q_{23}\\\
q_{31}&q_{23}&0\end{pmatrix}$

Note that the signs in the entry $q_{21}-q_{23}q_{31}$ are indeed correct: the
region associated to $q_{21}$ is oriented counter-clockwise and it uses a
shaded quadrant, thus it is positive, and the region associated to
$q_{23}q_{31}$ is oriented counter-clockwise and it uses two shaded quadrants,
thus it is negative. Let us now compare $R$ and $R^{\prime}$, which keep track
of the regions in the potential $W(\mathcal{C})$ before and after a triple
point move.

For that, consider the automorphism
$\phi\in\operatorname{Aut}(\mathbb{C}\langle Q(\mathcal{C})\rangle)$ of the
path algebra which is the identity on all arrows $p_{ij}$ except for
$(i,j)=(2,1)$ and sends the arrow $p_{21}$ to

$p_{21}\mapsto p_{21}-p_{23}p_{31}.$

After applying this automorphism, $R$ becomes

$\phi(R)=\begin{pmatrix}0&(p_{21}-p_{23}p_{31})+p_{23}p_{31}&p_{31}\\\
(p_{21}-p_{23}p_{31})&0&p_{23}\\\
p_{31}&p_{23}&0\end{pmatrix}=\begin{pmatrix}0&p_{21}&p_{31}\\\
p_{21}-p_{23}p_{31}&0&p_{23}\\\ p_{31}&p_{23}&0\end{pmatrix}.$

The matrix $R^{\prime}$ and $\phi(R)$ are now related by the trivial
relabeling $p_{ij}\longmapsto q_{ij}$. This shows that the quiver
$Q(\mathcal{C})$ remains invariant under this first case of the triple points
move and the right-equivalence class of $(Q(\mathcal{C}),W(\mathcal{C}))$ is
also invariant.

Second, we now consider the other local model for a triple move, as depicted
in Figure 5 (left). The local quiver is drawn in Figure 5 (right).

Figure 5. (Left) Local model for Case II before triple move. (Right) Local
quiver.

In the same notation as above, the matrix $R$ before the triple point move is

$R=\begin{pmatrix}0&p_{21}&p_{13}\\\ p_{21}&0&p_{32}\\\
p_{13}&p_{32}&0\end{pmatrix},$

except that in this case we also have the closed triangle region associated to
the monomial $p_{13}p_{32}p_{21}$, entirely contained in this local model.
Notice that this is a clockwise oriented triangle with none of its (interior)
quadrants being shaded: the sign is therefore positive.

Figure 6. (Left) Local model for Case II after triple move. (Right) Local
quiver.

After the triple point move we have the local model depicted in Figure 6
(left), whose quiver is illustrated in Figure 6 (right). Therefore, the quiver
$Q(\mathcal{C})$ also remains invariant under this second type of triple point
move. In the notation as above, the matrix $R^{\prime}$ after the triple point
move is

$R^{\prime}=\begin{pmatrix}0&q_{21}&q_{13}\\\ q_{21}&0&q_{32}\\\
q_{13}&q_{32}&0\end{pmatrix},$

and we also have the triangle region associated to the monomial
$q_{13}q_{32}q_{21}$, again entirely enclosed in this local model. The sign of
this triangle is indeed positive: it is oriented counter-clockwise and its
three (interior) quadrants are shaded. This accounts for a total of four
negative signs, and therefore a resulting positive sign. In this second type
of triple move the comparison before and after is given by the relabeling
$p_{ij}\mapsto q_{ij}$, which indeed maps $R$ to $R^{\prime}$ and the triangle
$p_{13}p_{32}p_{21}$ to the triangle $q_{13}q_{32}q_{21}$. Therefore the
quiver $Q(\mathcal{C})$ and the potential $W(\mathcal{C})$ are both identical
before and after this triple point move. This concludes the result. ∎

####  2.2.2. A property of local bigons.

An $\ell$-gon with $\ell=2$ will be referred to as a bigon $B\subset\Sigma$.
Note that a bigon $B\subset\Sigma$ must be oriented, either clockwise or
counter-clockwise.333If $\gamma_{i},\gamma_{j}\in\mathcal{C}$ bound an
“unoriented bigon”, then $Q(\mathcal{C})$ has two arrows from $\gamma_{i}$ to
$\gamma_{j}$, or viceversa. We do not consider this a bigon. It does not yield
a 2-cycle $Q(\mathcal{C})$ and (thus) it does not contribute a quadratic
monomial to $W(\mathcal{C})$. By definition, a bigon $B\subset\Sigma$ is said
to be local if $\mbox{int}(B)\cap\gamma_{i}=\emptyset$ for all
$\gamma_{i}\in\mathcal{C}$.

Given a local bigon $B\subset\Sigma$ as in Figure 7, we refer to the region
$\rho(B)$ drawn in red, resp. the region $\lambda(B)$ drawn in blue, as its
right region, resp. as its left region.

Figure 7. The regions $\rho(B)$ (red) and $\lambda(B)$ (blue). The bigons
depicted in orange.

######  Assumption 1.

Every local bigon $B\subset\Sigma$ is assumed to satisfy that $\lambda(B)$ is
a different region than $\rho(B)$, i.e. $\lambda(B)\neq\rho(B)$ for all local
bigons $B$. $\Box$

It will be proven in Section 3.4, specifically Proposition 3.15, that this
assumption is satisfied for the configurations of curves that we shall use,
i.e. for those configurations constructed in Section 3. For now we work under
Assumption 1: we suppose it holds for all local bigons discussed subsequently.

####  2.2.3. Behavior under bigon moves.

By definition, the local bigon moves are the local moves depicted in Figure 8.
This move applies to local bigons, i.e. bigons $B\subset\Sigma$ such that
$\mbox{int}(B)\cap\gamma_{i}=\emptyset$ for all $\gamma_{i}\in\mathcal{C}$.
This is the reason for referring to it as a local bigon move, instead of just
a bigon move.

Figure 8. The two local bigon moves.

In order to understand how $(Q(\mathcal{C}),W(\mathcal{C}))$ changes under
local bigon moves, we introduce the following:

######  Definition 2.6.

Let $(Q,W)$ be a QP and $a,b$ be two arrows in $Q$ such that $ab$ is a 2-cycle
and $ab$ appears as a quadratic monomial in $W$. The $ab$-reduction of
$(Q,W)$, or the local reduction $(Q,W)$ at $ab$, is the QP
$(Q^{\prime},W^{\prime})$ obtained as follows:

  * (i)

The quiver $Q^{\prime}$ coincides with $Q$ except that both arrows $a$ and $b$
have been erased.

  * (ii)

The potential $W^{\prime}$ is constructed as follows. Suppose that there exist
polynomials $U,V$, neither of them containing $a$ or $b$, such that

$W=(a-U)(b-V)+W^{\prime},$

where $W^{\prime}$ does not contain $a$ or $b$, and the equality is up to
cyclic permutation of each monomial. Then $W^{\prime}:=W-(a-U)(b-V)$.

If such polynomials $U,V$ do not exist, then the $ab$-reduction of $(Q,W)$ is
said not to exist. $\Box$

Definition 2.6 and the use of the word reduction for such an operation is a
direct influence of [DWZ10, Section 4]. Note that an oriented bigon
$B\subset\Sigma$ uniquely determines its two intersection points, and it is
uniquely determined by them if we know they bound a bigon. Equivalently, these
are two arrows $a,b$ in $Q(\mathcal{C})$ such that $ab$ is a 2-cycle and $ab$
is a monomial appearing in $W(\mathcal{C})$. In these cases, where $ab$ is the
2-cycle corresponding to an oriented bigon $B$, we also refer to an
$ab$-reduction as a $B$-reduction.

######  Lemma 2.7.

Let $(Q(\mathcal{C}),W(\mathcal{C}))$ be the curve QP associated to a curve
configuration $\mathcal{C}$, $B\subset\Sigma$ be a local bigon and
$\mathcal{C}^{\prime}$ the configuration $\mathcal{C}$ after a local bigon
move at $B$. Then the $B$-reduction of $(Q(\mathcal{C}),W(\mathcal{C}))$
exists and it equals $(Q(\mathcal{C}^{\prime}),W(\mathcal{C}^{\prime}))$.

###### Proof.

Let us consider a bigon $B\subset\Sigma$ bounded by two curves $\gamma_{1}$
and $\gamma_{2}$. There are two cases, depending on whether the bigon $B$ is
oriented clockwise or counter-clockwise. The two cases are almost identical
and thus we focus on that of a clockwise oriented bigon, as depicted in Figure
9 (left). The local quiver is drawn in Figure 9 (right) and contains the
2-cycle $p_{12}p_{21}$. Since the bigon is oriented clockwise and none of the
two quadrants of the bigon is shaded, its contribution to the potential
$W(\mathcal{C})$ is the monomial $p_{12}p_{21}$.444In the case of a counter-
clockwise oriented bigon, the bigon would contribute to the potential with
$-p_{12}p_{21}$. There would be a minus sign because the bigon would be
oriented counter-clockwise and use exactly two shaded quadrants.

Figure 9. (Left) Local model for a bigon before bigon move. The intersection
points are highlighted (red) and we have shaded the quadrants. (Right) Local
quiver.

First, let us prove that the $B$-reduction of
$(Q(\mathcal{C}),W(\mathcal{C}))$ exists.

Let $S_{21}$ be the set of polygons that use the south-west corner of
$p_{21}$, i.e. the set of polygons that come in from $\gamma_{2}^{in}$, turn
left at $p_{21}$ and exit via $\gamma_{1}^{out}$.555There are no polygons
using the north-west or south-east quadrants at $p_{21}$ because of
orientations. Similarly, there are no polygons using the north-east and south-
west quadrants at $p_{12}$. A polygon in $S_{21}$ has the opposite orientation
from the bigon $B$ and contains $p_{21}$ in its associated monomial in
$W(\mathcal{C})$. Therefore, the contribution from polygons in $S_{21}$ to the
potential $W(\mathcal{C})$ has the form $-Up_{21}$ for some polynomial $U$
that does not contain $p_{21}$. Similarly, let $S_{12}$ be the set of polygons
that use the south-east corner of $p_{12}$, i.e. the set of polygons that come
from $\gamma_{1}^{in}$, turn down at $p_{12}$ and exit via $\gamma_{2}^{out}$.
The contribution from the polygons in $S_{12}$ to the potential
$W(\mathcal{C})$ has the form $-Vp_{12}$ for some polynomial $V$ that does not
contain $p_{12}$.

By Assumption 1, the polynomial $U$ does not contain $p_{12}$ either. Indeed,
any polygon in $S_{21}$ uses $\lambda(B)$. If its contribution to the
potential contained $p_{12}$, then $\lambda(B)=\rho(B)$ as it would be using
the region $\rho(B)$ and polygons contributing to $W(\mathcal{C})$ are
embedded. Similarly $V$ does not contain $p_{21}$ either. Let us write the
potential as

$W(\mathcal{C})=p_{12}p_{21}-Up_{21}-Vp_{12}+\tilde{W}.$

for some $\tilde{W}$. Now, monomials in $W(\mathcal{C})$ are precisely given
by boundaries of embedded regions, and thus their associated cycles in the
quiver are irreducible, i.e. not the composition of two cycles. Therefore, the
construction above is such that $\tilde{W}$ does not contain $p_{12}$ nor
$p_{21}$. Following the formulation in Definition 2.6, we rewrite

$W(\mathcal{C})=(p_{12}-U)(p_{21}-V)-UV+\tilde{W},$

where $\tilde{W}$ does not contain $p_{12}$ or $p_{21}$. Therefore, we can
select $W^{\prime}:=-UV+\tilde{W}$ in Definition 2.6. Thus we conclude that
$(Q(\mathcal{C}),W(\mathcal{C}))$ is indeed $B$-reducible.

Second, let us now show that the $B$-reduction of
$(Q(\mathcal{C}),W(\mathcal{C}))$ equals
$(Q(\mathcal{C}^{\prime}),W(\mathcal{C}^{\prime}))$. After the bigon move at
$B$ we have the local configuration in Figure 10 (left). The local quiver
$Q^{\prime}(\mathcal{C})$ becomes two vertices with no arrows between them, as
there are no intersections in this local piece; it is depicted in Figure 10
(right). For the same reason, the potential $W(\mathcal{C}^{\prime})$ after
the bigon move has no contributions coming from this local configuration.

Figure 10. (Left) Local model after bigon move. (Right) Local quiver.

