# Legendrian Loops and Cluster Modular Groups

James Hughes  Duke University, Dept. of Mathematics, Durham, NC 27708, USA
<EMAIL_ADDRESS>
###### Abstract.

This work studies Legendrian loop actions on exact Lagrangian fillings of
Legendrian links in $(\mathbb{R}^{3},\xi_{\text{st}})$. By identifying the
induced action of Legendrian loops as generators of cluster modular groups, we
establish the existence of faithful group actions on the exact Lagrangian
fillings of several families of Legendrian positive braid closures, including
all positive torus links. In addition, we leverage a Nielsen-Thurston-like
classification of cluster automorphisms to provide new combinatorial and
algebraic tools for proving that a Legendrian loop action has infinite order.

##  1\. Introduction

In this work, we study actions of Legendrian loops of Legendrian links in
$(\mathbb{R}^{3},\xi_{\text{st}})$ on the Lagrangian concordance monoid. We
first show that by interpreting Legendrian loops as generators of cluster
modular groups, we can explicitly realize these group actions for several
families of Legendrian positive braid closures. Using this cluster-theoretic
perspective, we also provide new combinatorial and algebraic tools for
determining whether a Legendrian loop has infinite order. We then apply these
tools to show that a large class of Legendrian loop actions have infinite
order, recovering several previously known results.

###  1.1. Context

Recent work of Casals and Gao produced infinitely many exact Lagrangian
fillings for many Legendrian torus links [CG22], providing the first such
examples. They generate these fillings by understanding Legendrian loops, that
is, elements of the fundamental group of the space of Legendrian embeddings of
a given knot or link. Legendrian loops produce an action on exact Lagrangian
fillings by concatenating the trace of the loop with an existing filling. In
addition to the result cited above, Legendrian loops have been used to produce
various novel examples of families of Legendrian and Lagrangian submanifolds,
including the construction of infinitely many closed exact Lagrangian
submanifolds of Weinstein manifolds [CG22, CN22], infinitely many spherical
spun Legendrians [Gol22], and infinitely many Legendrian spheres in higher
dimensions [CSC23].

Despite the recent spate of results in this direction, the work of Casals and
Gao remains the only one to describe a non-cyclic group acting on the
Lagrangian concordance monoid. In particular, they obtain faithful
$\operatorname{PSL}_{2}(\mathbb{Z})$ and $\operatorname{Mod}(\Sigma_{0,4})$
actions in the case of Legendrian torus links $\Lambda(3,6)$ and
$\Lambda(4,4)$ [CG22, Theorem 1.1]. While their work predates the construction
of a cluster structure on the decorated sheaf moduli $\mathfrak{M}(\Lambda)$
[CW22, Theorem 1.1], they draw heavily on the work of Fraser [Fra18] in
understanding cluster automorphisms of top-dimensional positroid cells of the
Grassmannian.

Since the initial work of Casals and Gao, several works have appeared relating
cluster theory and the classification of exact Lagrangian fillings of positive
braid closures. These works include the proposal of a conjectural $ADE$
classification [Cas21, Conjecture 5.3], the existence of infinitely many exact
Lagrangian fillings for several families of Legendrians links [CZ21, GSW24],
computations of the cohomology of the augmentation variety [CGG+22], and the
existence of the conjectural number of fillings of several families of
Legendrian links [ABL22, Hug23b, CG23].

###  1.2. Main results

The main goal of this work is to initiate a systematic study of groups arising
from Legendrian loop actions on the Lagrangian concordance monoid. We do so by
interpreting Legendrian loops as _cluster_ automorphisms of the decorated
sheaf moduli $\mathfrak{M}(\Lambda)$. Cluster automorphisms form a group
called the cluster modular group, and the results of Casals and Gao can be
understood as producing a subgroup of the cluster modular group of the sheaf
moduli $\mathfrak{M}(\Lambda)$.

####  1.2.1. Cluster modular groups generated by Legendrian loops

For a given cluster algebra $\mathbb{A}$, the cluster modular group
$\mathcal{G}(\mathbb{A})$ is only known in a limited number of cases. To
describe explicit groups in these cases, we start by restricting our attention
to finite, affine, and extended affine Dynkin type cluster varieties arising
as the sheaf moduli of certain Legendrian links. These Legendrian links are
Legendrian isotopic to the rainbow closures $\Lambda(\beta)$ of certain
positive braids $\beta\in\operatorname{Br}_{n}^{+}$. See Figure 1 for a
depiction of these Legendrian braid closures as well as the Legendrian
isotopic $(-1)$-closures of $\beta\Delta_{n}^{2}$, as pictured in Figure 1,
with the full twist $\Delta_{n}^{2}$ given by the square of
$\Delta_{n}=\prod_{i=1}^{n}\prod_{j=1}^{n-i}\sigma_{j}$. Given a Dynkin type
$X$, we define the Legendrian link $\Lambda(X)$ to be the $(-1)$-framed
closure of the corresponding braid:

  * •

$A_{n-1}=\sigma_{1}^{n+2}$

  * •

$D_{n}=\sigma_{1}^{n-2}\sigma_{2}\sigma_{1}^{2}\sigma_{2}\Delta_{3}^{2}$ for
$n\geq 4$

  * •

$E_{6},E_{7},E_{8}=\sigma_{1}\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{i-3}\Delta_{3}^{2}$
for $i=6,7,$ and $8$, respectively

  * •

$\tilde{D}_{n}=(\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2}\sigma_{1}^{n-4}\Delta_{3}^{2}$
for $n\geq 4$

  * •

$\tilde{E}_{n}=(\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2}\sigma_{3}\sigma_{1}^{n-5}\Delta_{3}^{2}$
for $n\in\\{6,7,8\\}$

  * •

$E_{7}^{(1,1)}=(\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2}\sigma_{3}^{2}\sigma_{1}^{2}\Delta_{3}^{2}$

  * •

$E_{8}^{(1,1)}=(\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2}\sigma_{3}\sigma_{1}^{4}\Delta_{3}^{2}$

Here $\tilde{D}$ and $\tilde{E}$ denote affine Dynkin types $D$ and $E$.
Similarly, the superscript $(1,1)$ indicates extended affine type. Note that
$\Lambda(E_{7}^{(1,1)})$ and $\Lambda(E_{8}^{(1,1)})$ are Legendrian isotopic
to the Legendrian torus links $\Lambda(4,4)$ and $\Lambda(3,6)$, respectively
and that all of the above links are isotopic to rainbow closures
$\Lambda(\beta)$ for some $\beta\in\operatorname{Br}_{n}^{+}$.

Figure 1. Front projections of the Legendrian isotopic links given as the
rainbow closure (left) and $(-1)$-framed closure (right) of the positive
braids $\beta$ and $\beta\Delta^{2}$. Here $\Delta$ denotes a half twist of
the braid.

Let us denote by $\mathcal{H}$ the set of Legendrian links above. To obtain a
generating set for the cluster modular group, we must also include the cluster
automorphism known as the Donaldson-Thomas (DT) transformation, first
considered in this contact-geometric context in [GSW24, Section 4.1] and then
described geometrically by Casals and Weng in [CW22, Section 5.4]. The square
of the DT transformation can be realized as a Legendrian loop, but a single
power is defined as the composition of a Legendrian isotopy with a strict
contactomorphism. Studying the induced actions of Legendrian loops and the DT
transformation, we show that Legendrian loops generate a finite index subgroup
of the cluster modular group in the cases we consider.

######  Theorem 1.1.

For $\Lambda\in\mathcal{H}$, Legendrian loop actions generate a finite-index
subgroup of the cluster modular group $\mathcal{G}(\mathfrak{M}(\Lambda))$.
Moreover, for any $\Lambda\in\mathcal{H}$ excluding $\Lambda(\tilde{D}_{n})$,
Legendrian loop actions and the $\operatorname{DT}$ transformation generate
the group $\mathcal{G}(\mathfrak{M}(\Lambda))$.

For Legendrian links of affine and extended affine type, the cluster modular
groups are infinite, allowing us to recover the fact that these Legendrian
links admit infinitely many distinct exact Lagrangian fillings.

######  Remark 1.2.

As a consequence of [CG23, Theorem 1.1], any cluster automorphism of
$\mathfrak{M}(\Lambda(\beta))$ can be given as the induced action of a
sequence of Lagrangian disk surgeries applied to $\mathbb{L}$-compressing
cycles of $H_{1}(L)$ for any filling $L$ of $\Lambda$. However, realizing any
given sequence of mutations as a Legendrian loop is a nontrivial question, as
demonstrated in the case of $\Lambda(\tilde{D}_{n})$ in Appendix A. More
precisely, there is no known algorithm for showing that for a Legendrian loop
$\varphi$ whose induced action produces a sequence of mutations
$\mu_{n}\circ\dots\circ\mu_{1}$ the Lagrangian filling
$L^{\prime}=\mu_{n}\circ\dots\circ\mu_{1}$ produced by the sequence of
Lagrangian disk surgeries at the corresponding $\mathbb{L}$-compressing cycles
is Hamiltonian isotopic to the Lagrangian filling $\varphi(L)$.

The methods we use to prove Theorem 1.1 suggest ways to realize subgroups of
cluster modular groups as Legendrian loop actions, even in cases where the
full cluster modular group is not known. In the case of Legendrian torus links
$\Lambda(k,n)$, the sheaf moduli $\mathfrak{M}(\Lambda(k,n))$ is isomorphic to
the affine cone of the Grassmannian $\operatorname{Gr}^{\circ}(k,n+k)$. In
[Fra18, Conjecture 8.2], Fraser describes a conjectural presentation for the
cluster modular group of $\operatorname{Gr}^{\circ}(k,n+k)$ by studying braid
group actions on this space, giving us a roadmap for realizing these group
actions as these induced action of some Legendrian isotopy. See Subsection 5.3
for a more detailed description of this conjectural cluster modular group
$\mathcal{G}^{\prime}(k,n)$. By generalizing work of Casals and Gao in [CG22],
we show the following:

######  Theorem 1.3.

For any $k,n\geq 2$, Legendrian loop actions of $\Lambda(k,n)$ together with
the DT transformation generate a subgroup of
$\mathcal{G}(\mathfrak{M}(\Lambda(k,n))$ isomorphic to
$\mathcal{G}^{\prime}(k,n+k)$.

If we view Legendrian loops as elements of $\pi_{1}(\operatorname{Leg}(K))$
where $\operatorname{Leg}(K)$ is the space of Legendrian embeddings of a knot
or link $K$, then Theorems 1.1 and 1.3 can be interpreted as a statement about
actions of $\pi_{1}(\operatorname{Leg}(K))$ on $\mathfrak{M}(\Lambda)$ when
$\Lambda$ is a max-tb Legendrian representative of $K$ and
$\mathfrak{M}(\Lambda)$ admits a cluster structure. In particular, following
evidence given by Theorem 1.2.4 of [FMAP22], we conjecture that
$\pi_{1}(\operatorname{Leg}(K))$ is isomorphic to a finite-index subgroup of
the cluster modular group $\mathcal{G}(\mathfrak{M}(\Lambda(\beta)))$. Work of
Fernández and Min in this context also suggests a way to understand cluster
modular groups of more complicated cluster varieties by realizing the
corresponding Legendrian as a cable of a Legendrian $\Lambda$ with a known
cluster modular group $\mathcal{G}(\mathfrak{M}(\Lambda))$.

Following a remark in [Cas21, Section 5], we also consider Legendrian links
with exact Lagrangian fillings that are invariant under finite group actions.
Let $G$ be a finite group acting by exact symplectomorphisms or anti-exact
symplectomorphisms on $(\mathbb{R}^{4},\omega_{\text{st}})$, thereby inducing
an action by contactomorphisms on the boundary
$(\mathbb{R}^{3},\xi_{\text{st}})$. See Subsection 6.2 for explicit
constructions of the group actions that we consider. We define an exact
Lagrangian $G$-filling of $\Lambda$ to be an exact Lagrangian filling $L$ of
$\Lambda$ such that $G(L)=L$ and $G(\Lambda)=\Lambda$. Similarly, we refer to
a Legendrian loop $\varphi$ with trace $\operatorname{Tr}(\varphi)$ fixed by
$G$ as a $G$-loop.

In general, one can obtain a new cluster algebra by folding an existing one
along a $G$-action satisfying certain properties. Work of An, Bae, and Lee
[ABL22, Theorem 1.4] establishes the existence of the conjectured number of
$G$-fillings in this setting; see [Cas21, Conjecture 5.4] for a precise
formulation. We prove a $G$-equivariant version of Theorem 1.1 for the
following Legendrian links and $G$-actions:

$\displaystyle\mathcal{H}^{G}:=\\{(\Lambda(A_{2n-1}),\mathbb{Z}_{2}),(\Lambda(D_{4}),\mathbb{Z}_{3}),(\Lambda(\tilde{E}_{6}),\mathbb{Z}_{3}),(\Lambda(\tilde{D}_{2n}),\mathbb{Z}_{2}),(\Lambda(\tilde{D}_{4}),\mathbb{Z}_{2}),$
$\displaystyle(\Lambda(D_{n}),\mathbb{Z}_{2}),(\Lambda(E_{6}),\mathbb{Z}_{2}),(\Lambda(\tilde{E}_{6}),\mathbb{Z}_{2}),(\Lambda(\tilde{E}_{7}),\mathbb{Z}_{2}),(\Lambda(\tilde{D}_{4}),\mathbb{Z}_{2})\\}$

where the pair $(\Lambda,G)$ represents a Legendrian link and a $G$-action
fixing $\Lambda$. Note that while the positive braids we give above do not
necessarily admit obvious $G$-actions, the Legendrian links we consider admit
fronts that do exhibit the required symmetry. We compile the data of the
choice of such fronts in Table 3. Denote by $\mathfrak{M}(\Lambda)^{G}$ the
$G$-invariant subset of $\mathfrak{M}(\Lambda)$, i.e. the moduli of
$G$-invariant sheaves with singular support on $\Lambda$.

######  Theorem 1.4.

For $(\Lambda,G)\in\mathcal{H}^{G}$, the cluster modular group
$\mathcal{G}(\mathfrak{M}(\Lambda)^{G})$ is generated by $G$-invariant
Legendrian loops of $\Lambda$ and the DT transformation.

Implicit in the statement of Theorem 1.4 is the fact that the ring of regular
functions $\mathbb{C}[\mathfrak{M}(\Lambda)^{G}]$ is a skew-symmetrizable
cluster algebra for these choices of $\Lambda$ and $G$ with a
$\operatorname{DT}$ transformation obtained from the DT transformation on
$\mathfrak{M}(\Lambda)$. In such cases, imposing the $G$-invariance condition
on sheaves in $\mathfrak{M}(\Lambda)$ corresponds to the folding procedure
used to obtain skew-symmetrizable cluster algebras from skew-symmetric ones,
as is made precise in Theorem 6.6. See Subsection 6.3 for a contact-geometric
characterization of this cluster structure using the ingredients of [CW22,
Theorem 1.1]. We compile all of the information about the cluster modular
groups of Theorem 1.4 and specific $G$-actions in Table 3 below.

####  1.2.2. Properties of Legendrian loops

The second goal of this work is to use properties of cluster modular groups to
provide new methods for understanding Legendrian loop actions. While the
intimate connection between cluster modular groups and mapping class groups is
an area of ongoing research, work of Ishibashi gives a partial Nielsen-
Thurston classification for cluster automorphisms [Ish19]. By interpreting
Legendrian loops in this context, we obtain insight into certain fixed-point
properties of their actions. In addition, a process known as cluster reduction
gives us a method for linking many Legendrian loops actions to mapping classes
of decorated triangulations of punctured surfaces. We apply these insights to
give new tests for determining if a Legendrian loop $\varphi$ produces
infinitely many distinct exact Lagrangian fillings.

Given a cluster $\mathcal{A}$-space $\mathcal{A}$, the cluster-theoretic
analogue of Teichmüller space is its positive real part $\mathcal{A}_{>0}$,
obtained by declaring all cluster variables to be positive real numbers and
observing that the mutation formula is subtraction free. Analogous to the
fixed point property of periodic mapping classes, the following statement is
an immediate corollary of Ishibashi’s work on cluster modular groups.

######  Theorem 1.5.

Let $\Lambda$ be a Legendrian link whose sheaf moduli $\mathfrak{M}(\Lambda)$
admits a cluster structure. Any finite order Legendrian loop $\varphi$ of
$\Lambda$ has a fixed point in $\mathfrak{M}(\Lambda)_{>0}$.

As an immediate corollary, we obtain the fact that any Legendrian loop action
without fixed points in $\mathfrak{M}(\Lambda)_{>0}$ necessarily yields
infinitely many distinct exact Lagrangian fillings. See Example 7.9 for a
computation that recovers the fact that one of the Legendrian loops of
$\Lambda(3,6)$ considered by Casals and Gao has infinite order.

In addition to the above fixed point properties, we produce a combinatorial
method for determining when a Legendrian loop has infinite order by studying
the Legendrian loop action on the intersection quiver of a given exact
Lagrangian filling. We apply this method to study Legendrian
$\vartheta$-loops: loops of Legendrian links
$\Lambda=\Lambda(\beta\Delta^{2},i,\gamma)$ obtained by removing the $i$th
strand of the $(-1)$-framed closure of $\beta\Delta^{2}$ and replacing it with
the $(-1)$-framed closure of a positive braid $\gamma$; see [CN22, Definition
2.11] or Definition 2.1 below for a more precise description. While all of the
Legendrian links we apply this test to are already known to admit infinitely
many fillings, we also show that the induced action of $\vartheta$-loops on
$\mathfrak{M}(\Lambda)_{>0}$ is properly discontinuous. This allows us to
obtain a partial converse to Theorem 1.5.

######  Theorem 1.6.

Let $\Lambda=\Lambda(\beta\Delta^{2},i,\sigma_{1}^{k})$ with $k\geq 3$ and
assume that the induced action $\tilde{\vartheta}$ of the corresponding
$\vartheta$ loop has infinite order. Then $\tilde{\vartheta}$ has no fixed
points in $\mathfrak{M}(\Lambda)_{>0}$.

