# Infinitely many Lagrangian fillings

Roger Casals  University of California Davis, Dept. of Mathematics, Shields
Avenue, Davis, CA 95616, USA<EMAIL_ADDRESS>and  Honghao Gao
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East
Lansing, MI 48824, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that all maximal-tb Legendrian torus links $(n,m)$ in the standard
contact 3-sphere, except for $(2,m)$, $(3,3),(3,4)$ and $(3,5)$, admit
infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is
proven by constructing infinite order Lagrangian concordances which induce
faithful actions of the modular group $\operatorname{PSL}(2,\mathbb{Z})$ and
the mapping class group $M_{0,4}$ into the coordinate rings of algebraic
varieties associated to Legendrian links. In particular, our results imply
that there exist Lagrangian concordance monoids with subgroups of exponential-
growth, and yield Stein surfaces homotopic to a 2-sphere with infinitely many
distinct exact Lagrangian surfaces of higher-genus. We also show that there
exist infinitely many satellite and hyperbolic knots with Legendrian
representatives admitting infinitely many exact Lagrangian fillings.

###### 2010 Mathematics Subject Classification:

Primary: 53D10. Secondary: 53D15, 57R17.

##  1\. Introduction

We show that essentially all maximal-tb Legendrian torus knots in the standard
contact 3-sphere remarkably admit infinitely many non-Hamiltonian isotopic
exact Lagrangian fillings in the standard symplectic $4$-ball. Heretofore, the
existence of Legendrian links with infinitely many exact Lagrangian fillings
remained open.

In fact, the faithful $\operatorname{PSL}(2,\mathbb{Z})$ representation in our
Theorem 1.1 allows us to obtain several consequences. We present new results
for Lagrangian concordance monoids, including the first known example of a
Lagrangian concordance of infinite order, the existence of an exponential-
growth subgroup in the fundamental group of the space of Legendrian links
isotopic to $\Lambda(3,6)$, and the existence of Weinstein 4-manifolds
homotopic to the 2-sphere with infinitely many non-Hamiltonian isotopic exact
Lagrangian surfaces of higher-genus in the same smooth isotopy class. In
addition, we construct infinitely many instances of both satellite and
hyperbolic knots in the 3-sphere with Legendrian representatives with
infinitely many exact Lagrangian fillings in the standard symplectic 4-ball.

###  1.1. Context

Legendrian knots in contact 3–manifolds are instrumental to study the contact
geometry of 3–manifolds [6, 20, 21, 22, 24, 25, 32]. The classification of
Legendrian knots and their Lagrangian fillings has been one of the central
areas of research in low-dimensional contact topology [17, 18, 36, 47, 49, 50,
51]. The only Legendrian knot for which there exist a complete non-empty
classification of Lagrangian fillings is the Legendrian unknot [19].

The works [18, 47, 58] succeeded in constructing a Catalan number worth of
Lagrangian fillings for the maximal-tb Legendrian $(2,n)$-torus links. It is
also known that all positive braids admit at least one Lagrangian filling
[36]. A crucial question that remained open is the existence of Legendrian
links with infinitely many of exact Lagrangian fillings. This article
affirmatively resolves this question.

Figure 1. The Legendrian torus links $\Lambda(3,6)$ (left) and $\Lambda(4,4)$
(right).

In fact, we shall geometrically construct Lagrangian concordances which
themselves produce infinitely many Lagrangian fillings, which is a
significantly stronger statement than the existence of infinitely many
Lagrangian fillings. The constructions are explicit and can be readily drawn
in the front projection. The construction implies that these exact Lagrangian
surfaces are all smoothly isotopic. We will distinguish these Lagrangian
fillings by studying their action on part of the coordinate ring of the moduli
of framed constructible sheaves $\mathcal{M}(\Lambda)$ [34, 38, 59] for
certain Legendrian links $\Lambda\subseteq(S^{3},\xi_{\text{st}})$. The
techniques we use for our results illustrate the strength of applying methods
from the microlocal theory of sheaves [38, 59] and the theory of cluster
algebras [27, 29, 30] to 3-dimensional contact and symplectic topology.