By comparing the two quivers directly, $Q(\mathcal{C}^{\prime})$ is obtained
from $Q(\mathcal{C})$ by exactly removing the arrows $p_{12}$ and $p_{21}$. We
claim that the $B$-reduction of the curve potential $W(\mathcal{C})$ equals
the curve potential after the bigon removal, namely
$W^{\prime}=W(\mathcal{C}^{\prime})$, where $W^{\prime}=-UV+\tilde{W}$ as
above. Indeed, since the bigon region disappears, $p_{12}p_{21}$ is not in
$W(\mathcal{C}^{\prime})$. Also, any region in $S_{21}$ itself disappears
under the bigon move, and thus the polynomial term $-Up_{12}$ no longer
appears in $W(\mathcal{C}^{\prime})$. The argument for $-Vp_{21}$ is
identical. This justifies that $p_{12}p_{21}$, $-Up_{12}$ and $-Vp_{21}$ must
be subtracted from $W(\mathcal{C})$ to obtain the potential
$W(\mathcal{C}^{\prime})$. It suffices to prove that the term $UV$ must be
added (with signs) to $W(\mathcal{C}^{\prime})$. Indeed, when we remove the
local bigon, a polygon in $S_{21}$ and a polygon in $S_{12}$ will be connected
along the strip between $\gamma_{1}$ and $\gamma_{2}$, creating new polygons
that contribute $-UV$. This leads to the addition of $-UV$ in the potential
$W(\mathcal{C}^{\prime})$. In conclusion, the curve QP
$(Q(\mathcal{C}^{\prime}),W(\mathcal{C}^{\prime}))$ is the $B$-reduction of
$(Q(\mathcal{C}),W(\mathcal{C}))$, as required. ∎

Before moving forward to the next subsection, we introduce the following
definition:

######  Definition 2.8.

A configuration of curves $\mathcal{C}$ such that there is no bigon bounded by
$\mathcal{C}$ is said to be reduced. A configuration which is not reduced is
said to be non-reduced. $\Box$

####  2.2.4. Hass-Scott algorithm and the reduced part of
$(Q(\mathcal{C}),W(\mathcal{C}))$.

Consider a non-reduced configuration $\mathcal{C}_{0}$ with a collection of
bigons $\\{B_{1},\ldots,B_{m}\\}$. Note that these bigons $B_{i}$ might not be
local, i.e. the interior of each $B_{i}$ might intersect curves in
$\mathcal{C}_{0}$ in a non-empty set. The first step is to modify
$\mathcal{C}_{0}$ into a reduced configuration. The Hass-Scott algorithm
[HS85] allows us to do this:

######  Theorem 2.9 ([HS85]).

Let $\mathcal{C}_{0}$ be a non-reduced configuration with a collection of
bigons $\\{B_{1},\ldots,B_{m}\\}$. Then, for any $i\in[m]$, there exists a
sequence of triple point moves and one local bigon move on $\mathcal{C}_{0}$
that yields a new configuration $\mathcal{C}_{1}$ such that the collection of
bigons of $\mathcal{C}_{1}$ is $\\{B_{1},\ldots,B_{m}\\}\setminus\\{B_{i}\\}$.
$\Box$

The proof of Theorem 2.9 is local on a given bigon $B_{i}$, in that it only
modifies the configuration $\mathcal{C}_{0}$ in a neighborhood of $B_{i}$. It
is worth remarking that the Hass-Scott algorithm is able to remove a bigon
without introducing further (unnecessary) intersections: just with triple
points moves any bigon (possibly non-local) becomes a local bigon. In summary,
given a non-reduced configuration $\mathcal{C}\subset\Sigma$, the algorithm in
[HS85] implies that there exists a sequence of moves, both possibly containing
and intertwining triple point moves and local bigon moves, that when applied
to $\mathcal{C}$ yields a reduced configuration.

###  2.3. Curve QPs under $\gamma$-exchange

We introduce the following operation on a configuration $\mathcal{C}$.

####  2.3.1. Definition of $\gamma$-exchange of $\mathcal{C}$

Let $\gamma\in\mathcal{C}$ be a curve. By definition, the configuration of
curves $\mu_{\gamma}(\mathcal{C})$ is given by the configuration of curves

$\mu_{\gamma}(\mathcal{C}):=\\{\mu_{\gamma}(\gamma_{1}),\ldots,\mu_{\gamma}(\gamma_{b})\\},$

where the curves $\mu_{\gamma}(\gamma_{i})$ are obtained as follows,
$i\in[b]$.

Consider a neighborhood $U\subset\Sigma$ of $\gamma$ such that any curves in
$\mathcal{C}\setminus\\{\gamma\\}$ intersect $U$ as depicted in Figure 13.
That is, up to planar isotopy, each curve
$\gamma_{i}\in\mathcal{C}\setminus\\{\gamma\\}$ intersects $U$ at a collection
of intervals $\\{I_{i}^{p}\\}$, $p\in[q_{i}]$ for some $q_{i}\in\mathbb{N}$,
where each such interval only intersects $\gamma$ once and intervals do not
intersect each other; note that this collection $\\{I_{i}^{p}\\}$ might be
empty for some curves $\gamma_{i}\in\mathcal{C}$. The neighborhood $U$ must be
a cylinder, since it is the neighborhood of an embedded connected curve in an
oriented surface $\Sigma$.

The curves $\mu_{\gamma}(\gamma_{i})$ in $\mu_{\gamma}(\mathcal{C})$ are
constructed as follows. We apply a positive Dehn twist of the cylinder $U$
along the simple embedded curve $\gamma$ to all the segments in $U$ that
intersect $\gamma$ positively, depicted in blue in Figure 13, and the identity
map to all the segments in $U$ that intersect $\gamma$ negatively, depicted in
green in Figure 13. Note that a Dehn twist often refers to a mapping class,
i.e. an element of $\pi_{0}(\mbox{Diff}^{c}(U))$. In this construction we
explicit mean a representative of that mapping class: at this stage we make
one such choice of representative and continue.

Since both a Dehn twist and the identity are (represented by) compactly
supported diffeomorphisms, each resulting segment $f(I_{i}^{p})\subset U$, $f$
a Dehn twist or the identity, can be glued to the corresponding curve
$\gamma_{i}\cap(\Sigma\setminus U)$. By definition, the result of applying
such operation to $\gamma_{i}$ is the curve $\mu_{g}(\gamma_{i})$ if
$\gamma_{i}\neq\gamma$. We define $\mu_{\gamma}(\gamma):=-\gamma$. See Figure
14 for the result of applying $\mu_{\gamma}$ to Figure 13.

######  Remark 2.10.

The choices in the above construction will not affect any aspects of our
results. For instance, the choice of neighborhood $U$ with the required
properties only modifies the resulting configuration
$\mu_{\gamma}(\mathcal{C})$ by a global isotopy of $\Sigma$. Similarly, the
choice of representative of the Dehn twist class in
$\pi_{0}(\mbox{Diff}^{c}(U))$ results in the same configuration
$\mu_{\gamma}(\mathcal{C})$ up to global isotopy. $\Box$

By Remark 2.10 and the fact that we only consider configurations $\mathcal{C}$
up to a global diffeomorphism of $\Sigma$, the resulting configuration
$\mu_{\gamma}(\mathcal{C})$ is well-defined. We therefore define:

######  Definition 2.11.

The configuration of curves $\mu_{\gamma}(\mathcal{C})$ is said to be the
$\gamma$-exchange of $\mathcal{C}$. $\Box$

Note that applying a $\gamma$-exchange twice along the same curve, first to
$\gamma$ and then to $-\gamma$, leads to the same configuration; same up to an
overall compactly supported diffeomorphism of $\Sigma$. Indeed, consecutively
applying a $\gamma$-exchange at the same vertex yields the configuration
$\mu_{-\gamma}\mu_{\gamma}(\mathcal{C})=\tau_{\gamma}(\mathcal{C})$, given by
applying a (representative of a) Dehn twist
$\tau_{\gamma}\in\mbox{Diff}^{c}(\Sigma)$ to all the curves in $\mathcal{C}$.
In that sense, a $\gamma$-exchange is an involution.

Two comments on Definition 2.11:

  * (i)

Since we only apply a Dehn twist to some collection of
$\gamma_{i}\in\mathcal{C}$, it is possible (and it occurs) that the smooth
representatives we have constructed for the curves in
$\mu_{\gamma}(\mathcal{C})$ bound bigons. That is, $\mu_{\gamma}(\mathcal{C})$
might be non-reduced even if $\mathcal{C}$ is reduced.

  * (ii)

If a configuration $\mathcal{C}$ of simple embedded curves is non-reduced and
$\gamma,\gamma^{\prime}$ are curves bounding a bigon, then
$\mu_{\gamma}(\mathcal{C})$ contains the immersed curve
$\mu_{\gamma}(\gamma^{\prime})$.

These two comments highlight an important principle: if we want to perform a
sequence of $\gamma$-exchanges, we must guarantee that
$\mu_{\gamma}(\mathcal{C})$ is reduced so that we can iterate the procedure,
as $\gamma$-exchanges are not defined for immersed curves $\gamma$. The saving
grace is that we might be able to perform triple moves and bigon moves to
$\mu_{\gamma}(\mathcal{C})$ so that a configuration becomes reduced, following
Theorem 2.9. Our focus thus now shifts towards understanding how polygons
bounded by $\mathcal{C}$ change under a $\gamma$-exchange. In other words,
understanding how the right-equivalence class of the curve QP
$(Q(\mathcal{C}),W(\mathcal{C}))$ changes under a $\gamma$-exchange.

######  Remark 2.12 (Algebraic topological $\gamma$-exchange).

There is a variation on Definition 2.11 which is also natural. Namely, we
define $\mu_{\gamma}(\gamma_{i})=\tau_{\gamma}(\gamma_{i})$ if the algebraic
intersection number $\langle\gamma_{i},\gamma\rangle$ is strictly positive (or
$\gamma_{i}=\gamma$) and $\mu_{\gamma}(\gamma_{i})=\gamma_{i}$ otherwise. Here
$\tau_{\gamma}$ denotes a positive Dehn twist along $\gamma$ and we use the
skew-symmetric intersection pairing in $H_{1}(\Sigma,\mathbb{Z})$. For the
configurations $\mathcal{C}$ that we will study, which are always reduced,
have algebraic intersection numbers equal to geometric intersection numbers
and all $\gamma_{i}\in\mathcal{C}$ be primitive, this alternative definition
coincides with Definition 2.11. In general, it does not coincide: consider a
$\gamma$-exchange along a null-homologous curve $\gamma$ which intersects
another $\gamma_{i}$ exactly at a positive and negative intersection points,
bounding a bigon. Definition 2.11 would produce an immersed representative of
the homology class of $\gamma_{i}$, and the definition in this remark would
keep $\gamma_{i}$ identically the same. $\Box$

######  Remark 2.13.

To our knowledge, Definition 2.11 or its variation in Remark 2.12 do not
appear in the published literature. That said, the unpublished draft [KQ,
Section 3] indeed contains the definition in Remark 2.12. In their context it
is introduced as an analogue of simple tilting on hearts for $t$-structures in
triangulated categories, following the homological construction of forward and
backward tilts in [HRS96]. Similarly, the unpublished article [STW16,
Definition 2.3] states the definition in Remark 2.12. As far as we are aware,
no reference has previously defined a potential for such configurations. In
fact, studying such potentials is already stated as an open problem in [STW16,
Problem 1.15]. $\Box$

####  2.3.2. QP-mutation according to Derksen-Weyman-Zelevinsky

In order to understand how the right-equivalence class of the curve QP
$(Q(\mathcal{C}),W(\mathcal{C}))$ changes under a $\gamma$-exchange, we recall
the notion of QP-mutation introduced in [DWZ10, Section 5]. Let $Q$ be a
quiver with set of vertices $Q_{0}$, set of arrows $Q_{1}$ and let us denote
$h(a)\in Q_{0}$, resp. $t(a)\in Q_{0}$, the head, resp. the tail, of an arrow
$a\in Q_{1}$.

Let $(Q,W)$ be a quiver with potential. Consider a vertex $v_{k}\in Q_{0}$ and
assume that $v_{k}$ does not belong to an oriented 2-cycle and that no
monomial in $W$ starts and ends with $v_{k}$. The former hypothesis will
always be satisfied in the applications of this manuscript, the latter
hypothesis can be always guaranteed after cyclically changing some monomials
in the potential.

######  Definition 2.14 ([DWZ10]).

The non-reduced QP-mutation of $(Q,W)$ at $v_{k}$, satisfying the above
hypotheses, is the QP $(\mu_{k}(Q),\mu_{k}(W))$ defined as follows.

The quiver $\mu_{k}(Q)$ has the same set of vertices $\mu_{k}(Q)_{0}=Q_{0}$ as
$Q$ and its set of arrows $\mu_{k}(Q)_{1}$ is obtained from $Q_{1}$ according
to the following procedure:

  1. (1)

All arrows in $Q_{1}$ not incident to $v_{k}$ also belong to $\mu_{k}(Q)_{1}$.

  2. (2)

For each pair $a,b\in Q_{1}$ of incoming arrow $a$ and outgoing arrow $b$ at
$v_{k}$, create the composite arrow $[ba]\in\mu_{k}(Q)_{1}$.

  3. (3)

Replace each incoming arrow $a\in Q_{1}$ (resp. each outgoing arrow $b\in
Q_{1}$) at $v_{k}$ by a corresponding arrow $a^{\ast}\in\mu_{k}(Q)_{1}$ (resp.
$b^{\ast}\in\mu_{k}(Q)_{1}$) now oriented in the opposite way.

The potential $\mu_{k}(W)$ is defined as

$\mu_{k}(W):=[W]+\Delta_{k},\quad\Delta_{k}:=\sum_{a,b\in
Q_{1},t(a)=h(b)=k}[ba]b^{\ast}a^{\ast}$

where $[W]$ is obtained by substituting the composite arrow $[ab]$ for each
factor $ab$ with $t(a)=h(b)=k$ of any cyclic path occurring in the expansion
of $W$ that contains $ab$. $\Box$

By definition, the QP $(\mu_{k}(Q),\mu_{k}(W))$ is said to be obtained by non-
reduced QP-mutation of $(Q,W)$ at the vertex $v_{k}$. It is shown in [DWZ10,
Theorem 5.2] that the right-equivalence class of $(\mu_{k}(Q),\mu_{k}(W))$
depends only on the right-equivalence class of $(Q,W)$.