The class of Legendrian links that admit $\vartheta$-loops is quite broad,
containing positive torus links $\Lambda(k,kn)$, as well as all of the other
Legendrian links in the set $\mathcal{H}$ defined above. In fact, Theorem 1.1
can be restated using only $\vartheta$-loops as the generators of a finite
index subgroup of the cluster modular group.

The key technical result that allows us to conclude Theorem 1.6 is the cluster
reduction of the induced action of the $\vartheta$-loops in question to
(tagged) mapping classes of surface-type cluster algebras. As a result, the
combinatorial criterion we obtain for understanding the order of a loop can be
stated as follows:

######  Theorem 1.7.

The induced action of a Legendrian loop $\vartheta$ of
$\Lambda(\beta\Delta^{2},i,\sigma_{1}^{k})$ for $k\geq 3$ is infinite order if
and only if the cluster-reduced surface type quiver is that of an infinite
type cluster algebra.

We also give a recipe for obtaining the cluster reduction of $\vartheta$
directly from the front diagram, making the criterion of Theorem 1.7
relatively straightforward to apply.

Organization: Section 2 below reviews the necessary background material
related to constructions of exact Lagrangian fillings of Legendrian links,
while Section 3 covers their sheaf-theoretic invariants. Section 4 explains
how to build a cluster structure from these elements, following [CW22]. In
Sections 5 and 6 we prove Theorems 1.1, 1.3, and 1.4 and compile information
about the cluster modular groups associated to the specific Legendrian links
in Tables 1, 2, and 3. Finally, Section 7 contains proofs of Theorems 1.5 1.6,
and 1.7 as well as various example computations.

Acknowledgements:  Many thanks to Roger Casals for his support and
encouragement throughout this project. Thanks also to Daping Weng for
patiently answering all of my questions about cluster theory, as well as to
Lenny Ng, Eduardo Fernández, and Orsola Capovilla-Searle for helpful
conversations about Legendrian loops. Finally, thanks to Dani Kaufman for
useful discussions about cluster modular groups and to Eugene Gorsky for
comments on a very early draft of this work. This work was partially supported
by NSF grant DMS-1942363.

##  2\. Contact-geometric background

We begin with the necessary background on Legendrian links and their exact
Lagrangian fillings. The standard contact structure $\xi_{st}$ in
$\mathbb{R}^{3}$ is the 2-plane field given as the kernel of the 1-form
$\alpha_{\text{st}}=dz-ydx$. A link
$\Lambda\subseteq(\mathbb{R}^{3},\xi_{st})$ is Legendrian if $\Lambda$ is
always tangent to $\xi_{st}$. As $\Lambda$ can be assumed to avoid a point, we
can equivalently consider Legendrians $\Lambda$ contained in the contact
3-sphere $(\mathbb{S}^{3},\xi_{st})$ [Gei08, Section 3.2]. We consider
Legendrian links up to Legendrian isotopy, i.e. ambient isotopy through a
family of Legendrians.

The symplectization $\operatorname{Symp}(M,\ker(\alpha))$ of a contact
manifold $(M,\ker(\alpha))$ is the symplectic manifold $(\mathbb{R}_{t}\times
M,d(e^{t}\alpha))$. Given two Legendrian links
$\Lambda_{-},\Lambda_{+}\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$,

an exact Lagrangian cobordism
$L\subseteq\operatorname{Symp}(\mathbb{R}^{3},\ker(\alpha_{\text{st}}))$ from
$\Lambda_{-}$ to $\Lambda_{+}$ is a cobordism $\Sigma$ such that there exists
some $T>0$ satisfying the following:

  1. (1)

$d(e^{t}\alpha_{\text{st}})|_{\Sigma}=0$

  2. (2)

$\Sigma\cap((-\infty,T]\times\mathbb{R}^{3})=(-\infty,T]\times\Lambda_{-}$

  3. (3)

$\Sigma\cap([T,\infty)\times\mathbb{R}^{3})=[T,\infty)\times\Lambda_{+}$

  4. (4)

$e^{t}\alpha_{\text{st}}|_{\Sigma}=df$ for some function
$f:\Sigma\to\mathbb{R}$ that is constant on $(-\infty,T]\times\Lambda_{-}$ and
$[T,\infty)\times\Lambda_{+}$.

An exact Lagrangian filling of the Legendrian link
$\Lambda\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ is an exact Lagrangian
cobordism $L$ from $\emptyset$ to $\Lambda$ that is embedded in
$\operatorname{Symp}(\mathbb{R}^{3},\ker(\alpha_{\text{st}}))$. Equivalently,
we consider $L$ to be embedded in the symplectic 4-ball with boundary
$\partial L$ contained in contact $(\mathbb{S}^{3},\xi_{\text{st}})$.

We will depict a Legendrian link $\Lambda\subseteq(\mathbb{R}^{3},\xi_{st})$
in either of two projections; the front projection
$\Pi:(\mathbb{R}^{3},\xi_{\text{st}})\to\mathbb{R}^{2}$ given by
$\Pi(x,y,z)=(x,z)$ or the Lagrangian projection
$\pi:(\mathbb{R}^{3},\xi_{\text{st}})\to\mathbb{R}^{2}$ given by
$\pi(x,y,z)=(x,y)$.

###  2.1. Legendrian loops

Let $\Lambda$ be the $(-1)$-framed closure of a positive braid. A Legendrian
loop is, by definition, a Legendrian isotopy of $\Lambda$ that fixes $\Lambda$
setwise at time one. By construction, the trace of any Legendrian loop
$\varphi$ produces an exact Lagrangian concordance
$\operatorname{Tr}(\varphi):=\\{\\{t\\}\times\lambda_{t}|0\leq t\leq 1\\}$
from $\Lambda$ to itself in the symplectization
$\operatorname{Symp}(\mathbb{R}^{3},\alpha_{\text{st}})$. One then obtains an
action on the exact Lagrangian fillings of $\Lambda$ via concatenation; given
an exact Lagrangian filling $L$ of $\Lambda$, the filling $\varphi(L)$ is
defined to be the filling $L\cup_{\Lambda}\operatorname{Tr}(\varphi)$.

The Legendrian loops we consider here can be decomposed into elementary
Legendrian isotopies. In the front projection, these elementary isotopies
correspond to Reidemeister III moves, planar isotopies, and (counter)clockwise
rotations of a crossing. In terms of the braid word $\beta$, these are given
by the braid move
$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$,
commutation $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for $|i-j|\geq 2$, and
cyclic permutation $\sigma_{i}w=w\sigma_{i}$. We refer to the latter
Legendrian isotopy as a cyclic rotation and denote it by $\delta$, as in
[CG22].

Note that since $\Delta_{n}^{2}$ is a central in $\operatorname{Br}_{n}$, any
Legendrian link given as the $(-1)$-framed closure of $\beta\Delta_{n}^{2}$
admits a Legendrian loop $\delta^{|\beta|}$ described in the $(-1)$-closure as
a sequence of cyclic rotations, one for each crossing in $\beta$. We refer to
such a Legendrian loop as a full cyclic rotation. The full cyclic rotation
features heavily in [GSW24], where the authors used it to prove the existence
of infinitely many exact Lagrangian fillings of many Legendrians of the form
$\Lambda(\beta)$.

####  2.1.1. $\vartheta$-loops

We consider a particular class of Legendrian loops known as $\vartheta$-loops,
first appearing in the specific setting of Legendrian torus links considered
by Casals and Gao in [CG22]. We present the more general definition given in
[CN22, Section 2.4]. Any braid $\beta$ yields a permutation $w(\beta)\in
S_{n}$ via its Coxeter projection $w:\operatorname{Br}_{n}\to S_{n}$ encoding
where strands of the braid start and end. In the case that $w(\beta)$ admits a
fixed point $i\in\\{1,\dots,n\\}$, then the $(-1)$-framed closure of $\beta$,
which we will denote by $\Lambda$, contains a standard Legendrian unknot
$\Lambda_{i}$ corresponding to the $i$th strand of $\beta$. Since
$\Lambda_{i}$ is a Legendrian (un)knot, there is an open neighborhood
$\mbox{Op}(\Lambda_{i})$, disjoint from $\Lambda\backslash\Lambda_{i}$, that
is contactomorphic to $(J^{1}(\Lambda_{i}),\xi_{\text{st}})$. Let
$\gamma\in\operatorname{Br}_{m}^{+}$ be a positive braid in
$J^{1}\mathbb{S}^{1}$ and denote by $\Lambda(\gamma)_{i}$ the Legendrian link
obtained by satelliting $\gamma$ about $\Lambda_{i}$. Further denote by
$\Lambda(\beta,i,\gamma)$ the Legendrian link
$\Lambda\backslash\Lambda_{i}\cup\Lambda(\gamma)_{i}$, which we will refer to
as the Legendrian link obtained by satelliting $\gamma$ around the Legendrian
unknot $\Lambda_{i}$.

By construction, in the open neighborhood $\mbox{Op}(\Lambda_{i})$ there is a
contact flow along the radial coordinate of the companion unknot that yields a
compactly supported contact isotopy $\Theta_{t}$ of $\Lambda_{i}$. Denote by
$\tilde{\Theta}_{t}$ the extension of $\Theta_{t}$ by the identity to all of
$\mathbb{R}^{3}$.

######  Definition 2.1.

Let $\Lambda=\Lambda(\beta,i,\gamma)$ be a Legendrian satellite link with
$\gamma\in\operatorname{Br}_{m}^{+}$. A $\vartheta$-loop of $\Lambda$ is a
compactly supported contact isotopy $\tilde{\Theta}_{t}$ fixing
$\Lambda(\gamma)_{i}$ setwise by time one.

See Figure 15 for an example of a $\vartheta$-loop depicted in
$J^{1}\mathbb{S}^{1}$. Unless otherwise specified, we will impose the
additional condition that $\gamma\in\operatorname{Br}_{2}^{+}$ for the
remainder of this article.

####  2.1.2. Kálmán loops

We consider a class of loops of Legendrian torus links originally studied in
[Kál05]. Kálmán showed that the loops are nontrivial elements of
$\pi_{1}(\operatorname{Leg}(K(n,k)))$ by studying their induced action on the
augmentation variety of $\Lambda(k,n)$. These loops are readily understood by
observing that for $\Lambda(k,n)$, the full cyclic rotation $\delta^{(k-1)n}$
factors as $(\delta^{k-1})^{n}$. More precisely, in the front given as the
$(-1)$-framed closure of $\beta=(\sigma_{1}\dots\sigma_{k-1})^{n+k}$, the
Kálmán loop $\rho=\delta^{k-1}$ is defined to be the Legendrian loop given by
cyclic rotation of the first $k-1$ crossings. Kálmán’s main result [Kál05,
Theorem 1.3] is that the induced action of this loop on the augmentation
variety of $\Lambda(k,n)$ has order $n+k$.

####  2.1.3. Donaldson-Thomas transformations

Finally, we define the Donaldson-Thomas transformation, denoted by
$\operatorname{DT}$. While the DT transformation is not known to be induced by
a Legendrian loop, its square $\operatorname{DT}^{2}$ is Hamiltonian isotopic
to the full cyclic rotation in the case of rainbow closures $\Lambda(\beta)$
[GSW24, Lemma 4.2]. For a more general class of Legendrians, Casals and Weng
describe a Legendrian isotopy and a strict contactomorphism that together
induce the $\operatorname{DT}$ transformation [CW22, Section 5]. For rainbow
closures $\Lambda(\beta)$, this procedure can be described as first performing
half of a full cyclic rotation $(\delta^{|\beta|})^{1/2}$ by consecutively
rotating all of the crossings past the left cusps in a counterclockwise
direction. This isotopy is then composed with the strict contactomorphism
$x\mapsto-x,z\mapsto-z$ to return to the original front. See Subsection 4.4
for an alternative definition in terms of the combinatorics of plabic fences.

###  2.2. Legendrian weaves

Let us now describe Legendrian weaves, a geometric construction of Casals and
Zaslow that can be used to produce exact Lagrangian fillings of a Legendrian
link [CZ21]. The key idea of their construction is to combinatorially encode a
_Legendrian_ surface $\mathfrak{w}$ in the 1-jet space
$J^{1}\mathbb{D}^{2}=T^{*}\mathbb{D}^{2}\times\mathbb{R}_{z}$ by the
singularities of its front projection in $\mathbb{D}^{2}\times\mathbb{R}_{z}$.
The Lagrangian projection of $\mathfrak{w}$ then yields an exact Lagrangian
surface in $T^{*}\mathbb{D}^{2}$.

The contact geometric setup of the Legendrian weave construction is as
follows. We construct a filling of $\Lambda$ by first describing a local model
for a Legendrian surface $\mathfrak{w}$ in
$J^{1}\mathbb{D}^{2}=T^{*}\mathbb{D}^{2}\times\mathbb{R}_{z}$. We equip
$T^{*}\mathbb{D}^{2}$ with the symplectic form $d(e^{r}\alpha)$ where
$\ker(\alpha)=\ker(dy_{1}-y_{2}d\theta)$ is the standard contact structure on
$J^{1}(\partial\mathbb{D}^{2})$ and $r$ is the radial coordinate. This choice
of symplectic form ensures that the flow of $e^{r}\alpha$ is transverse to
$J^{1}\mathbb{S}^{1}\cong\mathbb{R}^{2}\times\partial\mathbb{D}^{2}$ thought
of as the cotangent fibers along the boundary of the 0-section. The Lagrangian
projection of $\mathfrak{w}$ is then a Lagrangian surface in
$(T^{*}\mathbb{D}^{2},d(e^{r}\alpha))$. Moreover, since
$\mathfrak{w}\subseteq(J^{1}\mathbb{D}^{2},\ker(dz-e^{r}\alpha))$ is a
Legendrian, we immediately obtain the function
$z:\pi(\mathfrak{w})\to\mathbb{R}$ satisfying
$dz=e^{r}\alpha|_{\pi(\mathfrak{w})}$, demonstrating that $\pi(\mathfrak{w})$
is exact.

The boundary of $\pi(\mathfrak{w})$ is taken to be a positive braid $\beta$ in
$J^{1}\mathbb{S}^{1}$ so that we may regard it as a Legendrian link in a
contact neighborhood of $\partial\mathbb{D}^{2}$. As the 0-section of
$J^{1}\mathbb{S}^{1}$ is Legendrian isotopic to a max-tb standard Legendrian
unknot, we can take $\partial\pi(\mathfrak{w})$ to equivalently be the
standard satellite of the standard Legendrian unknot. Diagramatically, this
implies that the braid $\beta$ in $J^{1}\mathbb{S}^{1}$ can be given as the
$(-1)$-framed closure of $\beta$ in contact $\mathbb{S}^{3}$.

####  2.2.1. $N$-Graphs and Singularities of Fronts

To construct a Legendrian weave surface $\mathfrak{w}$ in
$J^{1}\mathbb{D}^{2},$ we combinatorially encode the singularities of its
front projection in a colored graph. Local models for these singularities of
fronts are classified by work of Arnold [Ad90, Section 3.2]. The three
singularities that appear in our construction describe elementary Legendrian
cobordisms and are pictured in Figure 2.

Figure 2. Singularities of front projections of Legendrian surfaces. Labels
correspond to notation used by Arnold in his classification.

Since the boundary of our singular surface $\Pi(\mathfrak{w})$ is the front
projection of an $N$-stranded positive braid, $\Pi(\mathfrak{w})$ can be
pictured as a collection of $N$ sheets away from its singularities. We
describe the behavior at the singularities as follows:

  1. (1)

The $A_{1}^{2}$ singularity occurs when two sheets in the front projection
intersect. This singularity can be thought of as the trace of a constant
Legendrian isotopy in the neighborhood of a crossing in the front projection
of the braid $\beta\Delta^{2}$.

  2. (2)

The $A_{1}^{3}$ singularity occurs when a third sheet passes through an
$A_{1}^{2}$ singularity. This singularity can be thought of as the trace of a
Reidemeister III move in the front projection.

  3. (3)

A $D_{4}^{-}$ singularity occurs when three $A_{1}^{2}$ singularities meet at
a single point. This singularity can be thought of as the trace of a 1-handle
attachment in the front projection.

Having identified the singularities of fronts of a Legendrian weave surface,
we encode them by a colored graph $\Gamma\subseteq\mathbb{D}^{2}$. The edges
of the graph are labeled by Artin generators of the braid and we require that
any edges labeled $\sigma_{i}$ and $\sigma_{i+1}$ meet at a hexavalent vertex
with alternating labels while any edges labeled $\sigma_{i}$ meet at a
trivalent vertex. To obtain a Legendrian weave
$\mathfrak{w}(\Gamma)\subseteq(J^{1}\mathbb{D}^{2},\xi_{\text{st}})$ from an
$N$-graph $\Gamma$, we glue together the local germs of singularities
according to the edges of $\Gamma$. First, consider $N$ horizontal sheets
$\mathbb{D}^{2}\times\\{1\\}\sqcup\mathbb{D}^{2}\times\\{2\\}\sqcup\dots\sqcup\mathbb{D}^{2}\times\\{N\\}\subseteq\mathbb{D}^{2}\times\mathbb{R}$
and an $N$-graph $\Gamma\subseteq\mathbb{D}^{2}\times\\{0\\}$. We construct
the associated Legendrian weave $\mathfrak{w}(\Gamma)$ as follows:

  * •

Above each edge labeled $\sigma_{i}$, insert an $A_{1}^{2}$ crossing between
the $\mathbb{D}^{2}\times\\{i\\}$ and $\mathbb{D}^{2}\times\\{i+1\\}$ sheets
so that the projection of the $A_{1}^{2}$ singular locus under
$\pi:\mathbb{D}^{2}\times\mathbb{R}\to\mathbb{D}^{2}\times\\{0\\}$ agrees with
the edge labeled $\sigma_{i}$.

  * •

At each trivalent vertex $v$ involving three edges labeled by $\sigma_{i}$,
insert a $D_{4}^{-}$ singularity between the sheets
$\mathbb{D}^{2}\times\\{i\\}$ and $\mathbb{D}^{2}\times\\{i+1\\}$ in such a
way that the projection of the $D_{4}^{-}$ singular locus agrees with $v$ and
the projection of the $A_{2}^{1}$ crossings agree with the edges incident to
$v$.