###  1.2. Main Results

Let $\Lambda(n,m)\subseteq(S^{3},\xi_{\text{st}})$ be the maximal-tb
Legendrian positive $(n,m)$-torus link, $(n,m)\in\mathbb{N}\times\mathbb{N}$,
as depicted in Figure 1. Positive Legendrian torus knots are Legendrian simple
positive braids [23], and thus are uniquely determined by their Thurston-
Bennequin invariants and their rotation numbers. These Legendrian links can be
obtained by considering the positive braids
$\beta=(\sigma_{1}\sigma_{2}\cdot\ldots\cdot\sigma_{n-1})^{m+n}$ in
$(J^{1}S^{1},\xi_{\text{st}})$ and satelliting the zero section
$S^{1}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ to the standard Legendrian unknot
$\Lambda(1,1)\subseteq(S^{3},\xi_{\text{st}})$. Let $\mathcal{L}(n,m)$ be the
space of Legendrian links isotopic to the maximal-tb Legendrian torus link
$\Lambda(n,m)\subseteq(S^{3},\xi_{\text{st}})$, with base point an arbitrary
but fixed maximal-tb Legendrian representative.

Let $\mathcal{M}(\Lambda(n,m))$ be the moduli space of framed sheaves
associated to the Legendrian link $\Lambda(n,m)$, as we shall introduce in
Section 3. This $\mathcal{M}(\Lambda(n,m))$ is an algebraic variety [58], and
in our case it will be a quasi-projective subvariety of the projective
Grassmannian $\operatorname{Gr}(n,n+m)$. Since $\mathcal{M}(\Lambda(n,m))$ is
a Legendrian isotopy invariant [34, 58, 59], it defines a monodromy
representation

$\Gamma:\pi_{1}(\mathcal{L}(n,m))\longrightarrow\operatorname{Aut}(\mathcal{M}(\Lambda(n,m))),$

into the space of algebraic automorphisms of the algebraic variety
$\mathcal{M}(\Lambda(n,m))$. In turn, by pull-back, we obtain a representation

$\Gamma^{*}:\pi_{1}(\mathcal{L}(n,m))\longrightarrow\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda(n,m))]),$

into the automorphisms of the coordinate ring
$\mathbb{C}[\mathcal{M}(\Lambda(n,m))]$ of $\mathcal{M}(\Lambda(n,m))$. In
particular, a set of based loops
$C_{1},\ldots,C_{r}:S^{1}\longrightarrow\mathcal{L}(n,m)$, $r\in\mathbb{N}$,
gives rise to a monodromy representation

$\Gamma^{*}:\langle[C_{1}],\ldots,[C_{r}]\rangle\longrightarrow\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda(n,m))]).$

of the subgroup
$\langle[C_{1}],\ldots,[C_{r}]\rangle\leq\pi_{1}(\mathcal{L}(n,m))$ generated
by the homotopy classes of the based maps
$C_{1},\ldots,C_{r}:S^{1}\longrightarrow\mathcal{L}(n,m)$. The first result we
present is:

######  Theorem 1.1.

Let $\mathcal{L}(3,6)$ be the space of Legendrian links isotopic to the
maximal-tb Legendrian torus link
$\Lambda(3,6)\subseteq(S^{3},\xi_{\text{st}})$. Then there exist two based
loops $A,B:S^{1}\longrightarrow\mathcal{L}(3,6)$, and a regular function
$\Delta\in\mathbb{C}[\mathcal{M}(\Lambda(3,6))]$ such that the monodromy
representation