######  Remark 2.15.

The mutation $\mu_{k}(Q)$ of a quiver $Q$, without the potentials $W$ and
$\mu_{k}(W)$, was previously defined in [FZ02, Definition 4.2]. $\Box$

The result of a QP-mutation of $(Q,W)$ with no 2-cycles might result in a QP
$(\mu_{k}(Q),\mu_{k}(W))$ with 2-cycles. In [DWZ10] the notions of reduced and
trivial QPs are introduced, as follows. A QP $(Q,W)$ is said to be reduced if
the degree-2 homogeneous part $W^{(2)}$ of $W$ is trivial, i.e. $W^{(2)}=0$.
That is, $(Q,W)$ is reduced if $W$ contains no quadratic monomial terms, i.e.
no terms of the form $ab$, $a,b\in Q_{1}$. A QP $(Q,W)$ is said to be trivial
if $W$ is entirely quadratic, i.e. $W\in\mathbb{C}\langle Q\rangle^{(2)}$
belongs to the degree-2 homogenous part of the path algebra $\mathbb{C}\langle
Q\rangle$ and the Jacobian algebra of $(Q,W)$ is isomorphic to $\mathbb{C}$.

######  Remark 2.16.

By [DWZ10, Prop. 4.4.], there is a more pragmatic criterion to detect
triviality: $(Q,P)$ is trivial if and only if the set of arrows $Q_{1}$
consists of $2N$ distinct arrows $a_{1},b_{1},\ldots,a_{N},b_{N}$ such that
each $a_{k}b_{k}$ is a cyclic 2-path, and there is a change of arrows
$\varphi$ such that $\varphi(W)$ is cyclically equivalent to
$a_{1}b_{1}+\ldots+a_{N}b_{N}$. $\Box$

######  Remark 2.17.

Note that the quiver $Q$ is not enough to determine the reduced and trivial
parts of a QP $(Q,W)$. For instance, the quiver $Q$ consisting of two vertices
$Q_{0}=\\{v_{1},v_{2}\\}$ and two arrows $Q_{1}=\\{a,b\\}$ with
$h(a)=t(b)=v_{1}$, $t(a)=h(b)=v_{2}$ is trivial if the potential is chosen to
be $W=ab$, and it is reduced if $W=0$ is chosen to vanish. $\Box$

The following structural result is established in [DWZ10, Theorem 4.6]:

######  Theorem 2.18 ([DWZ10]).

For every $QP$ $(Q,W)$ with trivial arrow span $Q_{triv}$ and reduced arrow
span $Q_{red}$, there exist a trivial QP $(Q_{triv},W_{triv})$ and a reduced
QP $(Q_{red},W_{red})$ such that $(Q,W)$ is right-equivalent to the direct sum
$(Q_{triv},W_{triv})\oplus(Q_{red},W_{red})$. Also, the right-equivalence
class of each of the QPs $(Q_{triv},W_{triv})$ and $(Q_{red},W_{red})$ is
determined by the right-equivalence class of $(Q,W)$.

This allows us to finally define QP-mutation:

######  Definition 2.19 ([DWZ10]).

The QP-mutation of $(Q,W)$ at $v_{k}$ is the QP
$(\mu_{k}(Q)_{red},\mu_{k}(W)_{red})$ given by the reduced part of the non-
reduced mutation $(\mu_{k}(Q),\mu_{k}(W))$. $\Box$

This is an involutive operation if performed at the same vertex $v_{k}$
consecutively, i.e. performing QP-mutation of $(Q,W)$ at $v_{k}$ twice
consecutively leads to $(Q,W)$ again. Note also that Definition 2.6 of local
reduction is one step towards extracting the reduced part of a QP.

####  2.3.3. Reduced part for curve QPs

Let $\mathcal{C}$ be a curve configuration and
$(Q(\mathcal{C}),W(\mathcal{C}))$ its associated curve QP. The Hass-Scott
algorithm, as stated in Theorem 2.9 above, geometrically explains Theorem 2.18
in the case of curve QPs. Indeed, consider a non-reduced configuration
$\mathcal{C}_{0}$ with a non-empty collection of bigons
$\\{B_{1},\ldots,B_{m}\\}$. Then its associated curve QP
$(Q(\mathcal{C}),W(\mathcal{C}))$ is not reduced: by definition, it contains
one 2-cycle in $Q(\mathcal{C})$ for each bigon, and such 2-cycles each appear
as quadratic monomial in $W(\mathcal{C})$.

######  Definition 2.20.

Let $\mathcal{C}$ be a configuration with a collection of bigons
$\\{B_{1},\ldots,B_{m}\\}$. Any reduced curve configuration
$\mathcal{C}_{red}$ obtained by iteratively applying the Hass-Scott algorithm
(Theorem 2.9) $m$ times to $\mathcal{C}$, where at each time exactly one bigon
is eliminated, is said to be a reduction of $\mathcal{C}$. $\Box$

Namely, Theorem 2.9 allows us to remove one bigon at a time, by applying a
sequence of triple point moves and then a local bigon move. By Proposition
2.5, the sequence of triple point moves does not change the right-equivalence
class of $(Q(\mathcal{C}),W(\mathcal{C}))$. Each time that we apply Theorem
2.9 we need exactly one local bigon move. By Lemma 2.7, the QP
$(Q(\mathcal{C}),W(\mathcal{C}))$ then undergoes a $B$-reduction at a bigon
$B$. By iteratively applying Theorem 2.9, we obtain the following:

######  Lemma 2.21.

Let $\mathcal{C}$ be a configuration, $(Q(\mathcal{C}),W(\mathcal{C}))$ its
associated curve QP, $(Q(\mathcal{C})_{red},W(\mathcal{C})_{red}))$ its
reduced QP part, and $\mathcal{C}_{red}$ a reduction of the configuration
$\mathcal{C}$. Then

  * (i)

$(Q(\mathcal{C}_{red}),W(\mathcal{C}_{red}))=(Q(\mathcal{C})_{red},W(\mathcal{C})_{red}))$,
up to right equivalence.

  * (ii)

If $(Q(\mathcal{C}),W(\mathcal{C}))$ is non-degenerate, then
$Q(\mathcal{C}_{red})$ has no 2-cycles.

The proof of Lemma 2.21 uses the notion of a non-degenerate QP, discussed in
Section 3.3 below. This proof is thus postponed until Section 3.5. Note that
$Q(\mathcal{C}_{red})$ might a priori have 2-cycles, even if there are no
bigons bounded by $\mathcal{C}_{red}$, cf. Remark 2.17. The non-degeneracy of
a quiver with potential precisely rules out these type of situations, where a
2-cycle in $Q$ is not kept track by the potential $W$.

####  2.3.4. Curve QP under $\gamma$-exchanges change via QP-mutations

We conclude this section by relating the $\gamma$-exchanges in Definition 2.11
to the QP-mutation in Definition 2.19.

######  Proposition 2.22.

Let $(Q(\mathcal{C}),W(\mathcal{C}))$ be a curve QP associated to
$\mathcal{C}$ and $\gamma\in\mathcal{C}$. Then, possibly after applying a
sequence of triple point moves and bigon moves to $\mu_{\gamma}(\mathcal{C})$,
the QP $(Q(\mu_{\gamma}(\mathcal{C})),W(\mu_{\gamma}(\mathcal{C})))$ is right-
equivalent to the QP-mutation of the QP $(Q(\mathcal{C}),W(\mathcal{C}))$ at
the vertex $\gamma$.

###### Proof.

Given Definition 2.19, there are three pieces to justify:

  1. (1)

The change in the quiver $Q(\mathcal{C})$ under $\gamma$-exchange.

  2. (2)

The change in the potential $W(\mathcal{C})$ under $\gamma$-exchange.

  3. (3)

The reduced part of the non-reduced QP-mutation is indeed
$(Q(\mu_{\gamma}(\mathcal{C})_{red}),W(\mu_{\gamma}(\mathcal{C})_{red}))$.

Parts (1) and (2) will be argued directly, as they indeed need a new
computation. In fact, we will show that
$(Q(\mu_{\gamma}(\mathcal{C})),W(\mu_{\gamma}(\mathcal{C})))$ is the non-
reduced QP-mutation
$(\mu_{\gamma}(Q(\mathcal{C})),\mu_{\gamma}(W(\mathcal{C})))$ of
$(Q(\mathcal{C}),W(\mathcal{C}))$ at the vertex associated to $\gamma$. Part
(3) follows directly from Lemma 2.21 applied to the configuration
$\mu_{\gamma}(\mathcal{C})$, once
$(Q(\mu_{\gamma}(\mathcal{C})),W(\mu_{\gamma}(\mathcal{C})))=(\mu_{\gamma}(Q(\mathcal{C})),\mu_{\gamma}(W(\mathcal{C})))$
is proven.

First, we focus on the case of $\gamma$ and just two intersecting intervals
$\tau_{+}$ and $\tau_{-}$, which intersect $\gamma$ in two points $p_{+}$ and
$p_{-}$ with opposite signs. The core computations in the general case
essentially reduce to this situation. We have depicted this configuration in
Figure 11 (left), where the corresponding local quiver is drawn beneath the
configuration. The local quiver is the linear $A_{3}$-quiver, with a unique
arrow $p_{+}$ from (the vertex associated to) $\tau_{+}$ to $\gamma$, and a
unique arrow $p_{-}$ from $\gamma$ to $\tau_{-}$. In Figure 11 (and the
upcoming figures in this proof) we always identify the right hand side of the
figure with the left hand side via the identity map: these are all
configurations drawn in the resulting cylinder; indeed, $\gamma$ is a circle
and it is depicted as a flat horizontal segment with its right endpoint being
identified with its left endpoint.

Figure 11. The effect of $\gamma$-exchange in the case of exactly one positive
and one negative intersections. The configuration before the $\gamma$-exchange
is depicted on the left, and the configuration after the $\gamma$-exchange is
depicted on the right. The local quivers recording the intersection patterns
are drawn under the configurations.

After performing a $\gamma$-exchange, according to Definition 2.11, the
resulting configuration is as depicted in Figure 11 (right). The local quiver,
which is drawn right beneath the configuration, is the quiver obtained by
quiver mutation at (the vertex corresponding to $\gamma$), according to
Definition 2.19. Indeed, the previous two intersection points $p_{\pm}$
persist in the configuration but since the orientation of $\gamma$ is
opposite, their associated arrows go from $\tau_{-}$ to $\gamma$, for $p_{-}$,
and from $\gamma$ to $\tau_{+}$, for $p_{+}$. As illustrated in Figure 11, a
third intersection point $q$ is created after the $\gamma$-exchange: it is an
intersection between $\tau_{-}$ and $\tau_{+}$. This yields a new arrow $q$ in
the quiver which is precisely the composite arrow $q=[p_{-}p_{+}]$. In
conclusion, for these local configurations, we have verified that
$Q(\mu_{g}(\mathcal{C}))$ equals $\mu_{g}(Q(\mathcal{C}))$.

Let us now study the change of the potential $W(\mathcal{C})$ under
$\gamma$-exchange, also in these particular configurations just with $\gamma$,
$\tau_{+}$ and $\tau_{-}$. For that we need to understand how polygons bounded
by $\mathcal{C}$ change under $\gamma$-exchange. The key image is Figure 12,
that we now explain.

Figure 12. (Left) Potential pieces of polygons in $\mathcal{C}$ that have
$p_{-}$ and $p_{+}$ as vertices: the regions are highlighted in yellow. The
diagram in the third row of this left column represents an empty region.
(Right) Potential pieces of polygons in $\mathcal{C}$ that have $p_{-},p_{+}$
and $q$ as vertices, now after a $\gamma$-exchange.

Before the $\gamma$-exchange, the potential $W(\mathcal{C})$ records polygons
in $\mathcal{C}$. Consider the intersection of any such polygon recorded by
$W(\mathcal{C})$ with the region $U$ and assume again that we obtain the
configuration with just $\gamma$, $\tau_{+}$ and $\tau_{-}$, intersecting as
above. It suffices to focus on polygons that use $p_{-}$ and $p_{+}$. In this
configuration, there are exactly two, drawn in the first and second rows of
Figure 12 (left). The potential $W(\mathcal{C})$ records such polygons with
monomials (on the arrows of $\mathcal{C}$) that contain $p_{-}p_{+}$.