  * •

At each hexavalent vertex $v$ involving edges labeled by $\sigma_{i}$ and
$\sigma_{i+1}$, insert an $A_{1}^{3}$ singularity along the three sheets in
such a way that the origin of the $A_{1}^{3}$ singular locus agrees with $v$
and the $A_{1}^{2}$ crossings agree with the edges incident to $v$.

Figure 3. The weaving of singularities of fronts along the edges of the
$N$-graph. Gluing these local models according to the $N$-graph $\Gamma$
yields the weave $\mathfrak{w}(\Gamma)$.

If we take an open cover $\\{U_{i}\\}_{i=1}^{m}$ of
$\mathbb{D}^{2}\times\\{0\\}$ by open disks, refined so that any disk contains
at most one of these three features, we can glue together the resulting fronts
according to the intersection of edges along the boundary of our disks.
Specifically, if $U_{i}\cap U_{j}$ is nonempty, then we define
$\Pi(\mathfrak{w}(U_{1}\cup U_{2}))$ to be the front resulting from
considering the union of fronts
$\Pi(\mathfrak{w}(U_{1}))\cup\Pi(\mathfrak{w}(U_{j}))$ in $(U_{1}\cup
U_{2})\times\mathbb{R}$.

######  Definition 2.2.

The Legendrian weave
$\mathfrak{w}(\Gamma)\subseteq(J^{1}\mathbb{D}^{2},\xi_{st})$ is the
Legendrian lift of the front $\Pi(\mathfrak{w}(\cup_{i=1}^{m}U_{i}))$ given by
gluing the local fronts of singularities together according to the $N$-graph
$\Gamma$.

The immersion points of a Lagrangian projection of a weave surface
$\mathfrak{w}$ correspond precisely to the Reeb chords of $\mathfrak{w}$. In
particular, if $\mathfrak{w}$ has no Reeb chords, then its Lagrangian
projection $L(\mathfrak{w})$ is an embedded exact Lagrangian filling of
$\partial(\mathfrak{w})$. In the Legendrian weave construction, Reeb chords
correspond to critical points of functions giving the difference of heights
between sheets. Every weave surface in this work admits an embedding where the
distance between the sheets in the front projection grows monotonically in the
direction of the boundary, ensuring that there are no Reeb chords.

####  2.2.2. Homology of Weaves

In this subsection, we describe the homology of a Legendrian weave
$\mathfrak{w}(\Gamma)$. The smooth topology of $\mathfrak{w}(\Gamma)$ is that
of an $N$-fold branched cover over $\mathbb{D}^{2}$ with simple branched
points corresponding to each of the trivalent vertices of $\Gamma$. Assuming
that $\mathfrak{w}(\Gamma)$ is connected, the genus of $\mathfrak{w}(\Gamma)$
is then computed using the Riemann-Hurwitz formula:

$g(\mathfrak{w}(\Gamma))=\frac{1}{2}(v(\Gamma)+2-N\chi(\mathbb{D}^{2})-|\partial\mathfrak{w}(\Gamma)|)$

where $v(\Gamma)$ is the number of trivalent vertices of $\Gamma$ and
$|\partial\mathfrak{w}(\Gamma)|$ denotes the number of boundary components of
$\Gamma$.

We now describe a recipe for combinatorially identifying particular elements
of $H_{1}(\mathfrak{w}(\Gamma));\mathbb{Z})$ known as $\mathbb{L}$-compressing
cycles. These are cycles such that their Lagrangian projection in
$L(\mathfrak{w}(\Gamma))$ bounds an embedded Lagrangian disk in the complement
of $L(\mathfrak{w}(\Gamma))$. As described in Section 4,
$\mathbb{L}$-compressing cycles play a crucial role in defining the cluster
seed associated to $L(\mathfrak{w})$. We first consider an edge connecting two
trivalent vertices. Closely examining the sheets of our surface, we can see
that each such edge corresponds to a 1-cycle. We refer to such a 1-cycle as a
short I-cycle. Similarly, any three edges of the same color that connect a
single hexavalent vertex to three trivalent vertices correspond to another
1-cycle, which we refer to as a short Y-cycle. See Figure 4 for examples of a
short I-cycle (left) and a short Y-cycle (right). We can also consider a
sequence of edges starting and ending at trivalent vertices and passing
directly through any number of hexavalent vertices, as pictured in Figure 5.
Such a cycle is referred to as a long I-cycle. Finally, we can combine any
number of I-cycles and short Y-cycles to describe a wide class of 1-cycles as
trees with leaves on trivalent vertices and edges passing directly through or
branching at hexavalent vertices.

Figure 4. Two local models of $\mathbb{L}$-compressing cycles in
$H_{1}(\mathfrak{w})$.  Figure 5. A pair of long I-cycles. The cycle on the
left passes through an even number of hexavalent vertices, while the cycle on
the right passes through an odd number.

The intersection form $\langle\cdot,\cdot\rangle$ on
$H_{1}(\mathfrak{w}(\Gamma))$ plays a key role in understanding cluster
structures coming from Lagrangian fillings. If we consider a pair of 1-cycles
$\gamma_{1},\gamma_{2}\in H_{1}(\mathfrak{w}(\Gamma))$ with a fixed
orientation and a nonempty geometric intersection in $\mathfrak{w}$, then we
can assign a sign to their intersection. We refer to the signed count of the
intersection of $\gamma_{1}$ and $\gamma_{2}$ as their algebraic intersection
and denote it by $\langle\gamma_{1},\gamma_{2}\rangle.$

####  2.2.3. Mutations of weaves

In this section we describe Legendrian mutation, a geometric operation that we
can use to generate additional Legendrian weaves. Given a Legendrian weave
$\mathfrak{w}$ and a 1-cycle $\gamma\in H_{1}(\mathfrak{w};\mathbb{Z})$ that
bounds an embedded Lagrangian disk in the complement
$T^{*}\mathbb{D}^{2}\backslash L(\mathfrak{w})$, the Legendrian mutation
$\mu_{\gamma}(\mathfrak{w})$ outputs a Legendrian weave smoothly isotopic to
$\mathfrak{w}$ but that is generally not Legendrian isotopic to
$\mathfrak{w}$.

Combinatorially, we can describe Legendrian mutation in terms of the $N$-graph
associated to a weave. Figure 6 (left) depicts mutation at a short I-cycle,
while Figure 6 (right) depicts mutation at a short Y-cycle. See [CZ21, Section
4.9] for a more general description of mutation at long I\- and Y-cycles in
$N$-graphs. The geometric operation above coincides with the combinatorial
manipulation of the $N$-graphs. Specifically, for any two $N$-graphs, $\Gamma$
and $\Gamma^{\prime}$ related by either of the combinatorial moves described
in Figure 6, the corresponding Legendrian weaves $\mathfrak{w}(\Gamma)$ and
$\mathfrak{w}(\Gamma^{\prime})$ are mutation-equivalent relative to their
boundary [CZ21, Theorem 4.2.1].

Figure 6. Mutations at the orange short I-cycle (left) and short Y-cycle
(right) together with the associated intersection quivers. In both cases, the
dark green edge depicts the effect of mutation on an arbitrary cycle
intersecting the orange cycle.

####  2.2.4. Quivers from Weaves

We complete our discussion of Legendrian weaves by describing quivers and how
they arise via the intersection form on $H_{1}(\mathfrak{w};\mathbb{Z}).$ A
quiver is a directed graph without loops or oriented 2-cycles. In the
Legendrian weave setting, the data of a quiver can be extracted from a given
weave and a basis of its first homology via the intersection form. The
intersection quiver is defined as follows: for every basis element
$\gamma_{i}\in H_{1}(\mathfrak{w}(\Gamma);\mathbb{Z})$ we have a vertex
$v_{i}$ in the quiver; there are $k$ arrows pointing from $v_{j}$ to $v_{i}$
if $\langle\gamma_{i},\gamma_{j}\rangle=k$ for $k>0$.

The combinatorial operation of quiver mutation at a vertex $v$ is defined as
follows, see e.g. [FWZ20a, Definition 2.1.2]. First, for every pair of
incoming edges and outgoing edges, we add an edge starting at the tail of the
incoming edge and ending at the head of the outgoing edge. Next, we reverse
the direction of all edges adjacent to $v$. Finally, we cancel any directed
2-cycles. If we started with the quiver $Q$, then we denote the quiver
resulting from mutation at $v$ by $\mu_{v}(Q).$

The following theorem relates the two operations of quiver mutation and
Legendrian mutation:

######  Theorem 2.3 ([CZ21], Section 7.3).

Given an $N$-graph $\Gamma$, Legendrian mutation at an embedded cycle $\gamma$
induces a quiver mutation of the associated intersection quivers, taking
$Q(\Gamma,\\{\gamma_{i}\\})$ to $\mu_{\gamma}(Q(\Gamma,\\{\gamma_{i}\\})).$
$\Box$

##  3\. Microlocal theory of sheaves

In this section we define the moduli stacks $\mathfrak{M}(\Lambda,T)$ and
$\mathcal{M}_{1}(\Lambda)$ of microlocal rank-one sheaves with prescribed
singular support and describe the sheaf-theoretic components of the cluster
$\mathcal{A}$ and $\mathcal{X}$ spaces considered in [CW22].

###  3.1. Sheaves on Legendrian links

We discuss here the microlocal theory of sheaves as it appears in the setting
of Legendrian links and their exact Lagrangian fillings. In full generality,
the sheaf moduli we consider are constructed as the moduli of certain objects
of a particular subcategory of the dg-derived category of sheaves of
$k$-modules with singular support contained in a given Legendrian link. For a
brief review of the appropriate categorical framework, see [CW22, Section 2.7]
or [CL22, Appendix A].

Given a constructible sheaf $F\in Sh(M)$, the singular support $SS(F)$ of $F$
is a conical Lagrangian in $T^{*}M$. Quotienting by the natural
$\mathbb{R}$-action yields a Legendrian in the unit-cotangent bundle
$T^{\infty}M$. Fixing a co-orientation along any front projection
$\Pi_{xz}(\Lambda)\subseteq\mathbb{R}^{2}$ of a Legendrian $\Lambda$ allows us
to equivalently consider sheaves $F$ with $SS(F)$ as a subset of
$\mathbb{R}^{3}$ or $J^{1}\mathbb{S}^{1}$. In [STZ17], the authors analyze
singular support conditions in terms of a stratification of $\mathbb{R}^{2}$
induced by a front projection. By [STZ17, Proposition 5.17], as all Legendrian
links $\Lambda\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ we consider admit
front projections in $(J^{1}\mathbb{S}^{1},\xi_{\text{st}})$ without cusps, we
may consider sheaves of vector spaces concentrated in degree 0 rather than
working with sheaves of chain complexes. We then obtain a combinatorial
description of sheaves $F$ with singular support $SS(F)\subseteq\Lambda$ as
follows. For $\Pi_{xz}(\Lambda)$ a front of a Legendrian satisfying the
conditions given above, we associate to each region of
$\mathbb{R}^{2}\backslash\Pi_{xz}(\Lambda)$ a vector space – the stalk of $F$
within that region – such that the unbounded region of $\Pi_{xz}(\Lambda)$ is
assigned the zero vector space. The appropriate singular support conditions
are encoded by conditions on the maps induced by restriction near arcs,
crossings, and cusps.

######  Proposition 3.1 ([STZ17]).

Let $\Lambda$ be a front with binary Maslov potential. A sheaf $F$ on
$\mathbb{R}^{2}$ satisfying $SS(F)\subseteq\Lambda$ is given by the following
conditions on the vector spaces assigned to each region of
$\Pi_{xz}(\Lambda)$.

  1. (1)

The zero vector space is assigned to the unbounded region.

  2. (2)

The vector spaces assigned to any two regions separated by an arc differ in
dimension by $r$.

  3. (3)

At each cusp, the composition of maps in Figure 7 is the identity.

  4. (4)

At each crossing, the sequence $0\to S\to E\oplus W\to N\to 0$ is exact and
the four maps form a commuting square.

Figure 7. Local models of a front projection of $\Lambda$ and the maps induced
by restriction of sheaves.

Such sheaves are necessarily constructible with respect to a stratification
induced by a front projection of $\Lambda$. The crossing condition (4) can
also be phrased as the existence of an isomorphism between the cokernels of
the maps $S\to E$ and $W\to N$ – equivalently, $S\to W$ and $E\to N$. This
isomorphism at each crossing of $\Pi_{xz}(\Lambda)$ yields a rank-$r$ local
system on $\Lambda$ that we refer to as the microlocal monodromy of $\Lambda$.
The number $r$ is the microlocal rank of $F$. We require $r=1$ and denote by
$\mathcal{M}_{1}(\Lambda)$ the moduli of microlocal rank-one sheaves $F$ with
$SS(F)\subseteq\Lambda$. By [GKS12], $\mathcal{M}_{1}(\Lambda)$ is a
Legendrian isotopy invariant of $\Lambda$.

Following Proposition 3.1, we can describe the sheaf moduli as a space of
flags with certain transversality conditions given by the front projection as
follows. Let $\Lambda\subseteq J^{1}\mathbb{S}^{1}$ be the $(-1)$-framed
closure of an $N$-stranded positive braid $\beta\in\operatorname{Br}_{N}^{+}$.
The crossing singularities of the front $\Pi_{xz}(\Lambda)\subseteq
S^{1}\times\mathbb{R}$ divide $S^{1}\times\mathbb{R}$ into regions
$[a_{i},a_{i+1}]\times\mathbb{R}$. Given our description of
$\mathcal{M}_{1}(\Lambda)$ above, each sheaf $F\in\mathcal{M}_{1}(\Lambda)$
can be characterized as the assignment of a locally constant vector space in
each region of $[a_{i},a_{i+1}]\times\mathbb{R}$, divided by the strands of
$\beta$. The microlocal rank-one condition implies that the rank of these
vector spaces increases by one as we pass from a region to the one immediately
above it. This sequence of vector spaces $0\subseteq
V_{i}^{1}\subseteq\dots\subseteq V_{i}^{n}=\mathbb{C}^{n}$ can be thought of
as a flag $V_{i}^{\bullet}$ in $\mathbb{C}^{N}$. The singular support
condition at the crossing corresponding to the Artin generator $\sigma_{j}$
implies that the two flags $V_{i}^{\bullet}$ and $V_{i+1}^{\bullet}$ differ at
the $j$th position; in other words, $V_{i}^{j}$ is transverse to
$V_{i+1}^{j}$. The space $\mathcal{M}_{1}(\Lambda)$ can therefore be
understood as the space of flags satisfying these transversality conditions,
modulo a choice of basis.

###  3.2. Sheaves on Legendrian weaves

In order to build cluster structures on $\mathcal{M}_{1}(\Lambda)$ and the
decorated sheaf moduli $\mathfrak{M}(\Lambda)$ described in the following
subsection, we must explain how an embedded exact Lagrangian filling of
$\Lambda$ yields a toric chart $(C^{\times})^{n}$ in the (decorated) sheaf
moduli of $\Lambda$. By work of Jin and Treumann, an embedded exact Lagrangian
filling $L$ of $\Lambda$ has sheaf moduli equivalent to the space of local
systems on $L$ via the microlocal monodromy functor applied to the Legendrian
lift of $L$ [JT17, Section 1.7]. Restricting to rank-one local systems on $L$
yields an embedding of a toric chart
$(\mathbb{C}^{\times})^{b_{1}(L)}\hookrightarrow\mathcal{M}_{1}(\Lambda)$
where $b_{1}(L)$ denotes the first Betti number of $L$. We can compute these
local systems explicitly by analyzing the singularities of the fronts of
Legendrian weaves.

As described in [CZ21, Section 5.3] the data of
$\mathcal{M}_{1}(\mathfrak{w}(\Gamma))$ is equivalent to providing:

  1. (i)

An assignment to each region $R$ (connected component of
$\mathbb{D}^{2}\backslash\Gamma$) of a flag $V^{\bullet}(R)$ in the vector
space $\mathbb{C}^{N}$.

  2. (ii)

For each pair $R_{1},R_{2}$ of adjacent regions sharing an edge labeled by
$\sigma_{i}$, we require that the corresponding flags satisfy

$V^{j}(R_{1})=V^{j}(R_{2}),\qquad 0\leq j\leq N,j\neq i,\qquad\text{ and
}\qquad V^{i}(R_{1})\neq V^{i}(R_{2}).$

Finally, we consider the space of flags satisfying (i) and (ii) modulo the
diagonal action of $GL_{N}(\mathbb{C})$ on $V^{\bullet}$. By [CZ21, Theorem
5.3], the flag moduli space is isomorphic to the space of microlocal rank-one
sheaves $\mathcal{M}_{1}(\mathfrak{w}(\Gamma))$.

To better understand local systems on $\mathfrak{w}(\Gamma)$, we give examples
of the flag moduli space in a neighborhood of homology cycles of
$\mathfrak{w}(\Gamma)$. In the short I-cycle case, when the edges are labeled
by $\sigma_{1}$, the moduli space is determined by four lines $a\neq b\neq
c\neq d\neq a$, as pictured in Figure 8 (left). Around a short Y-cycle, the
data of the flag moduli space is given by three distinct planes $A\neq B\neq
C\neq A$ contained in $\mathbb{C}^{3}$ and three distinct lines $a\subsetneq
A,b\subsetneq B,c\subsetneq C$ with $a\neq b\neq c\neq a,$ as pictured in
Figure 8 (right).

Figure 8. The data of the flag moduli space given in the neighborhood of a
short I-cycle (left) and a short Y-cycle (right). Lines are represented by
lowercase letters, while planes are written in uppercase. The intersection of
the two lines $a$ and $b$ is written as $ab$.

As in the case of a Legendrian $\Lambda$ in $J^{1}\mathbb{S}^{1}$, Condition
(4) of Proposition 3.1 allows us to compute the microlocal monodromy about an
absolute cycle $\gamma\in H_{1}(L(\mathfrak{w}))$ as a composition of
isomorphisms between cokernels of maps of vector spaces. Since it is locally
defined, we can compute the microlocal monodromy about an I-cycle or Y-cycle
using the data of the flag moduli space in a neighborhood of the cycle. If we
have a short I-cycle $\gamma$ with flag moduli space described by the four
lines $a,b,c,d$, as in Figure 8 (left), then the microlocal monodromy about
$\gamma$ is given by the cross ratio

$\frac{a\wedge b}{b\wedge c}\frac{c\wedge d}{d\wedge a}$

Similarly, for a short Y-cycle with flag moduli space given as in Figure 8
(right), the microlocal monodromy is given by the triple ratio

$\frac{B(a)C(b)A(c)}{B(c)C(a)A(b)}$

where we interpret the plane $B$ as a covector in $\mathbb{C}^{3}$ to define
the pairing $B(a)$.