$\Gamma^{*}:\langle[A],[B]\rangle\longrightarrow\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda(3,6))])$

restricts to a faithful modular representation

$\Gamma^{*}|_{\mathcal{O}(\Delta)}:\operatorname{PSL}(2,\mathbb{Z})\longrightarrow\operatorname{Aut}(\mathcal{O}(\Delta))$

along the orbit $\mathcal{O}(\Delta)$ of the function $\Delta$. $\Box$

In Theorem 1.1, we choose the base point for the space $\mathcal{L}(3,6)$ to
be the Legendrian link in $(\mathbb{R}^{3},\xi_{\text{st}})$ whose front
projection is depicted in Figure 1, under an arbitrary but fixed choice of
Darboux chart
$(\mathbb{R}^{3},\xi_{\text{st}})\subseteq(S^{3},\xi_{\text{st}})$. In the
statement of Theorem 1.1,
$\langle[A],[B]\rangle\subseteq\pi_{1}(\mathcal{L}(3,6))$ denotes the subgroup
generated by the homotopy classes $[A],[B]\in\pi_{1}(\mathcal{L}(3,6))$, with
concatenation of based loops as its group operation. The modular group
$\operatorname{PSL}(2,\mathbb{Z})$ shall appear geometrically as the free
product $\mathbb{Z}_{3}*\mathbb{Z}_{2}$, with the factor $\mathbb{Z}_{3}$
generated by the restriction of $[A]$ and the factor $\mathbb{Z}_{2}$
generated by the restriction of $[B]$, and
$\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda(3,6))])$ denotes the group
of (cluster) automorphisms of $\mathbb{C}[\mathcal{M}(\Lambda(3,6))]$. Theorem
1.1 is remarkable in that $\operatorname{PSL}(2,\mathbb{Z})$ is an infinite
group and thus provides the first result of its kind in the study of
$3$-dimensional Legendrian links.

######  Remark 1.2.

The reason for the choice of the Legendrian torus link $\Lambda(3,6)$ is that
it is the geometric source of the affine root system $\widetilde{E_{8}}$.
Indeed, it can be understood as the maximal-tb Legendrian approximation of the
transverse link of the unimodal $J_{10}$ singularity [2]. The proof of Theorem
1.1 shall clarify how the $\widetilde{E_{8}}$ algebraic structure arises from
$\Lambda(3,6)$. $\Box$

We show that the Legendrian torus link $\Lambda(4,4)$ also exhibits a
noteworthy symmetry:

######  Theorem 1.3.

Let $\mathcal{L}(4,4)$ be the space of Legendrian links isotopic to the
maximal-tb Legendrian torus link
$\Lambda(4,4)\subseteq(S^{3},\xi_{\text{st}})$. Then there exist three based
loops $\Xi_{i}:S^{1}\longrightarrow\mathcal{L}(4,4)$, $1\leq i\leq 3$, and a
subset $F\subseteq\mathbb{C}[\mathcal{M}(\Lambda(4,4))]$ such that the
monodromy representation

$\Gamma^{*}:\langle[\Xi_{1}],[\Xi_{2}],[\Xi_{3}]\rangle\longrightarrow\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda(4,4))])$

restricts to a faithful representation

$\Gamma^{*}|_{\mathcal{O}(F)}:M_{0,4}\longrightarrow\operatorname{Aut}(\mathcal{O}(F))$

of the mapping class group $M_{0,4}$ along the orbit $\mathcal{O}(F)$. $\Box$

The mapping class group $M_{0,4}$ of the four-punctured 2-sphere contains a
subgroup isomorphic to $\operatorname{PSL}(2,\mathbb{Z})$ with finite index
and it is thus infinite. In Theorem 1.3, the base point for the space
$\mathcal{L}(4,4)$ is chosen to be the Legendrian link in
$(\mathbb{R}^{3},\xi_{\text{st}})$ with front projection as depicted in Figure
1, also under an arbitrary but fixed choice of Darboux chart
$(\mathbb{R}^{3},\xi_{\text{st}})\subseteq(S^{3},\xi_{\text{st}})$. The
subgroup
$\langle[\Xi_{1}],[\Xi_{2}],[\Xi_{3}]\rangle\subseteq\pi_{1}(\mathcal{L}(4,4))$
has loop concatenation as its group operation.