After the $\gamma$-exchange, as drawn in Figure 12 (right), those polygons
still exist but now must use the new intersection point $q$.666From the
perspective of $H_{1}(U;\mathbb{Z})$, this intersection point appears because
of the Picard-Lefschetz formula in its most elementary setting: explaining how
homology classes change under Dehn twists. Indeed, the boundary conditions for
the polygons are given (entering in $\tau_{\pm}$ and exiting in $\tau_{\mp}$)
and there is a unique manner in which those region can exist in the
configurations of Figure 12 (right). This is precisely the term
$[W(\mathcal{C})]$ in Definition 2.19 of the mutation of the potential: each
time we see $p_{-}p_{+}$ in a monomial in $W(\mathcal{C})$ we must substitute
it by the composite arrow $q=[p_{-}p_{+}]$. There is an additional
contribution to $W(\mu_{\gamma}(\mathcal{C}))$: there is a triangle, as drawn
in the third row of Figure 12. Since its vertices are $q,p_{+}$ and $p_{-}$,
it contributes the monomial $qp_{+}p_{-}$ to $W(\mu_{\gamma}(\mathcal{C}))$,
which is precisely the $\Delta_{\gamma}$ term in Definition 2.19. Therefore,
we have

$W(\mu_{\gamma}(\mathcal{C}))=[W(\mathcal{C})]+qp_{+}p_{-}=[W(\mathcal{C})]+\Delta_{\gamma}=\mu_{\gamma}(W(\mathcal{C})).$

Thus far we have argued
$(Q(\mu_{\gamma}(\mathcal{C})),W(\mu_{\gamma}(\mathcal{C})))=(\mu_{\gamma}(Q(\mathcal{C})),\mu_{\gamma}(W(\mathcal{C})))$
for the configuration with $\gamma$, $\tau_{+}$ and $\tau_{-}$ as in Figure
11. The general case is concluded as follows.

Consider a neighborhood $U$ of $\gamma$ such that777Such a neighborhood always
exists because $\mathcal{C}$ has finitely many curves and they all intersect
transversely. the configuration $\mathcal{C}$ intersected with $U$ is as
depicted in Figure 13. That is, only curves $\gamma_{i}\in\mathcal{C}$ that
intersect $\gamma$ do intersect $U$ and the intersections are (up to planar
isotopy) straight vertical segments that either point upwards or downwards and
only intersect $\gamma$ once (and these segments do not intersect each other
in $U$).

Figure 13. The general configuration $\mathcal{C}$ in a neighborhood of
$\gamma$ before a $\gamma$-exchange.

Choose an arbitrary but fixed point $g\in\gamma$. As we scan the arrows
starting at $g$ and move in the direction given by the orientation of
$\gamma$, there will be:

  * -

A number of blocks $U_{1},\ldots,U_{v}$, $v\in\mathbb{N}$, of arrows pointing
upward, each with $u_{i}:=|U_{i}|$ arrows, $i\in[v]$. The curves in $U_{i}$
will be labeled by $u_{i,j}$, $i\in[v]$, $j\in[u_{i}]$, where the index $j$ is
ordered left to right as we transverse $\gamma$ according to its orientation.

  * -

A number of blocks $D_{1},\ldots,D_{e}$, $e\in\mathbb{N}$, of arrows pointing
upward, each with cardinality $d_{i}:=|D_{i}|$, $i\in[e]$. The curves in
$D_{i}$ will be labeled by $d_{i,j}$, $i\in[e]$, $j\in[d_{i}]$, where the
index $j$ is ordered left to right as we transverse $\gamma$ according to its
orientation.

Figure 14 illustrates this notation in a specific example. Now, for every pair
of curves $u_{i,j}$ and $d_{l,k}$, the configuration of three curves $\gamma$,
$u_{i,j}$ and $d_{l,k}$ is precisely the local configuration we studied above,
as in Figure 11 (left). Since a $\gamma$-exchange fixes all the $u_{i,j}$ and
applies a Dehn twist to all the $d_{l,k}$, the behaviour of any such pair
under $\gamma$-exchange is identical and it coincides with the one studied
above, see Figure 11 (right). As emphasized in Subsection 2.3.1, a choice of
representative for the (mapping class of the) Dehn twist must be made: we
specifically choose a compactly supported diffeomorphism
$f\in\mbox{Diff}^{c}(U)$ which acts on the set of curves $d_{l,k}$ as
drawn888We drew the blue curves $d_{l,k}$ as PL-curves for convenience: the
curves are smooth, and we smooth this PL depiction by an arbitrarily smoothing
of the corners, so no further intersections are created.  in Figure 14.
Namely, each curve $f(d_{l,k})$ intersects each $u_{i,j}$ and $\gamma$ exactly
once.999Such a choice exists: the standard model for the Dehn twist defined
via the geodesic flow in the punctured disk bundle (extended by the antipodal
map to the zero section) has this property.

Figure 14. The general configuration $\mathcal{C}$ in a neighborhood of
$\gamma$ after a $\gamma$-exchange.

Since the only intersections before the $\gamma$-exchange are between
$u_{i,j}$ and $\gamma$, or $d_{l,k}$ and $\gamma$, the new intersections
afterwards will be precisely as recorded by the local configuration in Figure
11 (right). Note that this applies to every pair of $u_{i,j}$, $d_{l,k}$ and
hence for every pair of arrows in-and-out of $\gamma$ in $Q(\mathcal{C})$ we
will add the composite arrow (and flip that pair of arrows). Thus the quiver
changes according to a mutation at $\gamma$ also in this general case.
Similarly, since the only regions in Figure 13 using intersection points with
$\gamma$ are precisely those entering a $u_{i,j}$ and exiting a $d_{l,k}$, or
vice-versa, the change of polygons in $\mathcal{C}$ will be recorded by
tracking all the polygons that appear in the local configuration in Figure 11
(right) for all pairs of $u_{i,j}$ and $d_{l,k}$. We have already argued that
the potential in each local configuration precisely changes as a (non-reduced)
QP-mutation, and therefore it is also the case for the potential in the
general configuration. ∎

##  3\. Non-degeneracy of curve QPs for plabic fences

In this section we construct curve configurations $\mathcal{C}$ from a certain
type of plabic graphs and show rigidity for their associated curve QP
$(Q(\mathcal{C}),W(\mathcal{C}))$. Let us start with the plabic graphs that we
need:

######  Definition 3.1.

An embedded planar bicolored graph $\mathbb{G}\subset\mathbb{R}^{2}$ is said
to be a _plabic fence_ if it satisfies the following conditions.

  * (i)

The vertices of $\mathbb{G}\subset\mathbb{R}^{2}$ belong to the standard
integral lattice $\mathbb{Z}^{2}\subset\mathbb{R}^{2}$, and they are colored
in either black or white.

  * (ii)

The edges of $\mathbb{G}\subset\mathbb{R}^{2}$ belong to the standard integral
grid
$(\mathbb{Z}\times\mathbb{R})\cup(\mathbb{R}\times\mathbb{Z})\subset\mathbb{R}^{2}$.
Edges that are contained in $\mathbb{Z}\times\mathbb{R}$ are said to be
_vertical_ , and edges that are contained in $\mathbb{R}\times\mathbb{Z}$ are
said to be _horizontal_.

  * (iii)

A maximal connected union of horizontal edges is called a _horizontal line_.
All horizontal lines must start, on the left, at univalent white vertices with
the same $x$-coordinate and must end, on the right, at univalent white
vertices with the same $x$-coordinate.

  * (iv)

Each vertical edge must end at trivalent vertices of opposite colors, with
white on top and black on bottom, and the end points of a vertical edge must
be contained in the interior of a horizontal line. In addition, no two
vertical edges are contained in the same (vertical) line.

Figure 15. Two plabic fences. The fence on the right encodes
$\beta=(\sigma_{1}\sigma_{2})^{4}$.

These are a special type of plabic graphs, studied in [FPST17, Section 12],
[CW, Section 2.5], [STWZ19, Section 5], and also [GSW20a, SW21], the latter in
the context of triangulations of flag configuration diagrams. For a visual
instance, two plabic fences $\mathbb{G}$ are drawn in Figure 15.

######  Remark 3.2.

The definition of plabic fences in [FPST17, CW] is more general but, for our
purposes, it is without loss of generality that we can work with Definition
3.1. This is due to the fact that cyclic rotation is a quasi-cluster
transformation, as shown in [CW, CGG+22]; see also [SW21]. $\Box$

We use the following bijection between plabic fences with $n$ horizontal edges
and positive braid words in $n$-strands. For a plabic fence $\mathbb{G}$ we
construct a braid word $\beta(\mathbb{G})$ iteratively by scanning the plabic
fence left-to-right: starting with the empty word $\beta(\mathbb{G})$, when we
encounter a vertical edge between the $k$ and $(k+1)$st horizontal edges,
counting from the bottom, we add the Artin generator $\sigma_{k}$ to the right
of $\beta(\mathbb{G})$. The plabic fence that gives the braid word $\beta$
will be denoted by $\mathbb{G}(\beta)$.

###  3.1. Curve configuration $\mathcal{C}(\mathbb{G})$ of a plabic fence
$\mathbb{G}$

A plabic fence gives rise to a curve configuration $\mathcal{C}(\mathbb{G})$
as follows. Consider the conjugate surface $\Sigma(\mathbb{G})$ of [GK13,
Section 1.1.1]; see also [Gon17, Section 2], [STWZ19, Section 4] or [CL22,
Section 2.1] for definition and details on conjugate surfaces. For the
purposes of this manuscript, this is a (ribbon) surface obtained from a plabic
graph by using the three local models in Figure 16.101010The boundary of the
surface is given by the alternating strand diagram of $\mathbb{G}$, see
[Pos06, Gon17]. It retracts to $\mathbb{G}$ and thus
$H_{1}(\Sigma(\mathbb{G}),\mathbb{Z})\cong H_{1}(\mathbb{G},\mathbb{Z})$, the
latter being a free $\mathbb{Z}$-module of rank equal to the number of faces
of $\mathbb{G}$.111111A face is a bounded connected component of the
complement of the plabic fence in $\mathbb{R}^{2}$.

Figure 16. The three local models needed to draw a (projection of a) conjugate
surface associated to a plabic fence $\mathbb{G}$. The boundary of the surface
is in dark blue and the surface itself in light blue.

We allow the equivalence in Figure 17 for conjugate surfaces. The fact that
this move does not affect the combinatorics, nor the symplectic geometric in
Section 4, is verified rather simply121212The move removes/creates a trivial
$2$-cycle that would also feature in the potential, therefore the reduced part
of the QP quiver of the associated curve configuration will remain invariant.,
e.g. it is proven in [CL22, Section 2.1.3]. We always consider the diagram for
the conjugate surface $\Sigma(\mathbb{G})$ after we have applied this move
(left to right in Figure 17) to every local double-crossing configuration as
in Figure 17 (left). That is, we remove any instances of Figure 17 (left) in
the diagram of $\Sigma(\mathbb{G})$, substituting them by Figure 17 (right).

Figure 17. An equivalence of conjugate surfaces. This corresponds to a non-
dangerious tangency in the alternating strand diagram.

Therefore, near a face $F\subset\mathbb{G}$, the conjugate surface can be
drawn as in Figure 18; the face $F$ is the central face, immediately to the
left of the green vertical edge. Note that this is the general pattern near a
face $F$: bounded between two vertical edges at the same level, with a series
of vertical edges arriving with a black vertex from the level right above, and
a series of vertical edges arriving with a white vertex from the level right
below.

Figure 18. (Left) A plabic fence in black with the boundary of the conjugate
surface drawn in blue. (Right) The conjugate surface associated to the plabic
fence, with its interior drawn in turquoise.

The curve configuration $\mathcal{C}(\mathbb{G})$ from $\mathbb{G}$ is built
as follows:

######  Definition 3.3.

Let $\mathbb{G}$ be a plabic fence and $\Sigma(\mathbb{G})$ its conjugate
surface. The curve configuration $\mathcal{C}(\mathbb{G})$ is the
configuration of embedded closed curves in $\Sigma(\mathbb{G})$ constructed as
follows:

  * (i)

There is a curve $\gamma_{F}\in\mathcal{C}(\mathbb{G})$ for every face
$F\subset\mathbb{G}$.

  * (ii)

Each curve $\gamma_{F}\subset\Sigma(\mathbb{G})$ is obtained by applying the
local models in Figure 19 near each vertex of $\mathbb{G}$ and connecting the
resulting segments by following the boundary of (the diagram of)
$\Sigma(\mathbb{G})$ in a planar parallel manner.

The configuration $\mathcal{C}(\mathbb{G})$ of curves in $\Sigma(\mathbb{G})$
is said to be the (curve) configuration associated to $\mathbb{G}$. $\Box$

Figure 19. The two local models needed to associate a closed curve
$\gamma_{F}$ in $\Sigma(\mathbb{G})$ to every face of $\mathbb{G}$.

Figure 20 (left) depicts a plabic fence with a face $F$ and Figure 20 (right)
depicts its associated closed embedded curve $\gamma_{F}$ in
$\Sigma(\mathbb{G})$. Note again that this is the generic form of any plabic
fence near a face $F$, possibly adding a few more vertical edges at the levels
right above and below, which would in any case not change $\gamma_{F}$.

Figure 20. (Left) A plabic fence $\mathbb{G}$ in black with a face $F$
highlighted. (Right) The curve $\gamma_{F}$ in $\Sigma(\mathbb{G})$ associated
to the face $F$.

Note that we can (and do) draw the curves in $\mathcal{C}(\mathbb{G})$ such
that two curves $\gamma_{F_{1}}$ and $\gamma_{F_{2}}$ in
$\mathcal{C}(\mathbb{G})$ only intersect at the precise twist point of the
ribbon diagram for $\Sigma(\mathbb{G})$. That is, the only intersections of
curves in $\mathcal{C}(\mathbb{G})$ that occur are of the form depicted in
Figure 21.

Figure 21. The two local models near an intersection point of two curves in
$\mathcal{C}(\mathbb{G})$.