####  3.2.1. The decorated sheaf moduli

In preparation for describing cluster $\mathcal{A}$ spaces in the next
section, we introduce the decorated sheaf moduli $\mathfrak{M}(\Lambda,T)$. To
do so, we must first specify a trivialization of the abelian local system
given by the microlocal monodromy about each component of $\Lambda$. Let
$T=\\{t_{1},\dots,t_{k}\\}$ be a set of marked points on $\Lambda$ with every
component of $\Lambda$ carrying at least one marked point. Label the connected
components of $\Lambda\backslash T$ by the pair of endpoints of each segment
$(t_{i},t_{i+1})$ where indices are taken modulo the number of marked points
on the appropriate component. The decorated moduli stack
$\mathfrak{M}(\Lambda,T)$ is then given by the additional data of a
trivialization for every segment of each component of $\Lambda$.

$\mathfrak{M}(\Lambda,T):=\\{(F,\phi_{1},\dots,\phi_{k})|F\in\mathcal{M}_{1}(\Lambda),\phi_{i}\text{
is a trivialization of }\mathfrak{m}_{\Lambda}\text{ on }(t_{i},t_{i+1})\\}$

Two framings are equivalent if they differ by a global factor of
$\mathbb{C}^{\times}$. As in [CW22, Section 2.8] we define the moduli space
$\mathcal{M}_{1}(\Lambda,T)$ analogously by the addition of the trivialization
data associated with the marked points. In this work, we choose to suppress
$T$ from our notation when our discussion does not depend on the presence of
marked points.

The data of these trivializations have two primary motivations. First, it
allows us to describe $\mathfrak{M}(\Lambda,T)$ as a smooth affine scheme
rather than an Artin stack. Second, the trivializations give us the necessary
data for defining microlocal merodromies as a microlocal version of parallel
transport, yielding regular functions on $\mathfrak{M}(\Lambda,T)$, as we now
describe.

To describe cluster-$\mathcal{A}$ coordinates on $\mathfrak{M}(\Lambda,T)$, we
start with a basis $\\{\gamma_{i}\\}_{i=1}^{n}$ of $H_{1}(L,T)$ containing a
maximal linearly independent subset of $\mathbb{L}$-compressing cycles of $L$.
We then identify the lattice $H_{1}(L,T)$ with an isomorphic lattice
$H_{1}(L\backslash T,\Lambda\backslash T)$, and consider the dual basis of
cycles $\\{\gamma_{i}^{\vee}\\}_{i=1}^{n}$.111In order to more naturally
incorporate frozen cluster variables, one can also consider intermediate
lattices $M$ and $N$ satisfying $H_{1}(L)\hookrightarrow N\hookrightarrow
H_{1}(L,T)$ and $H_{1}(L,\Lambda)\twoheadleftarrow M\twoheadleftarrow
H_{1}(L\backslash T,\Lambda\backslash T)$. See [CW22, Remark 3.50] for more
details.

Given an oriented relative cycle $\gamma^{\vee}\in H_{1}(L\backslash
T,\Lambda\backslash T)$ starting at $s$ and ending at $t$ we first note that
the framing data of $\mathfrak{M}(\Lambda,T)$ specifies two vectors
$\phi_{s}\in\Phi_{s}$ and $\phi_{t}\in\Phi_{t}$ at $s$ and $t$. The result of
parallel transport along $\gamma^{\vee}$ yields a nonzero vector
$\gamma^{\vee}(\phi_{s})\in\Phi_{t}$.222In this section, we omit the
discussion of sign curves appearing in [CW22, Section 4.5].

######  Definition 3.2.

Given an $\mathbb{L}$ compressing cycle $\gamma\in H_{1}(L,T)$, the microlocal
merodromy along its dual relative cycle $\gamma^{\vee}\in H_{1}(L\backslash
T,\Lambda\backslash T)$ is the function
$A_{\gamma^{\vee}}=\gamma^{\vee}(\phi_{s})/\phi_{t}$.

One can compute $A_{\gamma^{\vee}}$ explicitly from a Legendrian weave
$\mathfrak{w}$ by lifting the framing data to a set of decorations. Given a
flag $V^{\bullet}$ with framing data $\phi_{i}=V^{i}/V^{i-1}$, we can
construct a volume form $\alpha_{i}\in\bigwedge^{i}V^{i}$ by first lifting
$\phi_{i}$ to a nonzero vector $\tilde{\phi}_{i}\in V_{i}$ and then setting
$\alpha_{i}=\tilde{\phi}_{i}\wedge\dots\wedge\tilde{\phi}_{1}.$ At an edge of
$\mathfrak{w}$ labeled by $\sigma_{i}$, we have flags $\mathcal{L}^{\bullet}$
and $\mathcal{R}^{\bullet}$ to the left and right of the $\sigma_{i}$ edge
with framing data $\\{\lambda_{i}\\}$ and $\\{\rho_{i}\\}$, respectively.
Denote by $\\{\alpha_{i}\\}$ and $\\{\beta_{i}\\}$ the decorations
corresponding to the framings on $\mathcal{L}^{\bullet}$ and
$\mathcal{R}^{\bullet}$. The parallel transport of $\lambda_{i}$ along an
oriented curve $\eta$ from the $i$th sheet on the left to the $i+1$st sheet on
the right can then be computed by

(1)
$\eta(\lambda_{i})=\left(\frac{\tilde{\lambda}_{i}\wedge\tilde{\rho}_{i}\wedge\alpha_{i-1}}{\tilde{\rho}_{i+1}\wedge\beta_{i}}\right)\rho_{i+1}.$

Similarly, the parallel transport of $\lambda_{i+1}$ along the oriented curve
$\eta^{\prime}$ from the $i+1$st sheet on the left to the $i$th sheet on the
right yields

(2)
$\eta(\lambda_{i+1})=\left(\frac{\alpha_{i+1}}{\tilde{\rho}_{i}\wedge\tilde{\lambda}_{i}\wedge\alpha_{i-1}}\right)\rho_{i}.$

Composing Equations 1 and 2, allows us to compute the microlocal merodromy
along any oriented curve in $\mathfrak{w}$. By [CW22, Proposition 4.29], the
microlocal merodromy $A_{\gamma^{\vee}}$ along a relative cycle
$\gamma^{\vee}\in H_{1}(\mathfrak{w}\backslash T,\Lambda\backslash T)$ dual to
an $\mathbb{L}$-compressing cycle $\gamma$ is a regular function on
$\mathfrak{M}(\Lambda,T)$. As we explain in the following section, this
regular function is actually a cluster variable, allowing us to define a
cluster-$\mathcal{A}$ structure on $\mathfrak{M}(\Lambda,T)$.

##  4\. Cluster algebras and cluster ensembles

In this section, we define cluster algebras and cluster varieties and describe
how they appear in a contact-geometric context. We start by introducing the
basic definitions and then discuss some specific cluster algebras related to
tagged triangulations of surfaces. The subsection ends with a description of
cluster modular groups. See [FWZ20a, FWZ20b] for an introductory reference on
cluster theory.

###  4.1. Cluster algebras

To an initial quiver $Q_{0}$ with $n$ vertices, we associate an initial set of
variables $a_{1},\dots a_{n}$, one for each vertex. Together, the $n-$tuple
$\bf{a}=(a_{1},\dots,a_{n})$ and the quiver $Q_{0}$ form a cluster seed
$(\bf{a},Q_{0})$. We designate a subset of vertices $Q_{0}^{mut}$ to be the
mutable part of the quiver. The vertices in $Q_{0}\backslash Q_{0}^{mut}$ are
designated as frozen and we require that there are no arrows between them. For
a cluster seed $(\bf{x},Q)$ we can denote by $b_{ij}$ the multiplicity of
arrows in $Q$ from vertex $i$ to vertex $j$. We then obtain a skew-symmetric
matrix, $B=(b_{ij})$, known as the exchange matrix, encoding the information
of the quiver. There are two types of cluster algebras, type $\mathbb{A}$ or
type $\mathbb{X}$, depending on the precise form of the mutation formula
relating different cluster variables.

######  Definition 4.1.

Let $(\bf{a},Q)$ be a cluster seed and $k\in Q^{mut}$ be a mutable vertex. The
cluster $\mathbb{A}$ seed mutation $\mu_{k}$ is an operation taking as input
the seed $(\bf{a},Q)$ and outputting the new seed
$(\bf{a}^{\prime},Q^{\prime})$ where $Q^{\prime}$ is related to $Q$ by quiver
mutation at vertex $k$ and $\bf{a}^{\prime}$ is related to $\bf{a}$ by
$a_{i}^{\prime}=a_{i}$ for all $i\neq k$ and

$a_{k}a_{k}^{\prime}=\prod_{b_{ik}>0}a_{i}^{b_{ik}}+\prod_{b_{ik<0}}a_{i}^{-b_{ik}}.$

Note that seed mutation is an involution, so that
$\mu_{k}^{2}(\bf{a},Q)=(\bf{a},Q)$.

Denote by $\mathcal{F}$ the field of rational functions
$\mathbb{C}(a_{1},\dots a_{n})$ and consider an initial seed
$(\bf{a},Q_{0})\subseteq\mathcal{F}$.

######  Definition 4.2.

The type $\mathbb{A}$ cluster algebra generated by $(\bf{a},Q_{0})$ is the
$\mathbb{C}$-algebra generated by all cluster variables arising in arbitrary
mutations of the initial seed.

The type $\mathbb{X}$ cluster algebra is generated from an initial seed
$(\bf{x},Q_{0})$ by the mutation formula

$x_{j}^{\prime}=\begin{cases}x_{j}^{-1}&i=j\\\ x_{j}(y_{k}+1)^{-b_{kj}}&j\neq
k,b_{kj}\leq 0\\\ x_{j}(x_{k}^{-1}+1)^{-b_{kj}}&j\neq k,b_{kj}\geq
0\end{cases}$

A cluster algebra is of finite type if it has only finitely many distinct
cluster seeds. Otherwise, it is of infinite type. Cluster algebras admit an
ADE classification.

######  Theorem 4.3 (Theorem 1.4, [FZ03]).

Cluster algebras are of finite type if and only if their quiver is mutation
equivalent to a Dynkin diagram of finite type with any orientation given to
its edges

Two quivers are mutation equivalent if one can be obtained by applying a
sequence of quiver mutations to the other. For the simply-laced cases, this
classification restricts our attention to ADE-type. We discuss some of the
combinatorial ingredients for understanding cluster algebras of types $A_{n}$
and $D_{n}$ in Subsection 4.3.1. Beyond cluster algebras of finite type, the
next simplest families are cluster algebras arising from quivers of finite
mutation type. These are cluster algebras with an underlying quiver that is
mutation equivalent to only finitely many quivers. Cluster algebras from
quivers of finite mutation type are classified in [FST12]. Among the finite
mutation type cluster algebras, we have types $\tilde{A}_{n},\tilde{D}_{n},$
and others corresponding to triangulations of surfaces (See e.g. [FST08]), as
well as types
$\tilde{E}_{6},\tilde{E}_{7},\tilde{E}_{8},E_{6}^{(1,1)},E_{7}^{(1,1)},E_{8}^{(1,1)}$
and two additional exceptional quivers.

###  4.2. Cluster ensembles from Legendrian links

Following [FG09], we give a brief description of the notion of a (skew-
symmetric)333For the skew-symmetrizable case, see Section 6. cluster ensemble.
We then give the necessary ingredients to understand how this structure arises
from the pair of sheaf moduli $\mathfrak{M}(\Lambda,T)$ and
$\mathcal{M}_{1}(\Lambda,T)$.

A cluster ensemble consists of a pair of schemes $\mathcal{A}$ and
$\mathcal{X}$ formed by birationally gluing algebraic tori according to
certain input data. The spaces $\mathcal{A}$ and $\mathcal{X}$ are dual in the
following sense. Consider an integer lattice $N$ with a skew-symmetric
bilinear form containing a saturated sublattice $N_{\operatorname{uf}}$ of $N$
known as the unfrozen sublattice. Denote the dual lattice
$\operatorname{Hom}(N,\mathbb{Z})$ by $M$. The lattice $N$ forms the character
lattice of a cluster torus in the $\mathcal{X}$ variety, while the lattice $M$
forms the character lattice of a cluster torus in the $\mathcal{A}$ variety.
Cluster tori are glued together by the birational map induced by the mutation
formulas given in the previous subsection, with the skew-symmetric bilinear
form giving the information of the exchange matrix. The ring of regular
functions $\mathcal{O}(\mathcal{A})$ forms an (upper) cluster algebra.

An abbreviated statement of the main theorem of Casals and Weng tells us that
for certain families of Legendrians $\Lambda$, the moduli
$\mathcal{M}_{1}(\Lambda,T)$ and $\mathfrak{M}(\Lambda,T)$ form a cluster
ensemble. The class of Legendrians they consider arises from a combinatorial
construction known as a (complete) grid plabic graph. This class includes all
Legendrian links considered in this work. A more precise summary of the main
result of Casals and Weng is as follows:

######  Theorem 4.4 (Theorem 1.1 [CW22]).

For $\Lambda$ a Legendrian arising from a complete grid plabic graph, the
decorated sheaf moduli $\mathfrak{M}(\Lambda,T)$ admits a cluster
$\mathcal{A}$ structure. Moreover, there is an explicitly constructed
Legendrian weave filling $L$ of $\Lambda$ with intersection quiver and sheaf
moduli $\mathfrak{M}(L)$ giving the data of the initial seed.

Recall that an $\mathbb{L}$-compressing cycle of a filling $L$ is a homology
cycle $\gamma\in H_{1}(L)$ that bounds an embedded Lagrangian disk in the
complement of $L$. Casals and Weng obtain mutable cluster $\mathcal{A}$
coordinates by computing microlocal merodromies along relative homology cycles
of $L$ dual to $\mathbb{L}$-compressing cycles [CW22, Section 4]. They also
obtain mutable cluster $\mathcal{X}$ variables of $\mathcal{M}_{1}(\Lambda)$
as microlocal monodromies about these $\mathbb{L}$-compressing cycles. These
two homology lattices $H_{1}(L,\mathfrak{t})$ and
$H_{1}(L\backslash\mathfrak{t},\Lambda\backslash\mathfrak{t})$ yield the dual
lattices appearing in the definition of a cluster ensemble. Legendrian
mutation of an $\mathbb{L}$-compressing cycle induces a cluster-$\mathcal{X}$
mutation on toric charts in $\mathcal{M}_{1}(\Lambda,T)$, while Legendrian
mutation of its dual induces a cluster-$\mathcal{A}$ mutation on toric charts
in $\mathfrak{M}(\Lambda,T)$. Frozen variables correspond to either marked
points or to homology cycles in a Legendrian weave filling $L$ that do not
bound embedded Lagrangian disks in the complement $\mathbb{D}^{4}\backslash
L$.

######  Remark 4.5.

The cluster structures of [CW22] are defined up to quasi-cluster equivalence.
In order to simplify exposition, we omit any consideration of quasi-cluster
equivalences and refer the interested reader to [CW22, Appendix A] for further
details.

###  4.3. Cluster modular groups

Given a cluster ensemble, one can consider maps that act on the ensemble by
permuting the cluster tori in a way that respects the cluster structure. More
precisely, a cluster automorphism of a cluster algebra $\mathbb{A}$ is a
permutation of the cluster variables of $\mathbb{A}$ that sends cluster seeds
to cluster seeds and commutes with mutation. Any cluster automorphism can be
defined by the image of a single seed and necessarily preserves the underlying
quiver up to a simultaneous change of orientation on all of the arrows [ASS12,
Proposition 2.4]. This leads to the following definition:

######  Definition 4.6.

The (orientation preserving) cluster modular group $\mathcal{G}(\mathbb{A})$
of a cluster algebra $\mathbb{A}$ is the group of maps $\pi$ permuting cluster
variables and commuting with mutation such that the induced map on quivers
$Q({\bf x})\to Q(\pi({\bf x}))$ is an (orientation preserving) quiver
automorphism.

######  Example 4.7.

For an $A_{2}$ cluster algebra, the cluster modular group $\mathcal{G}(A_{2})$
is isomorphic to $\mathbb{Z}_{5}$ and is generated by a $2\pi/5$ rotation of
the triangulation corresponding to the initial quiver. Viewing each diagonal
$D_{i,j}$ of the triangulation as the image of a Plücker coordinate
$\Delta_{i,j}$ in the top-dimensional positroid strata of the Grassmannian
$\operatorname{Gr}(2,5)$, we can see that this cluster automorphism is given
by the map $\Delta_{i,j}\mapsto\Delta_{i-1,j-1}$ for all $1\leq i<j\leq 5$.

For all classes of cluster algebras discussed in this work, any cluster
automorphism $\varphi$ can be given as a finite sequence of mutations
$\mu_{v_{1}},\mu_{v_{2}},\dots,\mu_{v_{m}}$ and a permutation $\pi\in S_{n}$
of the quiver vertex labels. In order to avoid confusion, we fix the notation
$\mu_{v_{1}},\mu_{v_{2}},\dots,\mu_{v_{m}}$ as denoting a sequence of
mutations starting with $\mu_{v_{1}}$ and ending with $\mu_{v_{m}}$. When we
need to specify the particular data of a cluster automorphism $\varphi$, we
denote it by the tuple
$\varphi=(\mu_{v_{1}},\mu_{v_{2}},\dots,\mu_{v_{m}};\pi)$ with the permutation
$\pi$ expressed in cycle notation. Following the conventions of [ASS12, KG21],
we allow for cluster automorphisms $\varphi$ defined solely as a permutation
of the quiver vertex labels without any mutations.