######  Remark 1.4.

The two groups $\operatorname{PSL}(2,\mathbb{Z})$ and $M_{0,4}$ featured in
Theorems 1.1 and 1.3 are akin to each other in that there are two group
isomorhisms $\operatorname{PSL}(2,\mathbb{Z})\cong B_{3}/Z(B_{3})$ and
$M_{0,4}\cong B^{s}_{4}/Z(B^{s}_{4})$, where $B_{3}$ denotes the braid group
in 3-strands, $B^{s}_{4}$ denotes the spherical braid group in 4-strands, and
$Z(B_{3})$ and $Z(B^{s}_{4})$ denote their respective centers. $\Box$

Let us now state implications of Theorems 1.1 and 1.3, all of which are new
results in low-dimensional contact and symplectic topology.

###  1.3. Lagrangian Fillings

Consider the subset

$\mathcal{H}:=\\{(n,m)\in\mathbb{N}\times\mathbb{N}:n\leq m,3\leq n,6\leq
m\\}\cup\\{(4,4),(4,5)\\}\subseteq\mathbb{N}\times\mathbb{N}.$

The first consequence of Theorems 1.1 and 1.3 is:

######  Corollary 1.5.

The Legendrian torus link $\Lambda(n,m)\subseteq(S^{3},\xi_{\text{st}})$,
$(n,m)\in\mathcal{H}$, admits infinitely many exact Lagrangian fillings in the
standard symplectic $4$-ball $(D^{4},\omega_{\text{st}})$. $\Box$

For each $\Lambda(n,m)$, these infinitely many exact Lagrangian fillings are
smoothly isotopic and not Hamiltonian isotopic. Note that both Theorem 1.1 and
Theorem 1.3 are needed in order to cover all $\Lambda(n,m)$ for
$(n,m)\in\mathcal{H}$. That said, Theorem 1.1 suffices in order to conclude
Corollary 1.5 for $(n,m)\in(\mathcal{H}\setminus\\{(4,4),(4,5)\\})$ and thus
Theorem 1.3 is included to achieve Corollary 1.5 for $(n,m)=(4,4)$ and
$(4,5)$. It should be noted that the article [58] succeeded in constructing
finitely many Lagrangian fillings of the maximal-tb Legendrian $(n,m)$-torus
link, as many as maximal pairwise weakly separated $n$-element subsets [44,
52] of $[1,n+m]$, a finite number which is bounded above by $nm+1$. Corollary
1.5 implies that these finitely many exact Lagrangian fillings do not exhaust
all possible, actually infinitely many, exact Lagrangian fillings.

Every knot $K\subseteq S^{3}$ is either a torus knot, a satellite knot or a
hyperbolic knot, as proven in [60, Theorem 2.3] by W.P. Thurston. Let us
consider a Legendrian representative
$\Lambda_{K}\subseteq(S^{3},\xi_{\text{st}})$ of the smooth type $K\subseteq
S^{3}$ and denote by $l(\Lambda_{K})\in\mathbb{N}\cup\\{\infty\\}$ the number
of orientable exact Lagrangian fillings $L\subseteq(D^{4},\omega_{\text{st}})$
of the Legendrian knot $\Lambda_{K}\subseteq(S^{3},\xi_{\text{st}})$, up a
Hamiltonian isotopy. Consider the smooth invariant

$\overline{l}(K):=\sup\\{l(\Lambda_{K}):\Lambda_{K}\subseteq(S^{3},\xi_{\text{st}})\mbox{
is a Legendrian representative of }K\\}\in\mathbb{N}\cup\\{\infty\\}$

for a smooth knot $K\subseteq S^{3}$. To our knowledge, there is not hitherto
known instance of a non-trivial knot $K\subseteq S^{3}$ for which
$\overline{l}(K)$ is known and non-vanishing. In addition, there are non-
trivial knots $K\subseteq S^{3}$ for which the invariant $\overline{l}(K)$
vanishes. For instance, $\overline{l}(m(8_{19}))=0$ is known to vanish since
the Kauffman upper bound is not sharp [31, 48]. We shall now use Theorem 1.1
to show that $\overline{l}(K)$ is actually infinite for infinitely many knots
within each of the three Thurston classes:

######  Corollary 1.6.

The equality $\overline{l}(K)=\infty$ holds for infinitely many torus knots,
infinitely many satellite knots and infinitely many hyperbolic knots
$K\subseteq S^{3}$ in the $3$-sphere.

The satellite knots in Corollary 1.6 can be chosen to be cable knots, and the
hyperbolic knots we will exhibit are also well-beloved [7, 33]. For instance,
we will show that $\overline{l}(K)=\infty$ already for $K=k(4_{3})$, the
simplest hyperbolic non-2-bridge knot [11].

###  1.4. Lagrangian Concordances

Now, let $\mathbb{L}(n,m)$ be the monoid of exact Lagrangian concordances, up
to Hamiltonian isotopy, for the Legendrian link
$\Lambda(n,m)\subseteq(S^{3},\xi_{\text{st}})$. Theorems 1.1 and 1.3 readily
imply:

######  Corollary 1.7.

There exists subgroups $\Gamma\subseteq\mathbb{L}(3,6)$ and
$\Gamma^{\prime}\subseteq\mathbb{L}(4,4)$ such that the group
$\operatorname{PSL}(2,\mathbb{Z})$ is a factor of $\Gamma$, and $M_{0,4}$ is a
factor of $\Gamma^{\prime}$. $\Box$

By definition, the groups $\Gamma$ and $\Gamma^{\prime}$ in Corollary 1.7 are
the subgroups generated by the exact Lagrangian concordances obtained by
graphing the Legendrian loops in Theorems 1.1 and 1.3. Corollary 1.7
emphasizes the relevance of Lagrangian concordances in the study of Legendrian
knots. In particular, the existence of Lagrangian concordances of infinite
order is a new result which itself provides a genuinely useful perspective for
the study of Lagrangian fillings. Indeed, there is no a priori reason for the
infinite Lagrangian fillings in Corollary 1.5 to be describable in terms of a
finite number of Lagrangian concordances. The present results show that this
is the case. Similarly,

######  Corollary 1.8.

There exists subgroups $\Gamma\subseteq\pi_{1}(\mathcal{L}(3,6))$ and
$\Gamma^{\prime}\subseteq\pi_{1}(\mathcal{L}(4,4))$ such that the group
$\operatorname{PSL}(2,\mathbb{Z})$ is a factor of $\Gamma$, and $M_{0,4}$ is a
factor of $\Gamma^{\prime}$. $\Box$

Corollary 1.7 and Corollary 1.8 are the first instances in contact topology of
infinite order elements in the concordance monoid $\mathbb{L}(\Lambda)$, and
the fundamental group $\pi_{1}(\mathcal{L}(\Lambda))$, for a Legendrian
$\Lambda\subseteq(S^{3},\xi_{\text{st}})$. Both
$\operatorname{PSL}(2,\mathbb{Z})$ and $M_{0,4}$ contain free groups of any
countable rank as subgroups, and thus many infinite order elements exist in
$\pi_{1}(\mathcal{L}(3,6))$ and $\pi_{1}(\mathcal{L}(4,4))$. In fact, $\Gamma$
and $\Gamma^{\prime}$ are exponential-growth subgroups of
$\pi_{1}(\mathcal{L}(3,6))$ and $\pi_{1}(\mathcal{L}(4,4))$.