In the next subsection we introduce a QP $(Q(\mathbb{G}),W(\mathbb{G}))$ that
will be useful to understand the properties of the curve QP associated
$\mathcal{C}(\mathbb{G})$.

######  Remark 3.4.

A weave $\mathfrak{w}$ naturally produces a curve configuration
$\mathcal{C}(\mathfrak{w})$ as well. If one chooses the weave
$\mathfrak{w}=\mathfrak{w}(\mathbb{G})$ associated to a plabic fence, as
constructed in [CW, Section 3], then the curve configuration
$\mathcal{C}(\mathfrak{w})$ is equivalent to $\mathcal{C}(\mathbb{G})$. This
can be proven with the same techniques as in [CL22, Section 3] but will not be
needed for the present manuscript. $\Box$

###  3.2. The QP associated to a plabic fence

There are several descriptions of quivers $Q(\mathbb{G})$ associated to a
plabic fence $\mathbb{G}$, see [BLL18, CW, FPST17, GK13, GSW20a, Pos06] for
some; they are all equivalent. For specificity, we explicitly describe the
quiver $Q(\mathbb{G})$ in the upcoming Defininion 3.5.

A piece of notation: suppose that $e\subset\mathbb{G}$ is a vertical edge at
level $k$, we denote by $F_{e}$ the face of $\mathbb{G}$ that has $e$ as its
right vertical edge. This face $F_{e}$ is unique or it does not exist. By
definition, a black pente-row (resp. white pente-row) is a consecutive
collection of black vertices in the same horizontal edge of $\mathbb{G}$ such
that:

  * -

There must be two white (resp. black) vertices bounding it: one white vertex
at its left, and one white vertex at its right.131313This is in the spirit of
pente capture configurations, following the board game Pente.

  * -

Each connected component of $\mathbb{R}^{2}\setminus\mathbb{G}$ whose closure
contains any segment between two vertices of the black vertices above or
between a black vertex and one of the two white vertices above must be a face.

The total number of black (resp. white) vertices in a black (resp. white)
pente-row is said to be its length. See Figure 22 for a length four black
pente-row and a length four white pente-row, where we have marked the
connected components of $\mathbb{R}^{2}\setminus\mathbb{G}$ that must be faces
with a blue dot. The rightmost face in a black (resp. white) pente-row, which
has a left corner at the rightmost black (resp. white) vertex, is said to be
its right face.

Figure 22. (Left) A black pente-row. (Right) A white pente-row.

######  Definition 3.5.

Let $\mathbb{G}$ be the plabic fence with $n$ horizontal edges, its associated
QP $(Q(\mathbb{G}),W(\mathbb{G}))$ is defined as follows. The quiver
$Q(\mathbb{G})$ has vertex set the set of faces of $\mathbb{G}$. The arrow set
of $Q(\mathbb{G})$ is inductively described as follows, scanning $\mathbb{G}$
left to right:

  * (i)

If $\mathbb{G}$ is the empty plabic fence, then the arrow set of
$Q(\mathbb{G})$ is empty.

  * (ii)

Choose a vertical edge $e\subset\mathbb{G}$ at level $k$ and assume that the
arrow set of $Q(\mathbb{G}_{<e})$ is $A_{<e}$, where $\mathbb{G}_{<e}$ is the
plabic subfence of $\mathbb{G}$ consisting of those vertical edges to the left
of $e$. If $F_{e}$ does not exist, the arrow set of $Q(\mathbb{G}_{\leq e})$,
where $\mathbb{G}_{\leq e}=\mathbb{G}_{<e}\cup\\{e\\}$, is defined to be
$A_{<e}$.

If $F_{e}$ exists, the arrow set of $Q(\mathbb{G}_{\leq e})$ is defined to be
$A_{<e}$ union the following possible arrows:

    * (a)

Let $d$ be the left vertical edge of $F_{e}$. If $F_{d}$ exists, then we add
an arrow $[de]$ from $F_{d}$ to $F_{e}$.

    * (b)

Let $d^{\uparrow}$ be the first vertical edge in $\mathbb{G}$ at level
$(k+1)$st to the right of $d$. If $d^{\uparrow}$ and $F_{d^{\uparrow}}$ exist,
then we add an arrow $[ed^{\uparrow}]$ from $F_{e}$ to $F_{d^{\uparrow}}$. See
Figure 23 (left) for such an arrow, marked with a pink $(Z)$ pattern.

    * (c)

Let $d^{\downarrow}$ be the first vertical edge in $\mathbb{G}$ at level
$(k-1)$st to the right of $d$. If $d^{\downarrow}$ and $F_{d^{\downarrow}}$
exist, then we add an arrow $[ed^{\downarrow}]$ from $F_{e}$ to
$F_{d^{\downarrow}}$. See Figure 23 (left) for such an arrow, marked with a
red $(S)$ pattern.

If the hypotheses in these cases are not met, no arrows are added: e.g. if
$F_{e}$ or $F_{d}$ do not exist in case (a), we do not add any arrows at that
stage.

See Figure 23 (right) for an instance of $Q(\mathbb{G})$. Note that
$Q(\mathbb{G})$ around a black pente-row of $\mathbb{G}$ has a (planar)
clockwise cycle, whereas $Q(\mathbb{G})$ around a white pente-row has a
(planar) counter-clockwise cycle. The lengths of these cycles in the quiver
are the lengths of the rows plus two.

Figure 23. (Left) The two local patterns for arrows in $Q(\mathbb{G})$ that
are not horizontal. (Right) An example of $Q(\mathbb{G})$.

The potential $W(\mathbb{G})$ is similarly described inductively. If
$\mathbb{G}$ is the empty plabic fence, then $W(\mathbb{G})=0$. For each
vertical edge $e$ added to the right of $\mathbb{G}_{<e}$ as above, the
potential is defined as

$W(\mathbb{G}_{\leq e})=W(\mathbb{G}_{<e})+P_{black}(e)-P_{white}(e),$

where the monomials in $P_{black}(e)$ and $P_{white}(e)$ are described as
follows. By definition, $P_{black}(e)$, resp. $P_{white}(e)$, is the (cyclic)
monomial in the arrows of $Q(\mathbb{G})$ encoding the planar cycle in
$Q(\mathbb{G})$ associated to the unique black (resp. white) pente-row with
right face equal $F_{e}$, if such pente-row exists and else it equals zero.
Note that monomials in $P_{white}(e)$ have an overall minus sign in front due
to the counter-clockwise orientation of their cycles. $\Box$

######  Remark 3.6.

Note that a pente-row of length one with right face $F_{e}$ gives rise to a
triangle cycle in $Q(\mathbb{G})$ and thus cubic monomial in the potential.
Thus this construction generalizes known quivers with potentials from
bipartite graphs, cf. [Gon17, Section 5.1.2]. For instance, the case of
$\beta=w_{0}$, the longest element in the symmetric group $S_{n}$, recovers
the potential associated to the $n$-triangulation of a triangle, see [Gon17,
Section 1.1] or [CZ22a, Section 3.1]. $\Box$

######  Remark 3.7.

For a non-inductive definition of $Q(\mathbb{G})$, cf. [GSW20b, Section 1].
The inductive nature of Definition 3.5 is useful in our inductive proof of
Propositions 3.8 and 3.12. Also, there is no need to keep track of frozen
vertices in Definition 3.5. There are natural generalizations to iced quivers,
e.g.. see [Pre20], [GSW20a] or [CW]. $\Box$

The main reason to introduce $(Q(\mathbb{G}),W(\mathbb{G}))$ is to be able to
describe the QP associated to $\mathcal{C}(\mathbb{G})$ combinatorially in
terms of $\mathbb{G}$. Indeed, we have:

######  Proposition 3.8.

Let $\mathbb{G}$ be a plabic fence and $\mathcal{C}=\mathcal{C}(\mathbb{G})$
its associated curve configuration. Then the curve QP
$(Q(\mathcal{C}),W(\mathcal{C}))$ equals the QP
$(Q(\mathbb{G}),W(\mathbb{G}))$.

###### Proof.

Let us scan the plabic fence $\mathbb{G}$ left to right and compare
$Q(\mathbb{G})$ and $Q(\mathcal{C})$ as we do so. Let $e\subset\mathbb{G}$ be
a vertical edge at level $k$ and $F=F_{e}$ its associated face.141414If
$F_{e}$ does not exist, there is no curve added nor the quiver gains vertices,
thus both coincide. Since $Q(\mathbb{G})$ and the curves in $\mathcal{C}$
associated to faces at level $k$ are local in the union of levels $(k-1),k$
and $(k+1)$st, it suffices to consider the piece of plabic fence in the
neighborhood of $F$ consisting of all faces that share a vertical edge (there
is at most one such face) or a piece of a horizontal edge with $F$. An example
of a neighborhood of $F$ is drawn in Figure 24, with the relevant set of
curves in $\mathcal{C}$ on its left and the quiver on its right. Note that we
are scanning left to right and assuming that the vertical edge $e$ is the
rightmost edge of $\mathbb{G}$ at this stage. For the plabic fence in Figure
24 (left), this is the edge depicted in green in Figure 18.

Figure 24. (Left) A neighborhood of a face $F$ with the conjugate surface
depicted and all the curves $\gamma_{F_{i}}$ for the faces $F_{i}$ of
$\mathbb{G}$ adjacent to $F$. (Right).

The curve $\gamma_{F}$ associated to the new vertex of $Q(\mathbb{G}_{\leq
e})$ not in $Q(\mathbb{G}_{<e})$ intersects the other curves $\gamma_{F_{i}}$,
$F_{i}$ faces in $\mathbb{G}_{<e}$, when there is a ribbon twist. (Recall
Figure 21.) The curve $\gamma_{F}$ only traverses twists that respectively
connect to:

  * -

The face $F_{d}$, where $d$ is the left vertical edge of $F_{e}$, if it
exists.

  * -

The face $F_{d^{\uparrow}}$, as in Definition 3.5, if it exists.

  * -

The face $F_{d^{\downarrow}}$, as in Definition 3.5, if it exists.

  * -

The region to the right of $e$ that, by hypothesis at this stage ($e$ being
the rightmost edge we have scanned) is unbounded and thus not a face.

The three intersections $\gamma_{d}\cap\gamma_{e}$,
$\gamma_{d^{\uparrow}}\cap\gamma_{e}$ and
$\gamma_{d^{\downarrow}}\cap\gamma_{e}$ are precisely recorded by the arrows
$[de]$, $[ed^{\uparrow}]$ and $[de],[ed^{\downarrow}]$, if they respectively
exist. The direction of the arrow is given by the intersection sign, which is
positive for $\gamma_{d}\cap\gamma_{e}$ and negative for
$\gamma_{d^{\uparrow}}\cap\gamma_{e}$ and
$\gamma_{d^{\downarrow}}\cap\gamma_{e}$. Therefore $Q(\mathbb{G})$ and
$Q(\mathcal{C})$ coincide.

In order to compare $W(\mathbb{G})$ and $W(\mathcal{C})$ it suffices to note
that a polygon bounded by $\mathcal{C}$ in the conjugate surface
$\Sigma(\mathbb{G})$ must have vertices be as in Figure 25. That is, a polygon
can use only one of the two regions at a given ribbon twist and, once a side
is chosen, it must be the interior region bounded by the two curves.151515The
other regions at one side of a twist have boundary components of
$\Sigma(\mathbb{G})$ as part of their boundary. Therefore, polygons bounded by
$\mathcal{C}$ must correspond to pente-rows. Indeed, a pente-row gives a
unique polygon bounded by $\mathcal{C}$ by drawing $\Sigma(\mathbb{G})$ near
the pente-row and cutting at the ribbon twists around the pente-row; each
ribbon twist corresponds to a vertex of the polygon. Such cuts give an
embedded planar region and there is a unique polygon embedded in it with
vertices given by the (locations where we cut the) ribbon twists. Note that we
cut as many vertical ribbon twists as the length of the pente-row and we
always cut two horizontal ribbon twists, i.e. the polygon has as many sides as
the length of the pente-row plus two. The polygon is oriented clockwise, resp.
counter-clockwise, if the row is black, resp. white.

Figure 25. In yellow, pieces of regions in the conjugate surface that have the
potential to give a polygon bounded by $\mathcal{C}$.

Conversely, the conjugate surface $\Sigma(\mathbb{G})$ has a ribbon twist in
any segment (horizontal or vertical) between a black and a white vertex, and
no twist if a segment is between two vertices of the same color. Embeddedness
of the polygon in $\Sigma(\mathbb{G})$ implies that it must lie within a
region bounded by twists (and no twists on the interior of that region). Since
the ends of a polygon must be as in Figure 25, the only possible polygons
bounded by $\mathcal{C}$ must have the form of those around a pente-row. In
conclusion, polygons bounded by $\mathcal{C}$ in $\Sigma(\mathbb{G})$ must
correspond to pente-rows and the potentials $W(\mathbb{G})$ and
$W(\mathcal{C})$ coincide as we scan $\mathbb{G}$ left to right.∎

We conclude this subsection with a combinatorial property satisfied by the
quivers $Q(\mathbb{G})$. This property will be used in the proof of
Proposition 3.12. The following proof was explained to us by D. Weng, to whom
we are grateful:

######  Lemma 3.9.

Let $\mathbb{G}$ be a plabic fence and $Q(\mathbb{G})$ its associated quiver.
Consider the rightmost vertical edge $e\in\mathbb{G}$ and the vertex $F_{e}\in
Q(\mathbb{G})_{0}$ associated to the face $F_{e}$ immediately to the left of
$e$, if it exists. Then there exists a sequence of vertices
$(v_{1},\ldots,v_{k})$ in $v_{i}\in Q(\mathbb{G}_{<e})_{0}$, $i\in[k]$ and
$k\in\mathbb{N}$ depending on $e$, such that
$\mu_{k}\cdots\mu_{1}(Q(\mathbb{G}))$ has $F_{e}$ be a source vertex.