######  Example 4.8.

Consider an $A_{2}$ quiver with vertices labeled 1 and 2 and an edge from 1 to
2. The cluster automorphism $\varphi=(\mu_{1};(1\,2))$ also generates
$\mathcal{G}(A_{2})$ and corresponds to a rotation of an initial triangulation
of the pentagon by $6\pi/5$.

In their initial work defining cluster automorphisms, [ASS12], Assem,
Schiffler and Shramchenko investigate cluster modular groups of finite,
affine, and surface-type cluster algebras. In [KG21], the authors give an
alternate presentation for affine type cluster modular groups and compute
cluster modular groups for extended affine types. In the extended affine case,
part of their work builds on Fraser’s investigation of cluster modular groups
of Grassmannians in [Fra18]. Results relevant to this manuscript are
summarized in Tables 1, 2, and 3.

We introduce the following notion in order to compare cluster automorphisms
defined on different initial seeds.

######  Definition 4.9.

Two cluster automorphisms $\varphi_{1}$ and $\varphi_{2}$ are conjugate if
they act identically on the set of cluster charts.

In our proof of Theorem 1.1, we use the combinatorics of tagged triangulations
to show that cluster automorphisms induced by Legendrian loops are conjugate
to cluster automorphisms coming directly from quiver combinatorics.

####  4.3.1. Combinatorics of (tagged) triangulations

The combinatorics of (tagged) triangulations of punctured surfaces play a key
role in defining and understanding many of the simpler classes of cluster
algebras. For $A_{n},D_{n}$ and $\tilde{D}_{n}$ type, many of the computations
of cluster modular groups are most accessible through the combinatorics of
tagged triangulations. More generally, we can define surface-type cluster
algebras as cluster algebras whose underlying quiver comes from a (tagged)
triangulation of a punctured surface. In this subsection, we discuss the
necessary combinatorial ingredients for understanding tagged triangulations of
surfaces in the context of cluster theory.

Let $\Sigma_{n_{1},\dots,n_{k}}$ be a surface with $n_{i}$ marked points on
the $i$th boundary component. We allow for $n_{i}=0$ and interpret this as a
puncture of the surface. Following [FST08], we define (tagged) arcs, and
(tagged) triangulations.

######  Definition 4.10.

An arc $\gamma\in\Sigma_{n_{1},\dots,n_{k}}$ is a curve in $\Sigma$ such that:

  * •

the endpoints of $\gamma$ lie on marked points;

  * •

the interior of $\gamma$ does not intersect itself;

  * •

the interior of $\gamma$ is disjoint from $\partial\mathbb{S}$ and marked
points;

  * •

$\gamma$ does not cut out an unpunctured monogon or an unpunctured bigon.

The last condition ensures that no arc is contractible to a point or into the
boundary of $\Sigma$. We consider arcs equivalent up to isotopy. Two (isotopy
classes of) arcs are said to be compatible if there are two arcs in their
respective isotopy classes that do not intersect in the interior of
$\Sigma_{n_{1},\dots,n_{k}}$. A triangulation of $\Sigma_{n_{1},\dots,n_{k}}$
is then a maximal pairwise compatible collection of (isotopy classes of) arcs.

From a triangulation $\mathcal{T}$, we can produce a cluster algebra as
follows. For every edge $\gamma_{i}$ in $\mathcal{T}$, we assign a vertex
$v_{i}$ in our quiver $Q_{\mathcal{T}}$. The vertices corresponding to
boundary edges of $\mathcal{T}$ are declared to be frozen. We add an edge from
$v_{i}$ to $v_{j}$ if there is a face where $\gamma_{j}$ is counterclockwise
from $\gamma_{i}$ and at least one of $v_{i}$ or $v_{j}$ is mutable. Note that
we must cancel any oriented two-cycles once we have accounted for all of the
edges in this fashion. To each vertex in the quiver, we assign a cluster
variable which can be interpreted as measuring the length of the arc in an
appropriate hyperbolic geometric context [FT18]. Mutation is given by
exchanging one diagonal of a quadrilateral for the other.

As defined, triangulations of $\Sigma_{n_{1},\dots,n_{k}}$ do not realize
every possible cluster seed in the corresponding cluster algebra if the number
of punctures is at least one. This is due to the appearance of self-folded
triangles, which produce arcs that cannot be mutated at. In order to represent
all possible cluster seeds as a triangulation, we require additional
decorations.

We arbitrarily divide an arc into two ends and allow each end to be either
tagged or untagged. To produce a tagged triangulation, we introduce additional
compatibility relations.

######  Definition 4.11 (Definition 7.4 [FST08]).

Two tagged arcs $\gamma_{1}$ and $\gamma_{2}$ in $\Sigma_{n_{1},\dots,n_{k}}$
are compatible if the following conditions hold:

  * •

the untagged arcs corresponding to $\gamma_{1}$ and $\gamma_{2}$ are
compatible;

  * •

if the untagged arcs corresponding to $\gamma_{1}$ and $\gamma_{2}$ represent
distinct isotopy classes and they share an endpoint $a$, then the tagging at
the ends of $\gamma_{1}$ and $\gamma_{2}$ incident to $a$ coincide;

  * •

if the untagged arcs corresponding to $\gamma_{1}$ and $\gamma_{2}$ lie in the
same isotopy class, then at least one end of $\gamma_{1}$ must be tagged in
the same way as the same end of $\gamma_{2}$.

To obtain a quiver from a tagged triangulation, we treat tagged arcs as normal
arcs. For an arc $\gamma$ sharing the same endpoints as $\gamma^{\prime}$, we
use the face obtained by deleting $\gamma^{\prime}$ to compute the direction
of the arrows in the quiver. See Figure 16 (left) for an example. The cluster
algebras constructed from (tagged) triangulations of a disk with either 0, 1,
or 2 punctures are of type $A_{n},D_{n},$ and $\tilde{D}_{n}$, respectively.

####  4.3.2. Surface type cluster modular groups

We highlight here results on cluster modular groups related to the theory of
mapping class groups and tagged triangulations, as they will reappear in key
technical arguments in Sections 5 and 7. We define the mapping class group
$\operatorname{Mod}(\Sigma_{n_{1},\dots,n_{k}})$ of a surface
$\Sigma_{n_{1},\dots,n_{k}}$ with $n_{i}$ marked points on the $i$th boundary
component as the group of orientation-preserving homeomorphisms of $\Sigma$
fixing the set of punctures up to homeomorphisms isotopic to the identity. We
interpret the case of $n_{i}=0$ as an interior puncture. The tagged mapping
class group $\operatorname{Mod}_{tag}(\Sigma_{n_{1},\dots,n_{k}})$ is defined
to be the semidirect product of
$\operatorname{Mod}(\Sigma_{n_{1},\dots,n_{k}})$ with $\mathbb{Z}_{2}^{p}$
where $p$ is the number of interior punctures of $\Sigma$. The product
structure is specified by the action of simultaneously swapping tags at all
arcs incident to a given puncture.

Denote by $\mathcal{A}(\Sigma_{n_{1},\dots,n_{k}})$ the surface-type cluster
algebra associated to $\Sigma_{n_{1},\dots,n_{k}}$. The following theorem
relates the tagged mapping class group of a surface and the cluster modular
group of the associated cluster algebra.

######  Theorem 4.12 (Proposition 8.5, [BS15]).

Assume that $\Sigma_{n_{1},\dots,n_{k}}$ is not a once or twice-punctured disk
with four or fewer marked points on the boundary. Then
$\operatorname{Mod}_{tag}(\Sigma_{n_{1},\dots
n_{k}})\cong\mathcal{G}(\mathcal{A}(\Sigma_{n_{1},\dots,n_{k}}))$.

###  4.4. Legendrian loops from plabic fences

In this subsection, we explain how to geometrically realize Kaufman and
Greenberg’s presentation of cluster modular groups for affine type. We
introduce plabic fences as a combinatorial means of obtaining a sequence of
mutations induced by a Legendrian loop. We then describe fronts with initial
quivers that are mutation equivalent to $T_{\bf{n}}$ quivers and Legendrian
loops that induce automorphisms conjugate to the $\tau_{i}$ generators of
$\Gamma_{\tau}$.

####  4.4.1. Plabic fences

Due to the variety of weave equivalence moves, it is sometimes difficult to
determine a sequence of mutations that induces the same cluster automorphism
as a Legendrian loop. Even when we can compute the sequence of mutations via
other combinatorial means, it can be challenging to show that the mutation
sequence in the weave agrees with the Legendrian loops. For example, see
Appendix A for a somewhat involved computation of a short mutation sequence
induced by a Legendrian loop. Therefore, in order to determine a sequence of
mutations corresponding to the induced action of most of the Legendrian loops
we consider, we require a combinatorial way to relate quivers to weaves. In
the case of $\Lambda(A_{n})$, this can be done by understanding the Kálmán
loop as rotations of triangulations dual to 2-weaves, as in [Hug23a]. For
Legendrian links of the form $\Lambda=\Lambda(\beta)$, we rely on the
combinatorics of plabic graphs.

Plabic graphs are a general combinatorial object related to cluster theory and
were first studied by Postnikov [Pos06]. The term ‘plabic fence’ that we use
for the family graphs we work with was coined in [FPST22] and refers to a
particular form of plabic graph, as we describe below. We follow the recipe of
[CW22] for producing a Legendrian link from a plabic fence and use the graph
to encode the combinatorics of Legendrian loop mutations.

######  Definition 4.13.

A plabic fence is a planar graph with univalent or trivalent vertices colored
either black or white constructed as follows:

  1. (1)

Stack $n$ horizontal lines of the same length on top of each other, each
starting with a white vertex on the left and ending with a black vertex on the
right.

  2. (2)

Add vertical edges between adjacent pairs of horizontal lines with trivalent
vertices where they meet colored so that each endpoint of the vertical edge is
a different color.

From the plabic fence $\mathbb{G}$, we can extract a quiver $Q_{\mathbb{G}}$
via the following process. First, assign a mutable vertex to each face of
$\mathbb{G}$. Then, for every edge $e\in\mathbb{G}$ connecting two faces, add
an arrow between the corresponding vertices oriented so that the white
endpoint of $e$ is to the right of the edge when traveling in the direction of
orientation.

To a plabic fence $\mathbb{G}$, we associate a Legendrian link
$\Lambda_{\mathbb{G}}$ and an initial Legendrian weave surface with boundary
$\Lambda_{\mathbb{G}}$, following the recipe from [CW22, Sections 2 and 3]. To
obtain a positive braid $\beta$ from $\mathbb{G}$ such that
$\Lambda_{\mathbb{G}}\cong\Lambda(\beta)$, first label the horizontal lines of
$\mathbb{G}$ from bottom to top with the numbers 1 through $n$ and label the
vertical edges between horizontal lines $i$ and $i+1$ by $\sigma_{i}$. We
refer to a vertical edge by the color of its top vertex, so that a vertical
edge between lines $i$ and $i+1$ is white if the vertex on line $i$ is colored
white. In the front projection, one should think of the black vertical edges
as denoting crossings appearing above cusps, while white vertical edges
correspond to crossings below cusps. More precisely, from $\mathbb{G},$ we
obtain braid words $\beta_{1}$ from white vertical edges and $\beta_{2}$ from
black vertical edges by scanning $\mathbb{G}$ from left to right and appending
a $\sigma_{i}$ (resp. $n-i$) to $\beta_{1}$ (resp. $\beta_{2}$) for every
white (resp. black) vertical edge between lines $i$ and $i+1$. The Legendrian
link is Legendrian isotopic to the rainbow closure of the braid
$\beta_{1}\beta_{2}^{\circ}$ (equivalently $\beta_{2}^{\circ}\beta_{1}$) where
$\beta_{2}^{\circ}$ is obtained from $\beta_{2}$ by reading the braid word
from right to left and replacing $\sigma_{i}$ with $\sigma_{n-i}$ for all $i.$

Figure 9. Legendrian link obtained from a plabic fence. The braid word
$\beta_{1}$ corresponds to white vertical edges, while the braid word
$\beta_{2}$ corresponds to black.

From $\Lambda_{\mathbb{G}}$, Casals and Weng construct an initial weave
$\mathfrak{w}(\mathbb{G})$ [CW22, Definition 3.24]. By [CL22, Theorem 1.2],
the combinatorial data of the seed corresponding to the plabic graph
$\mathbb{G}$ agrees with this choice of initial weave
$\mathfrak{w}(\mathbb{G})$. More precisely, the Legendrian
$\Lambda_{\mathbb{G}}$ given here is Legendrian isotopic to the Legendrian
obtained as the conormal lift of zig-zag strands and the conjugate Lagrangian
surface associated to an initial seed is Hamiltonian isotopic to the
Lagrangian projection of the initial weave. As a result, the intersection
quiver of $\mathfrak{w}(\mathbb{G})$ agrees with the initial quiver
$Q_{\mathbb{G}}$ coming from the plabic fence.

####  4.4.2. Mutations induced by elementary Legendrian isotopies

In order to compute the sequence of mutations induced by Legendrian loops we
decompose our Legendrian loops into a series of simple Legendrian isotopies
and describe how to combinatorially realize them in the plabic fence. The
first Legendrian isotopy we consider is a Reidemeister III move, which
interchanges crossings $\sigma_{i}\sigma_{i+1}\sigma_{i}$ with
$\sigma_{i+1}\sigma_{i}\sigma_{i+1}$. The plabic fences $\mathbb{G}_{\beta}$
and $\mathbb{G}_{\beta^{\prime}}$ corresponding to the two braid words $\beta$
and $\beta^{\prime}$ then differ by the local move pictured in Figure 10.
Combinatorially, we can see that $Q_{\mathbb{G}_{\beta}}$ differs from
$Q_{\mathbb{G}_{\beta^{\prime}}}$ by a mutation at the vertex corresponding to
the face. This elementary Legendrian isotopy also induces a map between
$\mathfrak{M}(\Lambda(\mathbb{G}_{\beta}),T)$ and
$\mathfrak{M}(\Lambda(\mathbb{G}_{\beta^{\prime}}),T)$ that is an isomorphism,
but generally not an automorphism. The following lemma states that this
induced isomorphism agrees with cluster mutation.

Figure 10. Local move of a plabic fence corresponding to the braid move
exchanging $\sigma_{i}\sigma_{i+1}\sigma_{i}$ and
$\sigma_{i+1}\sigma_{i}\sigma_{i+1}$.

######  Lemma 4.14 (Proposition 7.16, [CLSBW23]).

Let $\mathbb{G}_{\beta}$ and $\mathbb{G}_{\beta^{\prime}}$ be plabic fences
related by the local move pictured in Figure 10 and denote the quiver vertex
corresponding to the unique face of $\mathbb{G}$ by $k$. The initial seeds of
$\mathfrak{M}(\Lambda(\mathbb{G}_{\beta}),T)$ and
$\mathfrak{M}(\Lambda(\mathbb{G}_{\beta^{\prime}}),T)$ are related by mutation
at the cluster variable $a_{k}$.

In addition to the Reidemeister III move, we also consider a sequence of
isotopies given by the cyclic shift, which modifies the braid word by
conjugation. In a plabic fence with all white vertical edges, we can interpret
this isotopy as flipping the leftmost white edge to black, moving it past all
of the white vertical edges in the row to its right, and then flipping it back
to white. Each time we move a black edge past a white one, we perform the
local move pictured in Figure 11, known as a square move. As with the
Reidemeister III move, this local move induces an isomorphism between sheaf
moduli that corresponds to a cluster mutation between the initial seeds.

Figure 11. A square move in a plabic fence, corresponding to mutation at the
vertex represented by the face.

######  Lemma 4.15.

[CW22, Section 5.3] Let $\mathbb{G}_{\beta}$ and $\mathbb{G}_{\beta^{\prime}}$
be plabic fences related by the local move pictured in Figure 10 and denote
the quiver vertex corresponding to the unique face of $\mathbb{G}$ by $k$. The
initial seeds of $\mathfrak{M}(\Lambda(\mathbb{G}_{\beta}),T)$ and
$\mathfrak{M}(\Lambda(\mathbb{G}_{\beta^{\prime}}),T)$ are related by mutation
at the cluster variable $a_{k}$.

####  4.4.3. Legendrian loop mutation sequences

As noted above, an elementary Legendrian isotopy in general only induces an
isomorphism of $\mathfrak{M}(\Lambda,T)$. On the other hand, a Legendrian loop
necessarily induces an automorphism of $\mathfrak{M}(\Lambda,T)$. The
automorphism induced by a Legendrian $\varphi$ preserves intersection quivers
and extends to an action on cluster charts by considering the concatenation of
the trace of $\varphi$ with fillings of $\Lambda$. As a result, the
automorphism induced by a Legendrian loop is a _cluster_ automorphism.
Following the argument in the previous subsection, the sequence of mutations
giving the specific cluster automorphism induced by $\varphi$ can be computed
by decomposing $\varphi$ into elementary Legendrian isotopies.

For Legendrian satellite loops, we can obtain an explicit description of the
induced sequence of mutations. Consider
$\Lambda=\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})$ with $k\geq 3$. Here
$\beta\in\operatorname{Br}_{n}^{+}$, $\Delta_{n}^{2}$ is a full twist on $n$
strands, and $\sigma_{1}^{k}\in\operatorname{Br}_{2}^{+}$. As detailed in
Section 2.1, we obtain a Legendrian $\vartheta$-loop by rotating a single
crossing of the satellited braid $\sigma_{1}^{k}$ following the pair of
satellited strands in the contact neighborhood of the $i$th strand of the
$(-1)$-framed closure of $\beta$. Note that since $k\geq 3$, the link
$\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})\cong\Lambda(\beta^{\prime})$
for some braid $\beta^{\prime}\in\operatorname{Br}_{n+1}^{+}$. Therefore,
there is some grid plabic graph $\mathbb{G}(\beta^{\prime})$ such that
$\Lambda_{\mathbb{G}(\beta^{\prime})}\cong\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})$.