######  Remark 1.9.

Corollary 1.8 stands in contrast with A. Hatcher’s work [35, Theorem 1] in the
smooth category. Indeed, the fundamental group $\pi_{1}(\mathscr{K}_{(n,m)})$
of the space $\mathscr{K}_{(n,m)}$ of smooth knots in $S^{3}$ isotopic to the
$(n,m)$-torus knot is the finite Abelian group $\mathbb{Z}_{2}$. $\Box$

###  1.5. Stein surfaces

Finally, let $M(n,m)$ be the Stein surface obtained by a attaching Weinstein
$2$-handle [14, 63] to $(D^{4},\lambda_{\text{st}})$ along each of the
components of the Legendrian link
$\Lambda(n,m)\subseteq(S^{3},\xi_{\text{st}})$. For $\gcd(n,m)=1$, the
Weinstein 4-manifold $M(n,m)$ is homotopy equivalent to the 2-sphere. Theorems
1.1 and 1.3 imply the existence of infinitely many Lagrangian surfaces in the
following Stein surfaces:

######  Corollary 1.10.

Let $(n,m)\in\mathcal{H}$, $\gcd(n,m)=1$ and $M(n,m)$ the Weinstein 4-manifold
obtained by attaching a Weinstein 2-handle to $(D^{4},\omega_{\text{st}})$
along $\Lambda(n,m)$. Then $M(n,m)$ contains infinitely many smoothly isotopic
closed exact Lagrangian surfaces of genus $\frac{1}{2}(n-1)(m-1)$ which are
not Hamiltonian isotopic. $\Box$

To our knowledge, Corollary 1.10 presents the first known Stein surfaces
homotopic to the $2$-sphere with infinitely many non-Hamiltonian isotopic
exact Lagrangian surfaces of higher genus in the same smooth isotopy class.

Infinitely many distinct Lagrangian 2-spheres were known to exist in
$A_{k}$-Milnor fibres [55, Theorem 5.10], $k\geq 3$, and infinitely many exact
Lagrangian tori were known to exist in certain Stein surfaces by using either
of the articles [39, 57, 61, 62]. (These infinite families of genus $0$ and
$1$ are presently not known to come from infinitely many Lagrangian fillings
of a Legendrian link, nor are the ambient Weinstein 4-manifolds homotopic to
the $2$-sphere.)

In Corollary 1.10, the 1-dimensional intersection form of the Weinstein
4-manifold $M(n,m)$ is positive definite and equals $(nm-n-m-1)$, since
$tb(\Lambda(n,m))=nm-n-m$. In consequence, $M(n,m)$ does not admit any
Lagrangian surface of genus strictly less than $\frac{1}{2}(n-1)(m-1)$. Thus,
the genus in Corollary 1.10 is sharp.

######  Remark 1.11.

The Lagrangian 2-spheres in [55] differ by a composition of symplectic Dehn
twists [3, 54]. This is not the case for the exact Lagrangian higher-genus
surfaces in Corollary 1.10 since, by the paragraph above, the Weinstein
4-manifolds $M(n,m)$, $(n,m)\in\mathcal{H}$, do not contain embedded
Lagrangian 2-spheres. $\Box$

Organization. The article is organized as follows. Section 2 geometrically
constructs the loops in Theorem 1.1 and Theorem 1.3. Section 3 provides the
necessary aspects from the theory of Legendrian invariants constructed through
the study of microlocal sheaves. Sections 4 and 5 prove Theorem 1.1 and
Theorem 1.3, respectively, and Section 6 proves the Corollaries stated in the
introduction. $\Box$

Acknowledgements. R.C. is grateful to J.B. Etnyre and L. Ng for useful
conversations, and to I. Smith for helpful comments on Corollary 1.10. J.B.
Etnyre asked us an interesting question on our first manuscript, now answered
in Corollary 1.6. We also thank A. Keating, J. Sabloff, L. Starkston and U.
Varolgüneş for their interest on this project and a number of useful comments
and suggestions. Finally, both authors are indebted to E. Zaslow for many
valuable discussions on microlocal Legendrian invariants, and to C. Fraser for
helpful conversations on cluster modular groups. R. Casals is supported by the
NSF grant DMS-1841913 and a BBVA Research Fellowship. $\Box$

##  2\. The geometric construction

In this section we construct the Legendrian loops in Theorems 1.1 and 1.3
associated to the Legendrian links $\Lambda(3,6)$ and $\Lambda(4,8)$. This
construction is one of the central geometric contributions of the article.
This section also serves to setup the elements of contact geometry that we
shall need [21, 32].