###### Proof.

Let us consider the set of faces $F_{1},\ldots,F_{k},F_{k+1}$ at the same
level as the edge $e$. We order them left to right as we scan $\mathbb{G}$,
thus $F_{1}$ is the leftmost face of $\mathbb{G}$ at that level, $F_{2}$ is
the face adjacent to $F_{1}$ exactly to its right, and so on until
$F_{k+1}=F_{e}$. The claim is:

Assertion. $(v_{1},\ldots,v_{k})=(F_{1},\ldots,F_{k})$ is a sequence that
turns $F_{k+1}=F_{e}$ into a source.

Before we prove the assertion, three comments. First, we recall that [SW21],
or [GSW20a, Section 5], explain how to associate a quiver to a more general
type of plabic fence than that in Definition 3.1.161616These are instances of
the general theory of quivers for plabic graphs, see [Pos06]. Namely, to a
plabic fence that also allows vertical edges to have a black vertex at the top
and a white vertex at the bottom. See also [CW, Section 2] for such objects,
their quivers and their properties. These are essentially all rephrasings of
the theory of double wiring diagrams, cf. [FWZ, Section 2.4], now in the
plabic graph terminology and generalized to the non-reduced case. Just for
this proof, we refer to these more general plabic fences also as plabic
fences.

Now, the reflection move in [SW21, Section 2.3], see also [CW, Section 5.2] or
[GSW20a, Section 4], is as follows. A reflection move $r_{k}$ at level $k$ of
a plabic fence $\mathbb{G}$ is the operation that input $\mathbb{G}$ and
outputs a plabic fence $r_{k}(\mathbb{G})$ which coincides with $\mathbb{G}$
everywhere except that the leftmost vertical edge at level $k$ has been
flipped.171717Here flipping means: if it had a black vertex at the top, now it
has a black vertex at the bottom, and vice versa. A reflection move does not
change the quiver $Q(\mathbb{G})$, see [SW21].

Second, for each face $F$, let us denote by $\partial_{-}F$, resp.
$\partial_{+}F$ its left vertical edge, resp. right vertical edge. Suppose
that we have a square face in $\mathbb{G}$ which has a $\partial_{-}F$ with a
black vertex at the top and $\partial_{+}F$ with a black vertex at the bottom.
Then the square move, cf. [FWZ, Section 2.5], is the operation that exchanges
$\partial_{-}F$ and $\partial_{+}F$, i.e. the new plabic fence
$\mu_{F}(\mathbb{G})$ coincides with $\mathbb{G}$ everywhere except that
$\partial_{-}F$ now has a black vertex at the bottom and $\partial_{+}F$ now
has a black vertex at the top. A square move induces a mutation of
$Q(\mathbb{G})$ at the vertex $F$, i.e.
$Q(\mu_{F}(\mathbb{G}))=\mu_{F}(Q(\mathbb{G}))$.

Figure 26. Two moves on plabic fences that do not change the quiver. They
allows us to move a certain type of vertical edge to the right in the presence
of certain vertical edges above and below it.

Third, the two moves in Figure 26 allow us to slide right a vertical edge with
black at the top through vertical edges with black at the bottom to its upper
right or lower right. Let us now prove the assertion above, which will
conclude the proof for the lemma.

Proof of Assertion: Let us consider the initial plabic fence $\mathbb{G}$ and
apply a reflection $r_{k}(\mathbb{G})$, where $k$ is the level of
$e\in\mathbb{G}$. We still have $Q(r_{k}(\mathbb{G}))=Q(\mathbb{G})$. This
reflection creates a vertical edge $\delta$ with black on top at the leftmost
part of level $k$ of $r_{k}(\mathbb{G})$; all the remaining vertical edges in
$r_{k}(\mathbb{G})$ have black at the bottom. Then we slide $\delta$ to the
right through level $k$ by either applying the sliding moves in Figure 26, or
using square moves. The former has no effect on the quiver, the latter induce
quiver mutations.

Let us slide $\delta$ to the right until $\delta$ is exactly to the left of
$e$, so that (the new) $F_{e}$ has $\partial_{-}F_{e}=\delta$ and
$\partial_{+}F_{e}=e$. Denote by $\mathbb{G}_{\delta,e}$ this plabic fence.
Then $\delta$ will have slid through faces $F_{1},\ldots,F_{k}$ when going
from $\mathbb{G}$ to $\mathbb{G}_{\delta,e}$. Thus at this stage we have
performed a sequence of mutations at those faces, in this same order, and
$Q(\mathbb{G}_{\delta,e})=\mu_{F_{k}}\cdots\mu_{F_{1}}(Q(\mathbb{G}))$. To
conclude, it suffices to note that the quiver $Q(\mathbb{G}_{\delta,e})$
associated to the plabic fence $Q(\mathbb{G}_{\delta,e})$ has a unique arrow
out of the vertex associated to $F_{e}$, and therefore $F_{e}$ is a source, as
required. ∎

Lemma 3.9 states that we can find a sequence of mutations for $Q(\mathbb{G})$
so that the vertex corresponding to the face immediately left of rightmost
vertical edge of $\mathbb{G}$ becomes a source vertex, and such that this
sequence of mutations never includes a mutation at that particular vertex that
we want to turn into a source vertex.

######  Remark 3.10.

This is not needed for our applications, but note that the proof of Lemma 3.9
explicitly presents a mutation sequence that turns $F_{e}$ into a source.
$\Box$

###  3.3. Non-degeneracy of
$(Q(\mathcal{C}(\mathbb{G})),W(\mathcal{C}(\mathbb{G})))$

The notion of non-degeneracy of a QP was introduced in [DWZ10, Section 7]. It
reads as follows:

######  Definition 3.11.

Let $(Q,W)$ be a QP and $k_{1},\ldots,k_{l}\in Q_{0}$ be a finite sequence of
vertices, no two consecutive ones being equal. By definition, $(Q,W)$ is
$(k_{1},\ldots,k_{l})$-non-degenerate if all the QPs $(Q,W)$,
$\mu_{k_{1}}(Q,W),\mu_{k_{2}}\mu_{k_{1}}(Q,W),\ldots,\mu_{k_{l}}\cdots\mu_{k_{2}}\mu_{k_{1}}(Q,W)$
are 2-acyclic. By definition $(Q,P)$ is non-degenerate if it is
$(k_{1},\ldots,k_{l})$-non-degenerate for every such sequence of vertices.
$\Box$

Thus, in a non-degenerate $(Q,W)$ we can choose an arbitrary sequence of
vertices in $Q$ and mutate the QP $(Q,W)$ along that sequence of vertices.
That is, it is possible to perform QP-mutations to $(Q,W)$ just by naming the
quiver mutations of $Q$. There are some QPs that are non-degenerate and some
that are degenerate, see [DWZ10], especially Sections 7 and 8 therein, and,
for instance, [CZ22b, Section 2.2], where quivers associated to the top
positroid variety of a Grassmannians are shown to be non-degenerate.

Before establishing non-degeneracy of $(Q(\mathcal{C}),W(\mathcal{C}))$, we
recall that [DWZ10, Definition 6.10] defines a QP $(Q,W)$ to be rigid if the
trace space of its Jacobian algebra is equal to the ground ring $\mathbb{C}$.
Rather than the definition itself, the three following properties about rigid
QPs are most relevant to us:

  * -

Rigidity is preserved under QP mutation. That is, if $(Q,W)$ is rigid, then
$\mu_{v}(Q,W)$ is rigid for any vertex $v\in Q_{0}$. This is established in
[DWZ10, Corollary 6.11].

  * -

If $(Q,W)$ is rigid and $Q^{\prime}$ is obtained by adding one vertex $v$ to
$Q$, so that $Q^{\prime}_{0}=Q_{0}\cup\\{v\\}$, such that $v$ is a source (or
a sink) in $Q^{\prime}$, then $(Q^{\prime},W^{\prime})$ with $W^{\prime}=W$ is
also rigid.181818In the quiver literature, this operation is part of what is
known as a triangular extension of $Q(\mathbb{G}_{<e})$ and a point, cf.
[Ami09, Definition 3.8]. This is proven in [Lad13, Remark 4.4] or cf. [DWZ10,
Section 8].

  * -

A rigid QP $(Q,W)$ is non-degenerate. This is proven in [DWZ10, Corollary
8.2].

Let us now show that QP $(Q(\mathcal{C}),W(\mathcal{C}))$ associated to curve
configurations $\mathcal{C}$ such that $\mathcal{C}=\mathcal{C}(\mathbb{G})$
for a plabic fence are indeed non-degenerate.

######  Proposition 3.12.

Let $\mathbb{G}$ be a plabic fence and $\mathcal{C}=\mathcal{C}(\mathbb{G})$
its associated curve configuration. Then the QP
$(Q(\mathcal{C}),W(\mathcal{C}))$ is rigid, and thus non-degenerate.

###### Proof.

By Lemma 3.8, it suffices to argue that $(Q(\mathbb{G}),W(\mathbb{G}))$ as in
Definition 3.5 is rigid. As usual, we prove this by scanning the plabic fence
$\mathbb{G}$ left to right. First, the empty quiver
$(Q(\mathbb{G}),W(\mathbb{G}))$ is rigid and non-degenerate. Second, we assume
that we have scanned until the vertical edge $e$ and we have rigidity for
$W(\mathbb{G}_{<e})$. Let us consider two cases:

  * -

If $[de]$ does not exist, then $F_{e}$ is a source vertex and
$Q(\mathbb{G}_{\leq e})$ is obtained from $Q(\mathbb{G}_{<e})$ by adding a
source, thus no new cycles appear and the potential is still rigid by the
second property above. Similarly, if neither $[ed^{\uparrow}]$ and
$[ed^{\downarrow}]$ exist, then $F_{e}$ is a sink vertex and the same argument
applies.

  * -

If either $[ed^{\uparrow}]$ or $[ed^{\downarrow}]$ exist and so does $[de]$,
then $F_{e}$ is neither a sink nor a source. Nevertheless, we can apply Lemma
3.9 to turn $F_{e}$ into a source by mutating at vertices in
$Q(\mathbb{G}_{<e})$. Indeed, the sequence of mutations $\mu_{v}$ from Lemma
3.9 is at an ordered collection of vertices $v:=(v_{1},\ldots,v_{k})$ which
are already vertices in $Q(\mathbb{G}_{<e})$, i.e. it does not mutate at
$F_{e}$. By hypothesis, $W(\mathbb{G}_{<e})$ is rigid and, as stated above,
rigidity is preserved by QP mutation. Therefore $\mu_{v}(W(\mathbb{G}_{<e}))$
is also rigid. Since restriction commutes with mutation by [FIL21, Lemma 2.5]
and the mutations occur at vertices of $Q(\mathbb{G}_{<e})$, the mutated
potential $\mu_{v}(W(\mathbb{G}_{\leq e}))$ restricted to the subquiver
$Q(\mathbb{G}_{<e})$ is also rigid, as it equals
$\mu_{v}(W(\mathbb{G}_{<e}))$. Since the quiver $\mu_{v}(Q(\mathbb{G}_{\leq
e}))$ is given by adding a source to $\mu_{v}(Q(\mathbb{G}_{<e}))$, the
resulting potential $\mu_{v}(W(\mathbb{G}_{\leq e}))$, now without
restricting, is still rigid. The original potential $W(\mathbb{G}_{\leq e})$
is QP mutation-equivalent to this resulting potential, and thus also rigid.

In summary, in either case, the QP $(Q(\mathbb{G}),W(\mathbb{G}))$ stays rigid
as we scan $\mathbb{G}$ left to right. Therefore
$(Q(\mathcal{C}(\mathbb{G})),W(\mathcal{C}(\mathbb{G})))$ is non-degenerate. ∎

Note that there are many QP $(Q,W)$ that might not be rigid (or non-
degenerate). The class of quivers with potentials
$(Q(\mathbb{G}),W(\mathbb{G}))$ that we introduced and analyzed above is
particular enough that rigidity can be argued directly as in Proposition 3.12.
This is similar to the fact that the quivers in the Kontsevich-Soibelman class
$\mathcal{P}$191919This is the class generated by the one vertex quiver by
triangular extensions and mutations. are rigid and admit a unique non-
degenerate potential up to right-equivalence. In fact, though it will not be
needed for our application, it can also be proven that $W(\mathbb{G})$ is the
unique non-degenerate potential for $Q(\mathbb{G})$ up to right-equivalence.