By construction, every smallest subword of the form
$\sigma_{j}\dots\sigma_{j}$ of $\Lambda_{\mathbb{G}}$ yields a face of
$\mathbb{G}$. For
$\Lambda(\beta^{\prime})\cong\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})$,
certain faces of the plabic graph $\mathbb{G}(\beta^{\prime})$ correspond to
the satellited braid $\gamma=\sigma_{1}^{k}$. These faces can be identified
from $\beta$ and $i$ by describing the relationship between $\beta$ and
$\beta^{\prime}$: Let $\Lambda$ denote the $(-1)$-framed closure of
$\beta\Delta_{n}^{2}$. For any crossings $\sigma_{j}$ appearing above the
$i$th strand $\Lambda_{i}$ of $\Lambda$, we replace $\sigma_{j}$ by
$\sigma_{j+1}$ in $\beta^{\prime}$, as the appearance of the additional
satellited strand shifts the strands above it up by one. Similarly, any
crossings appearing below $\Lambda_{i}$ are fixed. If $\Lambda_{i}$ appears at
height $j$ at some point in the subword, then for any crossing $\sigma_{j}$
involving $\Lambda_{i}$, we replace $\sigma_{j}$ by $\sigma_{j}\sigma_{j+1}$.
Under these conditions, we also replace $\sigma_{j-1}$ by
$\sigma_{j-1}\sigma_{j}$.

Applying the above observation to the grid plabic graph $\mathbb{G}$ allows us
to understand the faces of $\mathbb{G}$ corresponding to $\gamma$ by analyzing
the braid word. In particular, for any two consecutive crossings of $\beta$
involving the $i$th strand the corresponding subword of the satellite braid
$\beta^{\prime}$ is given as follows:

  * •

$\sigma_{j}\sigma_{j+1}\mapsto\sigma_{j+1}\sigma_{j}\sigma_{j+2}\sigma_{j+1}$

  * •

$\sigma_{j-1}\sigma_{j}\mapsto\sigma_{j-1}\sigma_{j}\sigma_{j-2}\sigma_{j-1}$

  * •

$\sigma_{j}\sigma_{j}\mapsto\sigma_{j+1}\sigma_{j}\sigma_{j}\sigma_{j+1}$

  * •

$\sigma_{j-1}\sigma_{j-1}\mapsto\sigma_{j}\sigma_{j+1}\sigma_{j+1}\sigma_{j}$

As a result, we see that for any pair of crossings of $\beta$ involving
$\Lambda_{i}$, we have a distinguished face of the plabic fence
$\mathbb{G}(\beta^{\prime})$ that can be identified with crossings of the
satellited braid
$\Lambda(\sigma_{1}^{k})_{i}\subseteq\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})$.

Denote the subset of the quiver corresponding to these faces by $Q_{\gamma}$
and label the vertices from left to right by
$\\{v_{\gamma_{1}},\dots,v_{\gamma_{p}}\\}$. Denote also by $Q_{i}$ the subset
of the quiver corresponding to the faces in the $i$th row of $\mathbb{G}$
corresponding to pairs of crossings $\sigma_{i}\dots\sigma_{i}$ with vertices
$\\{v_{i_{1}},\dots,v_{i_{q}}\\}$. Note that $Q_{\gamma}$ and $Q_{i}$
necessarily have nonempty intersection so that some of the vertex labeling is
redundant. We denote the vertex labels in $Q_{\gamma}\cap Q_{i}$ by
$\\{i_{r_{j}}=\gamma_{r_{j}}\\}_{j=1}^{m}$ From the above description we can
extract a combinatorial recipe for the action of the Legendrian
$\vartheta$-loop.

######  Lemma 4.16.

Let $\Lambda=\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})$ for $k\geq 3$. The
$\vartheta$-loop described above induces the sequence of mutations

$\gamma_{p},\gamma_{p-1},\dots,\gamma_{1},\gamma_{1},\pi_{\gamma}^{-1}(i_{1}),\pi_{\gamma}^{-1}(i_{2}),\dots,\pi_{\gamma}^{-1}(i_{q})$

and the permutation of quiver vertex labels is given in cycle notation by

$(i_{1}\,\dots\,i_{r_{1}}\,\gamma_{r_{1}-1}\,\dots\,\gamma_{1})\dots(i_{r_{m-1}+1}\,\dots\,i_{r_{m}}\,\gamma_{r_{m}-1}\,\dots\,\gamma_{r_{m-1}+1}).$

###### Proof.

For $\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})$, the $\vartheta$-loop is
given by a sequence of Reidemeister III moves and a cyclic rotation that pass
a single crossing $c$ of $\Lambda(\sigma_{1}^{k})_{i}$ around the braid. The
first Reidemeister III move passes the crossing $c$ through the rightmost
strand passing over or under $\Lambda(\sigma_{1}^{k})_{i}$, forming a face of
$\mathbb{G}$. Each successive Reidemeister III move passes the crossing $c$
across a subsequent strand of $\Lambda(\beta\Delta_{n}^{2},i,\sigma_{1}^{k})$.
By Lemma 4.14, this Reidemeister III move corresponds to a mutation of the
quiver $Q_{\mathbb{G}}$ identified with the particular face of $\mathbb{G}$
formed by the strand and the crossing $c$, and yields a new plabic graph
$\mathbb{G}^{\prime}$. By construction we can readily identify the vertices
belonging to $Q_{\gamma}$ as a subset of $Q_{\mathbb{G}^{\prime}}$.

After performing all of the Reidemeister III moves, the mutation sequence up
to this point is precisely $\gamma_{p},\dots,\gamma_{1}$. The resulting plabic
fence is identical to $\mathbb{G}$ except that the vertical white edge
representing the crossing $c$ is now the leftmost edge of the $i$th row of the
plabic fence. Following the sequence of mutations, we see that the
corresponding face is labeled by the vertex $v_{\gamma_{1}}$. To complete the
$\vartheta$-loop, we perform a cyclic rotation by flipping the leftmost white
edge corresponding to $c$ to a black edge and performing a sequence of square
moves that push it as far to the right as possible. By Lemma 4.15, these
correspond to mutations of each vertex in row $i$, starting with the vertex
$v_{\gamma_{1}}$ of $Q_{\gamma}$ and then proceeding in order through the
vertices of in row $i$. For any vertex $r_{j}$ in $Q_{i}\cap Q_{\gamma}$, the
vertex in row $i$ after the mutations $\gamma_{p},\dots,\gamma_{1}$ is
$\pi_{\gamma}^{-1}(v_{r_{j}})$ while any other vertex in row $i$ is fixed by
$\gamma_{p},\dots,\gamma_{1}$ The resulting mutation sequence is therefore
given by
$\gamma_{p},\dots,\gamma_{1},\gamma_{1},\pi_{\gamma}^{-1}(i_{1}),\dots,\pi_{\gamma}^{-1}(i_{q})$,
as desired.

The permutation of quiver vertex labels resulting from this sequence of
mutations follows from the plabic fence computation. In particular, each
Reidemeister III move inducing a mutation in the sequence
$\gamma_{p},\dots,\gamma_{1}$ leads to replacing the quiver vertex label
$v_{\gamma_{j+1}}$ by the quiver vertex label $v_{\gamma_{j}}$. The sequence
of mutations $\pi_{\gamma}^{-1}(i_{1}),\dots,\pi_{\gamma}^{-1}(i_{q})$ induces
a similar relabeling, with the additional caveat that we need to account for
the nonempty intersection between $Q_{i}$ and $Q_{\gamma}$. As a result, each
cycle in our permutation corresponds to a single vertex $v_{r_{j}}$ in
$Q_{i}\cap Q_{\gamma}$ and is given by shifting the vertices in $Q_{i}$
between $v_{r_{j}}$ and $v_{r_{j-}}$ to the right by one and then shifting the
vertices in $Q_{\gamma}$ between $v_{r_{j}}$ and $v_{r_{j-1}}$ to the left by
one. Note that we include $v_{r_{j}}$ but not $v_{r_{j-1}}$ in this particular
cycle of the permutation. ∎

######  Example 4.17.

Consider the positive braid
$\beta=\sigma_{1}^{2}\sigma_{3}\sigma_{2}\sigma_{3}^{2}\sigma_{2}\sigma_{1}^{2}\Delta_{4}^{2}\in\operatorname{Br}_{4}^{+}$.
The Legendrian link $\Lambda(\beta,2,\sigma_{1}^{4})$ is isotopic to
$\Lambda(\sigma_{2}\sigma_{1}^{2}\sigma_{2}\sigma_{4}\sigma_{3}\sigma_{2}\sigma_{4}\sigma_{3}^{2}\sigma_{4}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}^{2}\sigma_{1}\sigma_{2}^{2})$
with a corresponding plabic fence $\mathbb{G}_{0}$, pictured in Figure 12. If
we label the faces from left to right as shown, we have that
$Q_{i}=\\{v_{2},v_{3},v_{6},v_{7},v_{9},v_{10},v_{11}\\}$, while
$Q_{\gamma}=\\{v_{1},v_{3},v_{5},v_{7},v_{8},v_{10}\\}$. The mutation sequence
induced by the $\vartheta$ loop is then given by

$\tilde{\vartheta}=(10,8,7,5,3,1,1,2,5,6,8,9,11;(2\,3\,1)(6\,7\,5)(9\,10\,8)$

Figure 12. A sequence of plabic fences depicting the mutation sequence of the
Legendrian $\vartheta$-loop. Starting with the initial plabic (top), we obtain
the result of performing the Reidemeister III moves inducing the mutation
sequence $\gamma_{p},\dots,\gamma_{1}$ (middle). Applying a cyclic rotation
results in the plabic fence depicted at the bottom.

Finally, we describe the sequence of mutations associated to the
$\operatorname{DT}$ transformation in the case of Legendrian links
$\Lambda(\beta)$. This description comes from giving a recipe for realizing
$\operatorname{DT}$ in terms of the combinatorics of plabic fences, following
[CW22, Section 5]. In the Legendrian, this procedure is roughly described as
rotating all of the crossings past the cusps and then composing with the
strict contactomorphism $x\mapsto-x,z\mapsto-z$; see Subsection 2.1. Starting
with a plabic fence of all white edges, the Legendrian isotopy can be
combinatorially realized by repeatedly flipping the rightmost white edge to
black and moving it past the remaining white edges in its row using square
moves until it becomes the rightmost black edge. Once all white edges are
flipped to black, the contactomorphism then takes all black edges to white
edges, returning to the initial plabic fence. Recording the square moves
involved in this process gives an explicit mutation sequence.

##  5\. Legendrian loops as generators of cluster modular groups

In this section, we describe presentations for cluster modular groups of of
skew symmetric cluster algebras of finite, affine, and extended affine Dynkin
type and match them with Legendrian loop actions. We start by describing the
quiver combinatorics used in [KG21] to give presentations for cluster modular
groups of affine and extended affine types. We then compare the induced action
of our Legendrian loops with the presentations of cluster modular groups given
below. Following our discussion of cluster modular groups for Dynkin types, we
also include a similar computation for cluster automorphisms arising from
Legendrian loops of torus links.

###  5.1. Presentations of cluster modular groups via $T_{\bf n}$ quivers

In [KG21], Kaufman and Greenberg use a particular family of quivers to
explicitly describe the cluster modular groups of affine and extended affine
cluster algebras.

We make use of their work by showing that the initial quivers from plabic
fences for affine and extended affine type yield nearly identical
combinatorial presentations. We begin by defining Kaufman and Greenberg’s
quivers for simply-laced type.

######  Definition 5.1.

Given a vector ${\bf n}=(n_{1},\dots,n_{k})$ of natural numbers $n_{i}\geq 2$,
a $T_{\bf n}$ quiver is a quiver with a pair of special vertices $v_{1}$ and
$v_{0}$ and a collection of $k$ ‘tails’ of vertices of length
$n_{1},\dots,n_{k}$, as pictured in Figure 13.444The quiver pictured in Figure
13 is actually opposite to the one considered in [KG21]. The pictured
orientation is chosen to match our previous conventions.

Figure 13. A $T_{n_{1},\dots n_{k}}$ quiver. Deleting the vertex $v_{0}$
results in a quiver with central vertex $v_{1}$ and tails of length $n_{i}-1$
(not including the vertex $v_{1}$) where the vertices alternate as either
sources or sinks for the two incident edges.

In this work, we will always consider the case of $k=3$. The particular
cluster types we consider are listed in Table 1 along with the corresponding
$T_{\bf n}$ quiver. $T_{\bf{n}}$ quivers admit a particular class of quiver
automorphisms, which we will denote $\tau_{1},\dots,\tau_{k}$. Recall that we
denote a cluster automorphism by a tuple with the first entry a sequence of
quiver mutations and the second entry a permutation describing the relabeling
of the quiver vertices. For a tail of length $n_{i}$ in $T_{\bf n}$, we denote
$i_{\text{odd}}=\\{i_{j}|3\leq j\leq n_{i},j\text{ odd}\\}$ and
$i_{\text{even}}\\{i_{j}|3\leq j\leq n_{i},j\text{ even}\\}.$ The automorphism
$\tau_{i}$ is then given by

$\tau_{i}=(\mu_{i_{\text{odd}}}\mu_{i_{\text{even}}}\mu_{i_{2}}\mu_{v_{0}}\mu_{v_{1}};(i_{2}\,i_{0}\,i_{1}))$

where $\mu_{i_{\text{odd}}}$ denotes a sequence of mutations starting at
$i_{j}$ for $j<n_{i}$ the largest number in $i_{\text{odd}}$ and ending at the
vertex $i_{3}$. The mutation sequence $\mu_{i_{\text{even}}}$ is defined
analogously. Denote by $\Gamma_{\tau}$ the subgroup of $\mathcal{G}$ generated
by $\tau_{1},\dots,\tau_{k}$.

######  Theorem 5.2 (Theorem 4.11, [KG21]).

$\Gamma_{\tau}$ is an abelian group with relations
$\tau_{i}^{n_{i}}=\tau_{j}^{n_{j}}$.

Using the relation $\tau_{i}^{n_{i}}=\tau_{j}^{n_{j}}$ from Theorem 5.2, we
can define the element $\gamma:=\tau_{i}^{n_{i}}$ of $\Gamma_{\tau}$. Kaufman
and Greenberg show that the Donaldson-Thomas transformation can then be given
in terms of $\gamma$ and the $\tau_{i}$ generators by
$\operatorname{DT}=\gamma^{2}\prod_{i=1}^{k}\tau_{i}\gamma^{-1}$ [KG21,
Theorem 4.14]. For the skew-symmetric cluster algebras we investigate here,
this simplifies to $\operatorname{DT}=\tau_{1}\tau_{2}\tau_{3}\gamma^{-1}$.

In addition to the $\tau_{i}$ generators, we must also consider graph
automorphisms $\sigma\in\operatorname{Aut}(T_{\bf{n}})$ acting on
$\Gamma_{\tau}$ by swapping tails $i$ and $j$ of length $n_{i}=n_{j}$. The
utility of the $T_{\bf{n}}$ quiver construction is demonstrated by the
following theorem.

######  Theorem 5.3 ([KG21], Theorems 5.2 and 6.1).

For a cluster algebra $\mathbb{A}$ of affine type,
$\mathcal{G}(\mathbb{A})\cong\Gamma_{\tau}\rtimes\operatorname{Aut}(T_{\bf{n}})$.
For a cluster algebra $\mathbb{A}$ of extended affine type,
$\Gamma_{\tau}\rtimes\operatorname{Aut}(T_{\bf{n}})$ is isomorphic to a finite
index subgroup of $\mathcal{G}(\mathbb{A})$.

Kaufman and Greenberg also conjecture that
$\mathcal{G}(\mathbb{A})\cong\Gamma_{\tau}\rtimes\operatorname{Aut}(T_{\bf{n}})$
for cluster algebras $\mathbb{A}$ that admit a $T_{\bf{n}}$ quiver and are not
of extended affine type [KG21, Conjecture 4.7].

###  5.2. Proof of Theorem 1.1

Let us now proceed to a proof of Theorem 1.1. We start by using Lemma 4.16 to
compare the induced action of Legendrian loops to the generating sets of
Kaufman and Greenberg described in the previous subsection. We then continue
with a case-by-case analysis of each Legendrian link corresponding to a skew-
symmetric cluster algebra of finite and affine type.

####  5.2.1. Legendrian loops conjugate to $\tau_{i}$

Let us define a family of positive braids $\beta_{k,n_{1},n_{2}}$ by

$\beta_{k,n_{1},n_{2}}:=(\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2k}\sigma_{1}^{n_{1}-2}\sigma_{3}^{n_{2}-2}$

for $k,n_{1},n_{2}\in\mathbb{N}$ and $n_{1},n_{2}\geq 2.$ For Legendrian links
corresponding to affine and extended affine type cluster algebras, we will
require $k=1$ and $n_{1}$ and $n_{2}$ agreeing with the corresponding values
for $T_{\bf{n}}$ quivers. See Table 1 below. Denote by
$\mathfrak{w}(k,n_{1},n_{2})$ the initial weave filling of
$\Lambda(\beta_{k,n_{1},n_{2}})$.

######  Lemma 5.4.

The intersection quiver $Q_{\mathfrak{w}(1,n_{1},n_{2})}$ is mutation
equivalent to $T_{n_{1},n_{2},2}.$

Figure 14. Plabic fence $\mathbb{G}$ corresponding to
$\Lambda(\tilde{D}_{n})$. The dots correspond to $n-6$ additional vertical
edges

###### Proof.

A plabic fence corresponding to $\Lambda(\beta_{1,n_{1},n_{2}})$ can be
obtained from the plabic fence in Figure 14 by the addition of $n_{2}-2$
vertical edges between the top two horizontal lines. We then get an initial
quiver by adding $n_{2}-2$ vertices to the quiver pictured in Figure 14 in a
manner identical to the tail labeled by vertices $v_{6},\dots,v_{n}$. Mutating
at vertices $v_{3}$ and $v_{4}$ of $Q_{\mathfrak{w}(1,n_{1},n_{2})}$ yields a
$T_{n_{1},n_{2},2}$ quiver up to the alternating behavior of the arrows in the
tails. This alternation can be obtained by observing that the tails are
$A_{n}-$type subquivers, which are mutation equivalent to any quiver given by
a choice of orientation on the underlying Dynkin diagram. ∎

To present Legendrian loops of $\Lambda(\beta_{k,n_{1},n_{2}})$, we will
consider the Legendrian isotopic link given as the $(-1)$-framed closure of
the braid
$(\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2k}\sigma_{1}^{n_{1}-2}\sigma_{3}^{n_{2}-2}\sigma_{1}^{2}\sigma_{3}^{2}(\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2}$.
Note that in either front, the Legendrian link
$\Lambda(\beta_{k,n_{1},n_{2}})$ is Legendrian isotopic to the link given by
simultaneously satelliting $\gamma_{1}=\sigma_{1}^{n_{1}}$ and
$\gamma_{2}=\sigma_{1}^{n_{2}}$ about the two strands of the Legendrian Hopf
link given as the $(-1)$-framed closure of the braid $\sigma_{1}^{4}$. With
this description, we immediately obtain two $\vartheta$-loops, depicted in
Figure 15. Note that when $n_{i}=2$, $\vartheta_{i}$ can still be identified
because of our choice of braid word for $\Delta^{2}$ even if no such loop is
apparent in the Legendrian isotopic rainbow closure
$\Lambda((\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2}\sigma_{1}^{n_{1}-2}\sigma_{3}^{n_{2}-2})$.