######  Remark 3.13.

It is likely that the more general class of curve configurations
$\mathcal{C}(\mathfrak{w})$ that we associated to the weaves $\mathfrak{w}$ in
[CGG+22] also have $W(\mathcal{C}(\mathfrak{w}))$ be non-degenerate; see
[CGG+22, Section 7.3]. In that case, the arguments in Section 4 would also
prove Theorem 1.1 for $(-1)$-closures. $\Box$

This is concludes our construction and study of the QPs
$(Q(\mathcal{C}),W(\mathcal{C}))$. We have established in Section 2 the
invariance of their right-equivalence classes under triple point moves and
local bigon moves, proven that they undergo a QP-mutation when a
$\gamma$-exchange is applied to $\mathcal{C}$, and now shown rigidity for
those configurations associated to plabic fences. There are two technical
pieces that still need justification: arguing that Assumption 1 holds and
proving Lemma 2.21. We conclude this section by presenting such proofs.

###  3.4. The assumption on bigons is satisfied

By definition, a cycle $v_{1}\ldots v_{n}$ in a QP $(Q,W)$ is said to be empty
if $v_{1}\ldots v_{n}$ does not appear as a monomial in $W$. The following
lemma shows that we have no empty cycles for QP associated to curve
configurations:

######  Lemma 3.14.

Let $(Q,W)$ be a QP and assume that it contains an empty cycle, i.e. a cycle
of arrows in $Q$ whose corresponding monomial does not appear in the potential
$W$. Then $(Q,W)$ is degenerate.

###### Proof.

Let us show this by induction on $n\in\mathbb{N}$, where $n$ is the number of
arrows of an empty $n$-cycle $v_{1}\ldots v_{n}$. The base case is $n=2$:
suppose there is a 2-cycle $v_{1}v_{2}$ in $Q$ such that $v_{1}v_{2}$ is not a
monomial in $W$. By Definition 3.11, without mutating at all, the quiver in
the reduced part of $(Q,W)$ still has the 2-cycle $v_{1}v_{2}$ and thus
$(Q,W)$ is degenerate.

The induction step is as follows. Consider an empty $n$-cycle $v_{1}\ldots
v_{n}$ for $(Q,W)$ and assume that the existence of an empty $n$-cycle in a QP
implies its degeneracy. Mutation of $(Q,W)$ at the vertex
$v=h(v_{1})=t(v_{2})$, creates the $(n-1)$-cycle $[v_{1}v_{2}]v_{3}\ldots
v_{n}$. By Definition 2.19, the potential $\mu_{v}(W)$ for the mutated QP
$\mu_{v}(Q,W)$ is obtained by substituting $v_{1}v_{2}$ by $[v_{1}v_{2}]$ in
$W$ and adding the cubic term $[v_{1}v_{2}]v_{1}^{\ast}v_{2}^{\ast}$.
Therefore, $\mu_{v}(W)$ contains the monomial $[v_{1}v_{2}]v_{3}\ldots v_{n}$
if and only if $W$ contains the monomial $v_{1}\ldots v_{n}$. By assumption,
$W$ does not contain $v_{1}\ldots v_{n}$ and thus $[v_{1}v_{2}]v_{3}\ldots
v_{n}$ is an empty $(n-1)$-cycle in $\mu_{v}(Q,W)$. By induction,
$\mu_{v}(Q,W)$ is degenerate and thus so is $(Q,W)$. ∎

Let us now use Lemma 3.14 to prove that Assumption 1 is satisfied for our non-
degenerate configurations. The precise statement reads as follows:

######  Proposition 3.15.

Let $\mathcal{C}$ be a curve configuration and $B\subset\Sigma$ a local bigon
bounded by $\mathcal{C}$. Suppose that $(Q(\mathcal{C}),W(\mathcal{C}))$ is a
non-degenerate potential. Then $\rho(B)$ and $\lambda(B)$ are different
regions.

###### Proof.

By contradiction, suppose that there exists a local bigon $B$ such that
$\rho(B)=\lambda(B)$. We will now argue that either:

  1. (1)

The potential $(Q,W)$ is degenerate, or

  2. (2)

The homology classes in $H_{1}(\Sigma;\mathbb{Z})$ of the curves
$\gamma_{1},\ldots,\gamma_{b}\in\mathcal{C}$ do not span, because there is a
sub-collection of them that give linearly dependent homology classes.

Let us consider a neighborhood of such local bigon $B$ with
$\rho(B)=\lambda(B)$. It is as depicted in Figure 27. Namely, the region
$\rho(B)=\lambda(B)$ is bounded by:

  1. (1)

A collection of curves $\gamma_{2},\ldots,\gamma_{s}\in\mathcal{C}$,
$s\in\mathbb{N}$, with intersection pattern exactly given by $\gamma_{i}$
intersects $\gamma_{i+1}$ negatively, at a point $g_{i+1,i}$ and $\gamma_{g}$
intersects $\gamma_{1}$ negatively at $g_{1,s}$.

  2. (2)

A collection of curves $\tau_{2},\ldots,\tau_{l}\in\mathcal{C}$,
$l\in\mathbb{N}$, with intersection pattern exactly given by $\tau_{i}$
intersects $\tau_{i+1}$ positively, at a point $t_{i,i+1}$ and $\tau_{l}$
intersects $\gamma_{1}$ negatively at $t_{l,1}$.

Figure 27 depicts a case with $s=4$ and $t=3$. Note that some curve
$\gamma_{i}$, resp. some curve $\tau_{i}$, might equal some of the other
curves $\gamma_{j}$, resp. the other curves $\tau_{j}$. The argument that now
follows works in that case as well.

Figure 27. A configuration $\mathcal{C}$ near bigon $B$, bounded by
$\gamma_{1},\tau_{1}$, with $\lambda(B)=\rho(B)$.

Now consider the resolution of the bigon $B$ with a local bigon move. This
yields the local curve configuration as in Figure 28. The arrows
$g_{2,1}\ldots g_{i+1,i}\ldots g_{1,s}$ in the quiver form a cycle $G$.
Similarly, the arrows $t_{1,2}\ldots t_{i,i+1}\ldots t_{s,1}$ in the quiver
form a cycle $T$.

Figure 28. A curve configuration $\mathcal{C}$ after performing a local bigon
move to Figure 27. The resulting region $\lambda(B)=\rho(B)$ is not simply-
connected.

Consider the cycles $G$ and $T$ in the quiver. If either $G$ or $T$ are empty,
then Lemma 3.14 implies that $(Q(\mathcal{C}),W(\mathcal{C}))$ is degenerate.
This contradicts the assumption. Therefore, both $G$ and $T$ must bound a
polygon. From the specific configuration we are studying, as in Figure 28, the
polygon for $G$ must be above $G$ and the polygon for $T$ must be below $T$.
Pieces of such a polygon bounded by $G$, above $G$, are depicted in Figure 28
in green, and pieces of such a polygon bounded by $T$, below $T$, are depicted
in Figure 28 in blue.

Consider now the smooth oriented representative $\overline{\gamma}$ of
$[\gamma_{1}+\gamma_{2}+\ldots+\gamma_{t}]$ in $H_{1}(\Sigma;\mathbb{Z})$
given by the oriented $\infty$-resolutions of the crossings
$g_{21},g_{32},\ldots,g_{t,t-1}$. Here the $\infty$ resolution is such that
the strand coming from the north-west continues to the north-east
strand.202020Similarly, the strand coming from the south-west continues south-
east. It is the only possible oriented resolution. Similarly, let
$\overline{\tau}$ be the smooth oriented representative of
$[\tau_{1}+\tau_{2}+\ldots+\tau_{l}]$ in $H_{1}(\Sigma;\mathbb{Z})$ given by
the oriented $\infty$-resolutions of the crossings
$t_{12},t_{12},\ldots,t_{l-1,l}$. Let $c^{+}\subset\Sigma$, resp.
$c^{-}\subset\Sigma$, be a smooth embedded core curve horizontally traversing
the yellow region in Figure 28 from left to right, resp. from right to left.
Now, the curve $\overline{\gamma}$ is homologous to $c^{+}$. Indeed, the
existence of a polygon above $G$ shows that the connected components of
$\overline{\gamma}$ not containing $\gamma_{1}$ are null-homologous. The
component containing $\gamma_{1}$ has an immersed point at $g_{1t}$. By
construction, this immersed point bounds a region above it, and thus it is
homologous to $c^{+}$. Similarly, the curve $\overline{\tau}$ is homologous to
$c^{-}$. This shows that
$[\overline{\gamma}]=[c^{+}]=-[c^{-}]=-[\overline{\tau}]$. Therefore
$[\overline{\gamma}]+[\overline{\tau}]=0$ and thus the set of curves in
$\mathcal{C}$ does not span $H_{1}(\Sigma;\mathbb{Z})$, as there are exactly
$b_{1}(\Sigma)$ of them and there is a non-empty subset of linearly dependent
classes. ∎

Proposition 3.12 and Proposition 3.15 show that Assumption 1 is always
satisfied for the curve configurations $\mathcal{C}(\mathbb{G})$ associated to
plabic fences $\mathbb{G}$.

###  3.5. Proof of Lemma 2.21

For Part (i), if $\mathcal{C}$ bounds bigons, we obtain $\mathcal{C}_{red}$ by
iteratively applying Theorem 2.9 to remove them. Lemmas 2.5 and 2.7 imply that
$(Q(\mathcal{C}),W(\mathcal{C}))$ undergoes a sequence of right-equivalences
and local reductions. Note that Lemma 2.7 implies that these reductions exist.
By Definition 2.6, each local reduction can be understood as splitting off a
trivial direct summand from $(Q(\mathcal{C}),W(\mathcal{C}))$. Indeed, an
$ab$-reduction of $(Q,W)$ yields a decomposition
$(Q_{ab},W_{ab})\oplus(Q^{\prime},W^{\prime})$, where $Q_{ab}$ has just two
arrows $a,b$ and $W_{ab}=ab$, and $(Q^{\prime},W^{\prime})$ is the
$ab$-reduction of $(Q,W)$. Thus $(Q_{ab},W_{ab})$ is trivial. Therefore, after
Theorem 2.9 is iteratively applied until we obtain $\mathcal{C}_{red}$,
$(Q,W)$ undergoes a sequence of local reductions until it becomes right-
equivalent to $(Q(\mathcal{C}_{red}),W(\mathcal{C}_{red}))$: in each such
local reduction a trivial summand splits off and we are left with
$(Q(\mathcal{C}_{red}),W(\mathcal{C}_{red}))$. Since $\mathcal{C}_{red}$ has
no bigons, the quadratic part of $W(\mathcal{C}_{red})$ vanishes and thus
$(Q(\mathcal{C}_{red}),W(\mathcal{C}_{red}))$ is reduced. Therefore, it is the
reduced part of $(Q(\mathcal{C}),W(\mathcal{C}))$, up to right-equivalence and
Lemma 2.21.(i) follows.

For Part (ii), Lemma 3.14, or directly Definition 3.11, implies that
$Q(\mathcal{C})$ has no empty 2-cycles, since
$(Q(\mathcal{C}),W(\mathcal{C}))$ is non-degenerate. Therefore, if
$\mathcal{C}$ bounds no bigons, then $Q(\mathcal{C})$ has no 2-cycles and thus
neither does its reduced part. $\Box$

##  4\. A Lagrangian filling for every cluster seed

The goal of this section is to prove Theorem 1.1. We use the results developed
in Sections 2 and 3.

###  4.1. Preliminaries

Let us consider a positive braid word $\beta$ on $n$-strands and its
associated Legendrian link $\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$.
By definition, $\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ is the
Legendrian link given by the front rainbow closure of $\beta$. Figure 29
depicts such a rainbow closure, i.e. a front for $\Lambda_{\beta}$ in the
$(x,z)$-plane $\mathbb{R}^{2}$. The box with the label $\beta$ contains
exactly the crossings of $\beta$. The ideal contact boundary
$(T^{\infty}\mathbb{R}^{2},\lambda_{st})$ of the cotangent bundle
$(T^{*}\mathbb{R}^{2},\lambda_{st})$ is contactomorphic to the 1-jet space
$(J^{1}S^{1},\xi_{st})$, where the Legendrian zero section $S^{1}\subset
J^{1}S^{1}$ is the fiber of the projection
$T^{\infty}\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ onto the base
$\mathbb{R}^{2}$. Let us consider a Legendrian embedding
$\iota_{0}:S^{1}\longrightarrow(\mathbb{R}^{3},\xi_{st})$ of the (unique) max-
tb Legendrian unknot $\Lambda_{0}\subset(\mathbb{R}^{3},\xi_{st})$.

Figure 29. The rainbow closure front projection of the positive braid word
$\beta$.

By the Weinstein neighborhood theorem [Gei08, Section 2.5], any Legendrian
link $\Lambda\subset(J^{1}S^{1},\xi_{st})$ can be satellited along the
Legendrian embedding $\iota_{0}$, as there is a neighborhood of the max-tb
Legendrian unknot $\Lambda_{0}\subset(\mathbb{R}^{3},\xi_{st})$
contactomorphic to $(J^{1}S^{1},\xi_{st})$, where the contactomorphism extends
$\iota_{0}$. We denote the resulting Legendrian link in
$(\mathbb{R}^{3},\xi_{st})$ by $\iota_{0}(\Lambda)$. If we consider the
Legendrian $\Lambda(\alpha)\subset(J^{1}S^{1},\xi_{st})$ given by the braid
diagram of a positive braid word $\alpha$ in $n$-strands, then
$\iota_{0}(\Lambda(\beta\Delta^{2}))\cong\Lambda_{\beta}$ are Legendrian
isotopic in $(\mathbb{R}^{3},\xi_{st})$, where $\Delta$ is the half-twist on
$n$ strands. See [CN22, Section 2.2] for further details. Note that the class
of Legendrian links $\Lambda_{\beta}$ is comprehensive, as it includes the
max-tb representatives of all algebraic links and also Legendrian
representatives of infinitely many satellite and hyperbolic links, cf. [CG22,
Section 6].

Let $L\subset(\mathbb{D}^{4},\lambda_{st})$ be an exact oriented Lagrangian
filling.212121Of some Legendrian link $\partial L$ in the contact boundary of
$(\mathbb{D}^{4},\lambda_{st})$. Consider an arbitrary but fixed convex
neighborhood ${\mathcal{O}}{\it p}(L)$ which is symplectomorphic to a convex
neighborhood of the zero section in $(T^{*}L,\lambda_{st})$ and the
hypersurface $\partial{\mathcal{O}}{\it
p}(L)\subset(\mathbb{D}^{4},\lambda_{st})$ is a contact hypersurface
contactomorphic to the ideal contact boundary $(T^{\infty}L,\lambda_{st})$ of
$(T^{*}L,\lambda_{st})$. An embedded (co)oriented connected curve
$\gamma\subset L$ lifts to a Legendrian knot
$\Lambda_{\gamma}\subset\partial{\mathcal{O}}{\it p}(L)$, as it defines a
front under the Legendrian projection
$(T^{\infty}L,\lambda_{st})\longrightarrow L$. Since there is a canonical
correspondence between oriented and co-oriented curves in an oriented surface,
we always discuss oriented curves rather than co-oriented curves. Following
[CW], we introduce the following:

######  Definition 4.1 ($\mathbb{L}$-compressing systems).

Let $L\subset(\mathbb{D}^{4},\lambda_{st})$ be an exact oriented Lagrangian
filling and $\gamma\subset L$ an embedded oriented curve. By definition,
$\gamma$ is said to be $\mathbb{L}$-compressible if there exists a properly
embedded Lagrangian 2-disk
$D\subset(T^{*}\mathbb{R}^{2}\setminus{\mathcal{O}}{\it p}(L))$ such that
$\partial\overline{D}\cap\partial{\mathcal{O}}{\it
p}(L)=\Lambda_{\gamma}\subset\mathbb{R}^{4}$ and the union of
$\overline{D}\cup\nu_{\gamma}$ is a smooth Lagrangian disk, where
$\nu_{\gamma}\subset{\mathcal{O}}{\it p}(L)$ is the Lagrangian conormal cone
of $\gamma$. This Lagrangian disk is said to be an $\mathbb{L}$-compressing
disk for $\gamma$.

A collection $\Gamma=\\{\gamma_{1},\ldots,\gamma_{b}\\}$ of such curves in
$L$, with a choice of $\mathbb{L}$-compressing disks
$\mathscr{D}=\\{D_{1},\ldots,D_{b}\\}$ for each curve, is said to be an
$\mathbb{L}$-compressing system for $L$ if $D_{i}\cap D_{j}=\emptyset$ for all
$i,j\in[b]$ and the (homology classes of the) curves in $\Gamma$ form a basis
of $H_{1}(L;\mathbb{Z})$. $\Box$

We often abuse notation and refer to the collection $\mathscr{D}$ as the
$\mathbb{L}$-compressing system for $L$.

###  4.2. Curve configurations and $\mathbb{L}$-compressing systems

Let $L\subset(\mathbb{D}^{4},\lambda_{st})$ be a Lagrangian filling and
$\Gamma$ be a complete $\mathbb{L}$-compressing system. By definition, the
curve configuration $\mathcal{C}(\Gamma)$ associated to $\Gamma$ is the
configuration of oriented closed embedded curves $\Gamma$ in $L$. If
$\mathscr{D}$ is the collection of $\mathbb{L}$-compressing disks associated
to $\Gamma$, we also write $\mathcal{C}(\mathscr{D})$ for
$\mathcal{C}(\Gamma)$. Note that $\mathcal{C}(\Gamma)$ is considered as a
collection of smooth oriented curves in a smooth surface, with no need to
record the Lagrangian condition on $L$ and the disks in $\mathscr{D}$. The
notation $\mathcal{C}(\Gamma)$, instead of just $\Gamma$, is in order to
emphasize the smooth embedded curves, rather than the symplectic topological
aspects of $\Gamma$.

######  Remark 4.2.

Suppose that a sequence of triple point moves and local bigons moves is
applied to an $\mathbb{L}$-compressing system $\mathcal{C}(\Gamma)$. This
yields a configuration $\mathcal{C}^{\prime}$ in $L$. Front homotopies, which
include triple point moves and local bigon moves, lift to Legendrian isotopies
in the ideal contact boundary. The trace of a Legendrian isotopy yields an
invertible Lagrangian concordance in the symplectization. By concatenating the
disks associated to $\Gamma$ with this Lagrangian concordance, we obtain an
$\mathbb{L}$-compressing system $\Gamma^{\prime}$ for $L$ such that
$\mathcal{C}^{\prime}=\mathcal{C}(\Gamma^{\prime})$. We will consider two such
configurations $\mathcal{C},\mathcal{C}^{\prime}$ equivalent and two such
$\mathbb{L}$-compressing systems $\Gamma,\Gamma^{\prime}$ equivalent. $\Box$

###  4.3. $\mathbb{L}$-compressing systems for $\Lambda_{\beta}$

Consider the plabic fence $\mathbb{G}(\beta)$ associated to $\beta$, as
introduced in Section 3. Then we have the following facts:

  1. (1)

The alternating strand diagram of $\mathbb{G}(\beta)$ is a front for
$\Lambda(\beta\Delta^{2})\subset(J^{1}S^{2},\xi_{st})$. This is proven in [CW,
Section 2] and see also [STWZ19, Section 5.1]. Thus, after including
$(J^{1}S^{1},\xi_{st})$ into $(\mathbb{R}^{3},\xi_{st})$ as a neighborhood of
the max-tb Legendrian unknot, it is Legendrian isotopic to the Legendrian link
$\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$.

  2. (2)

The conjugate surface $\Sigma(\mathbb{G}(\beta))$ gives rise to a
unique222222Up to compactly supported Hamiltonian isotopy in
$(T^{*}\mathbb{R}^{2},\lambda_{st})$. embedded exact Lagrangian filling
$L_{\beta}$ of $\Lambda_{\beta}$. By construction, $L_{\beta}$ is smoothly
isotopic to the (smooth) surface $\Sigma(\mathbb{G}(\beta))$. Uniqueness is
proven in [CL22, Prop. 2.4], existence in [STWZ19, Section 4.2]. In
particular, it gives an oriented embedded exact Lagrangian filling
$L_{\beta}\subset(\mathbb{R}^{4},\lambda_{st})$ in the symplectization of
$(\mathbb{R}^{3},\xi_{st})$, after we have identified the standard cotangent
bundle $(T^{*}\mathbb{R}^{2},\omega_{st})$ with the symplectic Darboux
$(\mathbb{R}^{4},\omega_{st})$.

  3. (3)

The conjugate surface $\Sigma(\mathbb{G}(\beta))$ also gives an
$\mathbb{L}$-compressing system $\Gamma(\beta)$ for $L$. The existence of such
$\mathbb{L}$-compressing system is proven in [CW, Section 3], see also [Cas22,
Section 2] and [CL22, Section 4.2]. By construction, the configuration of
curves in $\mathcal{C}(\Gamma(\beta))$ coincides with the configuration of
curves in $\mathcal{C}(\mathbb{G}(\beta))$. The Lagrangian
$\mathbb{L}$-compressing disks, showing that this is indeed an
$\mathbb{L}$-compressing system, are constructed in [CW, Section
3].232323Intuitively, the corresponding $\mathbb{L}$-compressing Lagrangian
disks are the faces of $\mathbb{G}$, which are disjoint Lagrangian pieces
(disks) of the Lagrangian zero section $\mathbb{R}^{2}$ in
$T^{*}\mathbb{R}^{2}$. See Figure 24 (left). Let $\mathscr{D}_{\beta}$ denote
its associated collection of $\mathbb{L}$-compressing Lagrangian disks.

The relevant fact about the $\mathbb{L}$-compressing system $\Gamma(\beta)$ is
that it can be used to produce new Lagrangian fillings from $L_{\beta}$.

######  Remark 4.3.

For context, let $\beta$ be a positive braid word and
$\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ its associated Legendrian
link. The union $\mathbb{L}_{\beta}\subset(\mathbb{R}^{4},\lambda_{st})$ of
the embedded exact Lagrangian filling $L_{\beta}$ and all the closures of the
Lagrangian disks of the collection $\mathscr{D}_{\beta}$ (extended by the
Lagrangian conormal cones of curves in $\Gamma(\beta)$) is an arboreal
Lagrangian skeleton for the Weinstein pair given by
$(\mathbb{D}^{4},\lambda_{st})$ and a Weinstein ribbon of $\Lambda_{\beta}$,
see [Eli18, Section 2]. $\Box$

###  4.4. Lagrangian disk surgery

Lagrangian disk surgery was introduced in [Yau17], following closely the
Lagrange surgery defined in [Pol91]. In our context, it is used as follows.
Consider a Legendrian link $\Lambda\subset(\mathbb{R}^{3},\xi_{st})$, seen as
$(S^{3},\xi_{st})$ minus a point, and an embedded exact Lagrangian filling
$L\subset(\mathbb{D}^{4},\lambda_{st})$ in the standard Darboux 4-ball
symplectic filling $(S^{3},\xi_{st})$. Suppose that there exists a properly
embedded Lagrangian disk $D\subset\mathbb{D}^{4}\setminus L$ such that
$\partial\overline{D}\subset\mbox{int}(L)$ is a smooth embedded connected
curve, where $\mbox{int}(L)$ in the interior of $L$.242424In the discussion of
Subsection 4.3, these Lagrangian disks are obtained by considering the union
of the disks $D_{\gamma}\in\mathscr{D}$ in an $\mathbb{L}$-compressing system
$\mathscr{D}$ and concatenating them with (a piece of) the Lagrangian conormal
cone in $T^{*}L$ of the corresponding (co)oriented curve $\gamma$. Then
Lagrangian disk surgery is an operation that inputs the pair $(L,D)$ and
outputs another pair $(L^{\prime},D^{\prime})$, with the same properties:
$L^{\prime}$ is an embedded exact Lagrangian filling of $\Lambda$ and
$D^{\prime}$ is an embedded Lagrangian disk in the complement of $L^{\prime}$
with embedded boundary on $L^{\prime}$. In this process, it is crucial that
$D$ is an embedded Lagrangian disk, and not just immersed. Two facts about
Lagrangian disk surgery are:

  * -

The Lagrangians $L$ and $L^{\prime}$ are not necessarily exact Lagrangian
isotopic. That is, there might not be a compactly supported Hamiltonian
isotopy from $L$ to $L^{\prime}$.

  * -

The Lagrangians $L$ and $L^{\prime}$ are smoothly isotopic, relative to their
boundaries. The Lagrangian disk surgery of $L^{\prime}$ along $D^{\prime}$
yields $(L,D)$ back, up to compactly supported Hamiltonian isotopy.

The first item above precisely indicates that we can potentially produce a new
Lagrangian filling by using a given Lagrangian filling and a Lagrangian disk
as above. See [CW, Pol91, Yau17] and references therein for these facts and
more details.

######  Remark 4.4.

Lagrangian disk surgery is not known to exist if the boundary
$\partial\overline{D}\subset L$ is an immersed curve, rather than embedded.
Similarly, the disk $D$ must be embedded.252525It is not just a lack of
available constructions, [CW, Section 4.10] presents examples of immersed
disks that one cannot perform Lagrangian disk surgery to, due to the existence
of frozen vertices coming from absolute 1-cycles in $L$. $\Box$

###  4.5. Effect of Lagrangian surgery on curve configurations

Let $L\subset(\mathbb{D}^{4},\lambda_{st})$ be an exact Lagrangian filling and
$\Gamma$ an $\mathbb{L}$-compressible system for $L$. Consider a disk
$D\in\mathscr{D}$. Lagrangian disk surgery on $D$ leads to another exact
Lagrangian filling $\mu_{D}(L)\subset(\mathbb{D}^{4},\lambda_{st})$ endowed
with a curve configuration $\mu_{D}(\Gamma)$ and a collection of Lagrangian
disks $\mu_{D}(\mathscr{D})$ bounding the curves in $\mu_{D}(\Gamma)$. There
is a natural bijection between the disks in $\mathscr{D}$ and those in
$\mu_{D}(\mathscr{D})$ and a diffeomorphism between $L$ and $\mu_{D}(L)$, as
stated above. Now, the configuration $\mu_{D}(\Gamma)$ might not be an
$\mathbb{L}$-compressible system because it might contain immersed curves.

In general, these new curve configurations $\mu_{D}(\Gamma)$ obtained by
Lagrangian disk surgery on a disk associated to $\Gamma$ can be understood via
the following:

######  Lemma 4.5.

Let $L\subset(\mathbb{D}^{4},\lambda_{st})$ be an exact Lagrangian filling,
$\Gamma$ an $\mathbb{L}$-compressible system for $L$ with
$\mathbb{L}$-compressing disks $\mathscr{D}$, and $\mathcal{C}(\Gamma)$ its