Figure 15. The Legendrian loop $\vartheta_{1}$ for the braid
$\beta_{1,n_{1},n_{2}}$ pictured in $J^{1}\mathbb{S}^{1}$. The loop takes one
of the $n_{1}-2$ crossings appearing in the box and drags it around the front,
following the path of the dotted arrows.

Recall that, as stated in Theorem 5.2, the group $\Gamma_{\tau}$ is Abelian.
The following result, along with the relationship between $\vartheta$ loops
and the $\tau_{i}$ generators discussed below, geometrically realizes this
property.

######  Proposition 5.5.

Let $\Lambda$ be a Legendrian link that is isotopic to a Legendrian link
obtained from simultaneously satelliting patterns $\gamma_{1}$ and
$\gamma_{2}$ about two distinct strands of some companion Legendrian
$\Lambda^{\prime}$. For any exact Lagrangian filling $L$ of $\Lambda$, we have
$\vartheta_{1}\circ\vartheta_{2}(L)\cong\vartheta_{2}\circ\vartheta_{1}(L)$.

###### Proof.

Consider the concatenation of $\operatorname{Tr}(\vartheta_{2})$ with
$\operatorname{Tr}(\vartheta_{1})$. We can produce the desired Hamiltonian
isotopy between concordances by producing showing that the composition of
Legendrian loops $\vartheta_{2}\circ\vartheta_{1}$ is homotopic to
$\vartheta_{1}\circ\vartheta_{2}$. The first composition can be described as
first performing $\vartheta_{1}$ during time $t_{1}$ and then performing
$\vartheta_{2}$ during time $t_{2}$ for $0\leq t_{1}<\frac{1}{2}$ and
$\frac{1}{2}\leq t_{2}\leq 1$. The required homotopy is then defined by
gradually increasing $t_{1}$ and gradually decreasing $t_{2}$.

∎

We now show that our description of Legendrian loops coincides with
$\Gamma_{\tau}$.

Figure 16. A pair of tagged triangulations of $\mathbb{D}^{2}_{n-2,0,0}$. The
triangulation on the left corresponds to the quiver $T_{n-2,2,2}$, while the
triangulation on the right corresponds to the quiver
$\mathbb{G}(\Lambda(1,n_{1},n_{2}))$ and is obtained from the triangulation on
the left by mutation at edges labeled 3 and 4. The dotted lines on the left
represent a zig-zag pattern of $n-6$ edges, while the dotted lines on the
right represent $n-6$ edges all sharing the top vertex.

######  Lemma 5.6.

For any Legendrian $\Lambda(\beta_{1,n_{1},n_{2}})$, the cluster automorphisms
$\tilde{\vartheta}_{1}$ and $\tilde{\vartheta}_{2}$ induced by the
corresponding Legendrian loops are conjugate to $\tau_{1}$ and $\tau_{2}$.

###### Proof.

We first establish the statement in the case of
$\Lambda(\tilde{D}_{n})\cong\Lambda(\beta_{1,n-2,2})$ using the combinatorics
of tagged triangulations of a twice-punctured disk. We then leverage these
combinatorics for the case of $n_{2}>2$.

Consider the plabic fence $\mathbb{G}(\tilde{D}_{n})$ depicted in Figure 14,
corresponding to the Legendrian link
$\Lambda(\tilde{D}_{n})\cong\Lambda_{1,n-2,2}$. The sequence of mutations
corresponding to $\vartheta_{1}$ can be determined from
$\mathbb{G}(\tilde{D}_{n})$ using Lemma 4.16. In particular, $\vartheta_{1}$
induces the mutation sequence

(3) $\tilde{\vartheta}_{1}=(5,0,3,1,1,4,6,\dots,n;(1\,4\,5\,0\,3))$

while the loop $\vartheta_{2}$ induces the mutation sequence
$\tilde{\vartheta}_{2}=(1,4;(1\,4\,0\,3))$. This second mutation sequence is
verified by explicit computation in the initial weave filling
$\mathfrak{w}(\mathbb{G}(\tilde{D}_{n}))$ in Appendix A. Note that the use of
Legendrian weaves for this computation appears to be necessary, as the
crossings involved in the Legendrian loop $\vartheta_{2}$ do not appear in the
rainbow closure
$\Lambda((\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2})^{2}\sigma_{1}^{n-3})$ and
therefore are not straightforwardly captured by the combinatorics of the
plabic fence.

Figure 16 depicts two tagged triangulations of a twice-punctured disk. The
triangulation $\mathcal{T}_{1}$ (left) corresponds to a quiver identical to
$Q_{\mathbb{G}(\tilde{D}_{n})}$, while the triangulation $\mathcal{T}_{2}$
(right) corresponds to a quiver identical to $T_{n-2,2,2}$. The generator
$\tau_{1}$ corresponds to a rotation of the boundary of the disk by $2\pi/n$
[KG21, Lemma B.3]. Therefore, to show that $\tau_{1}$ and
$\tilde{\vartheta}_{1}$ are conjugate, we need to show that
$\tilde{\vartheta}_{1}$ also corresponds to a rotation of the boundary of the
disk by $2\pi/n$. This is done explicitly in Figure 17.

Figure 17. Sequence of mutations induced by the Legendrian loop
$\vartheta_{1}$ in the tagged triangulation corresponding to
$Q_{\mathbb{G}(\tilde{D}_{n})}$. The second triangulation is obtained from the
first by mutating at edges labeled $5,0,$ and $3$ in order. The third
triangulation is obtained from the second performing the remaining mutations
of $\vartheta_{1}.$

Similar to the case of $\tau_{1}$, the generator $\tau_{2}$ corresponds to a
half twist about the two punctures in the triangulation $\mathcal{T}_{2}$.
Figure 18 shows the computations for $\tilde{\vartheta}_{2}$.

Figure 18. Sequence of mutations induced by the Legendrian loop
$\vartheta_{2}$ in the tagged triangulation corresponding to
$Q_{\mathbb{G}(\tilde{D}_{n})}$.

For $n_{2}>2$, we observe that the sequence of mutations induced by
$\tilde{\vartheta}_{1}$ or $\tilde{\vartheta}_{2}$ fixes the quiver vertices
corresponding to the other tail. Freezing or deleting these vertices therefore
yields a $\tilde{D}_{n}$ quiver and we can simply apply our above reasoning to
show that cluster automorphisms are conjugate. ∎

We now give a case-by-case description of $\vartheta$ loops as $\tau_{i}$
generators of $\Gamma_{\tau}$. We consider the cases of finite and affine type
separately and postpone the cases of $\Lambda(E_{7}^{(1,1)})\cong\Lambda(4,4)$
and $\Lambda(E_{8}^{(1,1)})\cong\Lambda(3,6)$ to the following subsection as a
subcase of Legendrian torus links.

###### Proof of Theorem 1.1.

Throughout this proof, we freely reference the specific cluster modular groups
and Legendrian links corresponding to a given Dynkin type as listed in Table
1.

Finite type:

  1. $A_{n}$:

The Kálmán loop of $\Lambda(A_{n})$ has order $n+3$ and generates
$\mathcal{G}(\mathfrak{M}(\Lambda(A_{n})))$, as stated in [Hug23a, Corollary
4.2].

Figure 19. Plabic fence $\mathbb{G}(D_{n})$ with corresponding quiver. The
dots represent $n-5$ additional vertical edges.

  2. $D_{n}$:

We use the combinatorics of tagged triangulations of once-punctured $n$-gons
to understand the cluster modular group of $\mathfrak{M}(\Lambda(D_{n}))$. By
[ASS12, Theorem 1.2], the generators of the cluster modular group correspond
to rotation of the $n$-gon by $2\pi/n$ and simultaneous changing of the tags
at the puncture when $n\geq 5$. Let $\mathbb{G}(D_{n})$ be the plabic fence
pictured in Figure 19. The corresponding Legendrian
$\Lambda(\mathbb{G}(D_{n}))$ is Legendrian isotopic to the link
$\Lambda(D_{n})$ defined in the introduction and admits a description as a
Legendrian satellite $\Lambda(\sigma_{1}^{4},1,\sigma_{1}^{n-2})$ of the Hopf
link. We denote the corresponding loop by $\vartheta$ and use Lemma 4.16 to
compute the induced sequence of mutations. In Figure 20 we give a tagged
triangulation with a quiver mutation equivalent to $Q_{\mathbb{G}(D_{n})}$ and
show by explicit computation that the sequence of mutations induced by
$\vartheta$ corresponds to rotation of the punctured $n$-gon by $2\pi/n$. In
addition, the $\operatorname{DT}$ transformation induces a sequence of
mutations that corresponds to a rotation of the once-punctured $n$-gon by
$2\pi/n$ and a simultaneous changing of all the taggings at the puncture.
Therefore, the pair $\vartheta$ and $\operatorname{DT}$ generate the cluster
modular group $\mathcal{G}(\mathfrak{M}(\Lambda(D_{n})))$.

Figure 20. Once-punctured disks depicting the mutation sequence
$(3,2,4,\dots,n;(1\,2\,3))$ induced by the Legendrian loop $\vartheta$ of
$\Lambda(D_{n})$.

  3. $E_{6},E_{7},E_{8}$

For $\Lambda(E_{6}),\Lambda(E_{7}),\Lambda(E_{8})$, the cluster modular group
is generated solely by the $\operatorname{DT}$ transformation, so the
statement follows immediately from the characterization of $\operatorname{DT}$
given in [CW22, Theorem 5.8].

Affine type:

  1. $\tilde{D}_{n}$, $n>4$:

Recall from Subsection 5.1 that the cluster modular group $\mathcal{G}$ of an
affine type cluster algebra is isomorphic to
$\Gamma_{\tau}\rtimes\operatorname{Aut}(T_{\bf{n}})$ and that $\Gamma_{\tau}$
is generated by $\tau_{1}$, $\tau_{2},$ and $\tau_{3}$. By Lemma 5.6, we can
realize $\tau_{1}$ and $\tau_{2}$ as Legendrian loops, and the relation
$\tau_{1}^{n_{1}}=\tau_{2}^{n_{2}}=\tau_{3}^{n_{3}}$ implies that the subgroup
generated by $\tau_{1}$ and $\tau_{2}$ is a finite index subgroup of
$\mathcal{G}$. When $n>4,$ Kaufman and Greenberg’s presentation contains a
quiver automorphism of order 2 that does not obviously appear as an
automorphism induced by a Legendrian loop, hence leading to the exclusion of
$\Lambda(\tilde{D}_{n})$ in the statement of the theorem.

  2. $\tilde{D}_{4}$:

The case of $\Lambda(\tilde{D}_{4})$ is analogous to $\Lambda(\tilde{D}_{n})$
when $n>4$ except that the quiver automorphism group for the $T_{2,2,2}$
quiver that Kaufman and Greenberg consider is the symmetric group $S_{3}$. We
can produce an order-two generator of this $S_{3}$ factor by realizing
$\Lambda(\tilde{D}_{4})$ as the $(-1)$-closure of the braid
$\beta=(\sigma_{2}\sigma_{1}^{3}\sigma_{2}\sigma_{1}^{3}\sigma_{2}\sigma_{3}\sigma_{1}^{n-2})^{2}$.
The $n+8$th power $\delta^{n+8}$ of the cyclic rotation then induces an
involutive cluster automorphism, where we conjugate by the induced action of a
Legendrian isotopy taking our original front for $\Lambda(\tilde{D}_{4})$ to
our particular choice of $(-1)$-closure.

  3. $\tilde{E}_{7},\tilde{E}_{8}$:

In the case of $\tilde{E}_{7}$ or $\tilde{E}_{8}$, an inspection of the quiver
$T_{\bf{n}}$ verifies that that $\operatorname{Aut}(T_{\bf{n}})$ is trivial.
Therefore, we need only show that we can generate $\Gamma_{\tau}$ by
Legendrian loops and $\operatorname{DT}$. By [KG21, Theorem 4.14], we have
that $\operatorname{DT}=\tau_{1}\tau_{2}\tau_{3}\gamma^{-1}$. Solving for
$\tau_{3}=\operatorname{DT}\tau_{1}^{-1}\tau_{2}^{-1}\gamma$ therefore gives
the remaining generator of $\Gamma_{\tau}$.

  4. $\tilde{E}_{6}$:

In the case of $\tilde{E}_{6}$, we require a cluster automorphism of order
three induced by a Legendrian loop. The Legendrian loop in question is
obtained by performing a Legendrian isotopy $\psi$ to obtain the rotationally
symmetric braid $(\sigma_{2}\sigma_{1}^{4})^{3}$, composing with $\delta^{5}$
and then performing $\psi^{-1}$ to return to the original front projection.
The induced action of this loop has order three, as desired.

∎

Cluster Type | Cluster modular group |  $T_{\bf{n}}$ quiver type
---|---|---
$A_{n}$ | $\mathbb{Z}_{n+3}$ | –
$D_{n}$ $n>4$ | $\mathbb{Z}_{n}\times\mathbb{Z}_{2}$ | –
$E_{6},E_{7},E_{8}$ | $\mathbb{Z}_{14},\mathbb{Z}_{10},\mathbb{Z}_{16}$ | –
$\tilde{D}_{n}$ $n>4$ | $\operatorname{Mod}_{tag}(\mathbb{D}^{2}_{n-2,0,0})$ | (n-2, 2, 2)
$\tilde{E}_{6}$ | $(S_{3}\times\mathbb{Z})$ | (3, 3, 2)
$\tilde{E}_{7}$ | $(\mathbb{Z}_{2}\times\mathbb{Z})$ | (4, 3, 2)
$\tilde{E}_{8}$ | $\mathbb{Z}$ | (5, 3, 2)
$E_{7}^{(1,1)}$ | $\pi_{1}(Conf(\mathbb{S}^{2},4))\rtimes\mathbb{Z}_{2}$ | (4, 4, 2)
$E_{8}^{(1,1)}$ | $\operatorname{PSL}_{2}(\mathbb{Z})\rtimes\mathbb{Z}_{6}$ | (6, 3, 2)
Table 1.

######  Remark 5.7.

The arguments employed in the affine type case apply to the slightly more
general family of Legendrian links $\Lambda(\beta_{1,n_{1},n_{2}})$. These
links correspond to $T_{n_{1},n_{2},2}$ quivers with $n_{1}\geq n_{2}>2$ and
Kaufman and Greenberg conjecture that the cluster modular group obtained from
such quivers is isomorphic to $\Gamma_{\tau}$ when $n_{1}\neq n_{2}$ and
$\Gamma_{\tau}\rtimes\mathbb{Z}_{2}$ when $n_{1}=n_{2}$. By Lemma 5.6 and the
fact that $\tau_{3}$ can be obtained from $\tau_{1},\tau_{2}$ and
$\operatorname{DT}$, we are able to generate $\Gamma_{\tau}$ by the induced
action of Legendrian loops. In addition, the anti-symplectic involution
$\iota$ induces the quiver automorphism generating
$\operatorname{Aut}(T_{\bf{n}})$ when $n_{1}=n_{2}$. As a result, we obtain a
symplectic-geometric description of the conjectured cluster modular group of
Kaufman and Greenberg [KG21, Conjecture 4.7] in this special case.

###  5.3. Torus links and cluster modular groups of
$\operatorname{Gr}^{\circ}(k,n+k)$

In this subsection, we show that Legendrian loops of torus links
$\Lambda(k,n)$ generate an index-two subgroup of a conjectural cluster modular
group of $\operatorname{Gr}^{\circ}(k,n+k)$, the affine cone of the
Grassmannian. We start by reviewing the relationship between
$\mathfrak{M}(\Lambda(k,n))$ and $\operatorname{Gr}^{\circ}(k,n+k)$. We then
describe the conjectural cluster modular group $\mathcal{G}^{\prime}(k,n)$ and
an explicit generating set given in [Fra18]. Finally, we show that the
Legendrian loops $\Sigma_{i}$ and $\rho=\delta^{k}$ induce generators of
$\mathcal{G}^{\prime}(k,n)$.

####  5.3.1. Relationship between $\mathfrak{M}_{1}(\Lambda(k,n),T)$ and
$\operatorname{Gr}^{\circ}(k,n+k)$

We now briefly describe an explicit isomorphism between
$\mathfrak{M}_{1}(\Lambda(k,n),T)$ and $\operatorname{Gr}^{\circ}(k,n+k)$. For
more details, we refer the interested reader to [CLSBW23, Section 6] for a
detailed description in the more general setting of Positroid strata of
$\operatorname{Gr}^{\circ}(k,n+k)$.

Consider a front projection of $\Lambda(k,n)\subseteq J^{1}\mathbb{S}^{1}$
corresponding to the braid word
$\beta_{k,n}=(\sigma_{1}\dots\sigma_{k-1})^{n+k}$ and add marked points
following the convention given by [CLSBW23, Theorem A.(iii)]. Following the
description of $\mathfrak{M}(\Lambda,T)$ in Subsection 3, the sheaf moduli of
$\Lambda(k,n)$ is then given by a sequence of flags $V_{1},\dots,V_{k+n}$ each
of which satisfies $V_{i}^{(j)}\pitchfork V_{i+1}^{(j)}$ for every $i$ and
$j$. In the particular case of $\Lambda(k,n)$, the choices of (complex) lines
$v_{1},\dots,v_{n+k}$ corresponding to the regions between pairs of
$\sigma_{1}$ crossings determine all of the other higher-dimensional vector
spaces $V^{(j)},j\geq 2$ making up the remaining data of each of flags.
Indeed, the flag $V_{i}$ is given by $v_{i}\subseteq v_{i}\wedge
v_{i+1}\subseteq\dots\subseteq v_{i}\wedge\dots\wedge v_{i+k-1}$ with indices
taken modulo $n+k$. See Figure 21 for an example computation in the case of
$\Lambda(4,4)$.

Figure 21. Flags making up $\mathfrak{M}(\Lambda(4,4))$. Here $V_{i}V_{j}$
denotes the subspace $V_{i}\wedge V_{j}$.

Let $\operatorname{Gr}(k,n)$ the space of $k$ dimensional subspaces of
$\mathbb{C}^{n}$ and consider its Plücker embedding. We denote by
$\operatorname{Gr}^{\circ}(k,n)\subseteq\mathbb{C}^{n\choose k}$ the affine
cone of the Grassmannian, i.e. the affine subvariety of $\mathbb{C}^{n\choose
k}$ whose points satisfy the Plücker relations. By the transversality
conditions defining $\mathfrak{M}(\Lambda(k,n),T)$ and the framing data given
by the choice of $T$, the vectors $v_{1},\dots,v_{n+k}$ form the columns of a
full rank $k\times n+k$ matrix. This determines an explicit isomorphism
between $\mathfrak{M}(\Lambda(k,n),T)$ and $\operatorname{Gr}^{\circ}(k,n+k)$
by interpreting the minors of the resulting matrix $(v_{1}\dots v_{n+k})$ as
the Plücker coordinates of a point in the affine cone of the Grassmannian.

The affine variety $\operatorname{Gr}^{\circ}(k,n)$ admits a cluster structure
with an initial seed given by a maximal weakly-separated collection of Plücker
coordinate functions [Sco06]. For certain choices of $k$ and $n,$ we recover
Legendrian links of finite and affine type. In these cases, one can verify
that the results below recover the cluster modular groups described in the
previous subsection.

####  5.3.2. Conjectural presentation of
$\mathcal{G}(\operatorname{Gr}^{\circ}(k,n+k))$

We give two types of cluster automorphisms of
$\operatorname{Gr}^{\circ}(k,n+k)$. The first, known as the cyclic shift, is
relatively well-studied in the cluster literature and played a key role in
proving that the totally non-negative Grassmannian is homeomorphic to a closed
ball [GKL22]. Given vectors $v_{1},\dots,v_{n+k}$ satisfying the
transversality conditions of $Gr^{\circ}(k,n+k)$, the cyclic shift $\rho$ acts
by $(v_{1},\dots,v_{n+k})\mapsto(v_{2},\dots v_{n+k},(-1)^{k-1}v_{1})$. From
the construction, we can immediately observe that $\rho$ has order $n+k$.

The second type of cluster automorphism is the main object of study in
[Fra18], where Fraser produces a braid group action on $Gr^{\circ}(k,n+k)$. We
first denote $d=\gcd(k,n+k)$ and assume that $d>1$. Given
$v_{1},\dots,v_{n+k}$ as above and $i\in\\{1,\dots,d-1\\}$, the action of
$\sigma_{i}$ on the first $d$ vectors is

(4)
$(v_{1},\dots,v_{d})\mapsto(v_{1},\dots,v_{i-1},v_{i+1},w_{1},v_{i+2},\dots,v_{d})$

with $w_{1}$ defined by the conditions

(5) $v_{i}\wedge v_{i+1}=v_{i+1}\wedge w_{1}\text{ and
}w_{1}\in\text{span}\\{v_{2},\dots,v_{i+k}\\}$

Note that Equations 4 and 5 together imply that
$w_{1}\in\text{span}\\{v_{i},v_{i+1}\\}\cap\text{span}\\{v_{2},\dots,v_{i+k}\\}$
with the normalization given by Equation 4 ensuring that $w_{1}$ is uniquely
defined. The cluster automorphism $\sigma_{i}$ is defined on the remaining
$n+k-d$ vectors by adding multiples of $d$ to the indices of the vectors and
defining $w_{j}$ analogously.

The extended affine braid group $\widehat{\operatorname{Br}}_{d-1}$ on $d$
strands is generated by elements $\rho,\sigma_{1},\dots,\sigma_{d-1}$ with
relations
$\sigma_{i+1}\sigma_{i}\sigma_{i+1}=\sigma_{i}\sigma_{i+1}\sigma_{i}$,
$\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for $|j-i|\geq 2$, and
$\rho\sigma_{i}\rho^{-1}=\sigma_{i}+1$ for $i\in\\{1,\dots,d-2\\}$ and
$\rho\sigma_{d-1}\rho^{-1}=\rho^{-1}\sigma_{1}\rho$. Fraser’s main result
states that $\sigma_{1},\dots,\sigma_{d-1}$ defined above together with the
cyclic shift generate an extended affine braid group action on
$\operatorname{Gr}^{\circ}(k,n+k)$ [Fra18, Theorem 5.3]. Denote by
$G_{d}^{\prime}=\widehat{Br}_{d-1}$ when $d<k$ and $G_{k}^{\prime}=Br_{k}$
when $d=k$. Fraser obtains a homomorphism
$G_{d}^{\prime}\to\mathcal{G}(\operatorname{Gr}^{\circ}(k,n+k))$ and
conjectures that for $n\geq 2k$, the kernel of this homomorphism is generated
by $\rho^{n}$. For $n=k$, he identifies the additional relation
$\sigma_{1}\dots\sigma_{k-1}^{2}\dots\sigma_{1}$. These yield finite index
subgroups of the conjectured cluster modular groups, as the addition of the DT
transformation to the generating set adds a $\mathbb{Z}_{2}$ factor given by
the relation $\operatorname{DT}^{2}=\rho^{k}$. In particular, for $n=k$,
Fraser’s conjectural cluster modular groups are isomorphic to
$\pi_{1}(Conf(\mathbb{S}^{2},k))\rtimes\mathbb{Z}_{2}$ where
$Conf(\mathbb{S}^{2},k)$ is the configuration space of $k$ points on the
sphere. For the purpose of comparing with the results of [CG22], note that
$Conf(\mathbb{S}^{2},k)$ modulo its center is isomorphic to
$\operatorname{Mod}(\Sigma_{0,k})$. See Table 2 for a summary of which values
of $k$ and $n$ yield the groups mentioned above.

$\Lambda(k,n)$ | $d$ | $\mathcal{G}^{\prime}(k,n)$
---|---|---
$\Lambda(2,n)\,n\geq 3$ | $n\mod 2$ | $\mathbb{Z}_{n+2}$
$\Lambda(k,k)\,k\geq 3$ | $k$ | $\pi_{1}(Conf(\mathbb{S}^{2},k))\rtimes\mathbb{Z}_{2}$
$\Lambda(k,n)\,n\geq 2k$ | $k$ | $(\operatorname{Br}_{k}\rtimes\mathbb{Z}_{2})/\langle\rho^{n+k}\rangle$
$\Lambda(k,n)\,n\geq k+1$ | $<k$ | $(\widehat{Br}_{d-1}\rtimes\mathbb{Z}_{2})/\langle\rho^{n+k}\rangle$
Table 2.

####  5.3.3. Legendrian loops induce Fraser’s automorphisms

In [CG22], the authors show that the Kálmán loop induces the cyclic shift
automorphism $\rho$ on $\mathcal{M}_{1}(\Lambda(3,6))$ and
$\mathcal{M}_{1}(\Lambda(4,4))$ while the $\vartheta$ loops they describe
induce $\sigma_{i}$ actions. Their work predates any proof of a cluster
structure on these sheaf moduli, so they rely on combinatorial tools to verify
that the induced actions of these Legendrian loops yield faithful group
actions on the sheaf moduli. Our goal in this subsection is to describe the
relevant Legendrian loops as cluster automorphisms and extend their work to
the general setting of $\operatorname{Gr}^{\circ}(k,n+k)$.

We begin with the following lemma relating the Kálmán loop $\rho$ to the
cyclic shift automorphism.

######  Lemma 5.8.

The Legendrian loop $\rho$ induces the cyclic shift on
$\mathfrak{M}(\Lambda(k,n),T)\cong\operatorname{Gr}^{\circ}(k,n+k)$.

###### Proof.

The statement immediately from the description of the isomorphism between
$\mathfrak{M}(\Lambda(k,n),T)$ and $\operatorname{Gr}^{\circ}(k,n+k)$ and the
definition of the Kálmán loop in the front given by the braid word
$(\sigma_{1},\dots,\sigma_{k-1})^{n+k}$. Indeed, the induced action of $\rho$
sends the flag $V_{i}$ to the flag $V_{i-1}$, which, up to sign, is the
desired map $v_{i}\mapsto v_{i-1}$ on 1-dimensional subspaces. ∎

To realize the braid group generators of
$\mathcal{G}^{\prime}(\operatorname{Gr}^{\circ}(k,n+k))$ as Legendrian loops,
we need to expand the class of loops we consider to include the following
generalization of Definition 2.1: Let $d=\gcd(k,n)$, $m=\frac{k}{d}$, and
assume that $d>1$. While the Coxeter projection $\pi:\operatorname{Br}_{n}\to
S_{n}$ has no fixed points, it does admit a fixed set
$\\{i,i+d,\dots,i+(m-1)d\\}$ of size $m$ where we consider indices $\mod k$.
For an adjacent pair of strands, $i$, $i+1$ corresponding to a set of fixed
orbits, $\Lambda(k,n)$ – presented as the $(-1)$-framed closure of
$(\sigma_{1}\dots,\sigma_{k-1})^{n+k}$ for specificity – is Legendrian
isotopic to the satellite of a positive braid about these fixed strands; in
other words $\Lambda(k,n)\cong\Lambda(\beta,\\{i,\dots,i+(m-1)d\\},\gamma\\}$
for some $\beta\in\operatorname{Br}_{k-1}^{+}$ and
$\gamma\in\operatorname{Br}_{2}^{+}$, where we have slightly abused notation
to replace the natural number $i$ in the definition from the previous section
with the set of natural numbers $\\{i,\dots,i+(m-1)d\\}$. We define
$\Sigma_{i}$ analogously to a $\vartheta$-loop of $\Lambda(\beta,i,\gamma)$ as
the isotopy given by the flow of the vector field in the open neighborhood of
the strands $\\{i,\dots,i+(m-1)d\\}$. In the front given by the $(-1)$-closure
of $(\sigma_{1}\dots,\sigma_{k-1})^{n+k}$, the loop $\Sigma_{i}$ can be
represented by a sequence of Reidemeister III describing a single crossing
orbiting the link $m$ times, as described below.

Note that in the case of $d=k$, the fixed set of the Coxeter projection is
simply $i$ and the $\Sigma_{i}$ loop coincides with a $\vartheta$-loop of
$\Lambda(\beta,i,\gamma)$.

The following lemma describing the action of the $\Sigma_{i}$ loops involves a
somewhat more detailed computation.

######  Lemma 5.9.

The Legendrian loop $\Sigma_{i}$ induces the cluster automorphism $\sigma_{i}$
of $\operatorname{Gr}^{\circ}(k,n+k)$ under the isomorphism between
$\mathfrak{M}(\Lambda(k,n),T)$ and $\operatorname{Gr}^{\circ}(k,n+k)$.

###### Proof.

Let $d=\gcd(k,n+k)$ and consider the induced action of the Legendrian loop
$\Sigma_{i}$ on $\mathfrak{M}(\Lambda(k,n))$. Following the computations in
Sections 4 and 5 of [CG22], we decompose the Legendrian isotopy $\Sigma_{1}$
into a sequence of Reidemeister III moves.

We restrict our consideration of the front to the first $d(k-1)$ crossings in
the front projection
$\Pi_{xz}(\Lambda((\sigma_{1}\dots\sigma_{k-1})^{n+k}))\subseteq
J^{1}\mathbb{S}^{1}$ and extend by symmetry to the remaining $(n+k-d)(k-1)$
crossings. We label the strands from bottom to top at their left endpoint.
Observe that in the first $k(k-1)$ crossings of the initial front projection
any fixed pair of strands labeled $i$ and $i+1$ cross exactly twice. In the
braid $(\sigma_{1}\dots,\sigma_{k-1})^{k}$, these two crossings are exactly
the $i$th $\sigma_{1}$ and the $i+1$st $\sigma_{k-1}$ with indices taken
modulo $k$.

The $\Sigma_{i}$ loop is then given by a sequence of Reidemeister III moves
that take the $i$th $\sigma_{1}$ crossing and pass it through the other
strands until we obtain a $\sigma_{i}$ as the leftmost crossing. Repeating
this process symmetrically for each block of $d(k-1)$ crossings and performing
a cyclic rotation then produces a $\sigma_{i}$ crossing at the right of the
first block of $d(k-1)$ crossings. If $d=k$, then we can perform another
series of Reidemeister III moves to pass this crossing through the other
strands until we encounter the $i-1$th $\sigma_{k-1}$ crossing. We then
perform a sequence of Reidemeister III moves to pass the $\sigma_{k-1}$
crossing through the remaining strands until we arrive back at the front that
we started with. If $d<k$, then after passing the $\sigma_{k-1}$ crossing
through the remaining strands, we obtain $\sigma_{i+d}$ as the leftmost
crossing. We perform another cyclic shift and repeat this process $m$ times in
total for $m=\frac{k}{d}$.

The induced action on $\mathfrak{M}(\Lambda(k,n),T)$ can be described as
follows. We can view each Reidemeister III move as transferring a region of
the front diagram from the right of a strand to the left of the same strand,
fixing all the remaining regions. Under the natural identification of sheaves
before and after this isotopy, the vector spaces corresponding to the fixed
regions are all preserved and we need only compute the vector space
corresponding to the new region. Following our description of
$\mathfrak{M}(\Lambda((\sigma_{1}\dots\sigma_{k-1})^{n+k}),T)$ above, the
vector space assigned to the new region is determined in the cases where it is
of dimension at least two. In the case where the region appears between a pair
of $\sigma_{1}$ crossings and the assigned vector space $W_{j}$ has rank one,
we determine $W_{j}$ as follows. By construction, a new region $W_{j}$ appears
whenever we replace the $i+jd$th $\sigma_{1}$ of
$(\sigma_{1}\dots\sigma_{k-1})^{k}$ with the $i+jd+1$th $\sigma_{k-1}$
crossing for $j\in\\{1,\dots,m-1\\}$. This has the effect of replacing
$v_{i+jd}$ by $v_{i+jd+1}$ and $v_{i+jd+1}$ by $W_{j}$ while preserving
$v_{i+jd+2}$. The singular support conditions then imply that $W_{j}$ lies in
the two-dimensional subspace $V_{i+jd}^{(2)}$ assigned to the region northwest
of it. Similarly, it must also lie in the $k-1$-dimensional subspace
$V_{i+jd+1}^{(k-1)}$ that is assigned to the unique upper region directly to
the northeast of the $W$ region. Therefore, the span of $W$ is determined by
the intersection $V_{i+jd}^{(2)}\cap V_{i+jd+1}^{(k-1)}$ and the particular
choice of framing data can be given by the identification of $V_{i+jd}^{(2)}$
with the span of $v_{i}$ and $W_{i}$, as required by the singular support
conditions. Up to indexing, these are equivalent to the conditions given by
Equation 5 determining $w_{1}$. Thus, the induced action of $\Sigma_{i}$
precisely matches Equation 4, as desired. ∎

##  6\. Cluster modular groups from folding $\mathfrak{M}(\Lambda)$

In this section, we study Legendrian loops and Legendrian links that exhibit
symmetry under certain finite group actions. These correspond algebraically to
folded cluster algebras, skew-symmetrizable cluster algebras that are
constructed by identifying symmetries of quivers that respect cluster
mutation. The process of folding a skew-symmetric cluster algebra is somewhat
delicate in general, as symmetries of a quiver are not necessarily preserved
by mutation. In this section, we discuss the algebraic process of folding a
cluster algebra by a $G$-action and then give a contact-geometric description
of it in terms of $G$-invariant Legendrian links and their $G$-fillings. We
then produce generators of cluster modular groups of the folded cluster
algebras using our results from Section 5.

###  6.1. Folding clusters via G-actions

We begin with the properties of a foldable quiver. See [FWZ20b, Section 4.4]
or [ABL22, Section 2.3] for more details. Let $Q$ be a quiver and $G$ be a
finite group action with $g\cdot Q=Q$ for all $g\in G$. We write $i\sim
i^{\prime}$ for vertices $v_{i}$ and $v_{i^{\prime}}$ lying in the same
$G$-orbit and denote by $\\#\\{i\rightarrow j\\}$ the number of edges from
$v_{i}$ to $v_{j}$.

######  Definition 6.1.

The quiver $Q$ is $G$-admissible if the following conditions hold:

  1. (1)

If $i\sim i^{\prime}$, then $i$ and $i^{\prime}$ are either both mutable or
both frozen.

  2. (2)

For all $i\sim i^{\prime}$ and any $g\in G$, we have $\\#\\{i\rightarrow
j\\}=\\#\\{g\cdot i\rightarrow g\cdot j\\}$.

  3. (3)

For all $i\sim i^{\prime}$, we have $\\#\\{i\rightarrow i^{\prime}\\}=0$.

  4. (4)

For all mutable $i\sim i^{\prime}$, we have $\\#\\{i\rightarrow
j\\}\\#\\{i^{\prime}\rightarrow j\\}\geq 0$.

Let $I$ be the $G$-orbit of a vertex $v_{i}$. We denote by $\\#\\{I\rightarrow
J\\}$ the sum $\sum_{i\in I}\\#\\{i\rightarrow j\\}$ for some arbitrary $j\in
J$. To package the data $\\#\\{i\rightarrow j\\}$ as part of a graph, we
produce a weighted quiver. That is, given a $G$-admissible quiver $Q$, we
consider the weighted graph $Q^{G}$ whose vertices are $G$-orbits of vertices
in $Q$ where $v_{i}$ and $v_{j}$ have an edge of weight $\\#\\{I\rightarrow