The Legendrian loops $\Sigma_{1},\delta^{2},\Xi_{1},\Xi_{2}$ and $\Xi_{3}$
that we construct can be equivalently considered as exact Lagrangian
concordances in the symplectization
$(S^{3}\times\mathbb{R}(t),d(e^{t}\alpha_{\text{st}}))$ with no critical
points with respect to the projection onto the $\mathbb{R}$-factor [14, 32].
These Lagrangian concordances are obtained by graphing concatenations of the
Legendrian isotopies described in Subsection 2.2.

###  2.1. The standard contact 3-space

The Legendrian links $\Lambda\subseteq(S^{3},\xi_{\text{st}})$ in this article
will be considered inside the standard contact 3-space
$(\mathbb{R}^{3},\xi_{\text{st}})$, considered as a standard Darboux chart
within the contact 3-sphere $(S^{3},\xi_{\text{st}})$ [2, 4].

In discussing Lagrangian fillings, the inclusion
$\Lambda\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ will be composed with the
inclusion $(\mathbb{R}^{3},\xi_{\text{st}})\subseteq(S^{3},\xi_{\text{st}})$
given by the one-point compactification. In this identification, the
Lagrangian fillings of a Legendrian link
$\Lambda\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ will be exact Lagrangian
surfaces in $(D^{4},\lambda_{\text{st}})$ considered up to Hamiltonian
isotopy.

The constructions in this article give rise to contact geometric objects in
$(\mathbb{R}^{3},\xi_{\text{st}})$, including Legendrians links and contact
isotopies. Nevertheless, it is enlightening to focus on a small neighborhood
of the standard Legendrian unknot
$\Lambda_{un}\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ which is
contactomorphic to $(J^{1}S^{1},\xi_{\text{st}})$, and work in the solid torus
$(J^{1}S^{1},\xi_{\text{st}})$. Thus, in this article, Legendrian links
$\Lambda\subseteq(J^{1}S^{1},\xi_{\text{st}})$ and compactly supported contact
isotopies in
$\\{\varphi_{t}\\}_{t\in[0,1]}\in\operatorname{Cont}(J^{1}S^{1},\xi_{\text{st}})$
shall implicitly be understood as Legendrian links
$\Lambda\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ and contact isotopies
$\\{\varphi_{t}\\}_{t\in[0,1]}\in\operatorname{Cont}(\mathbb{R}^{3},\xi_{\text{st}})$
by satelliting the zero section $S^{1}\subseteq(J^{1}S^{1},\xi_{\text{st}})$
to $\Lambda_{un}\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$.

###  2.2. Legendrian Loops

By definition, a Legendrian loop in $(J^{1}S^{1},\xi_{\text{st}})$ is a
Legendrian isotopy
$\\{\Lambda_{t}\\}_{t\in[0,1]}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ such that
$\Lambda_{0}=\Lambda_{1}$. Let $(\theta,p_{\theta},z)\in
S^{1}\times\mathbb{R}^{2}$ be global coordinates in $J^{1}S^{1}$ and
$\xi_{\text{st}}=\ker{(dz-p_{\theta}d\theta)}$. The description of our
Legendrian loops shall use the front projection

$(J^{1}S^{1},\xi_{\text{st}})\longrightarrow
S^{1}\times\mathbb{R},\quad(\theta,p_{\theta},z)\longmapsto(\theta,z),$

which is indeed a valid front as the fibers are Legendrians. By definition,
the Legendrian $\Lambda(\mathscr{B})\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$
associated to a positive Legendrian braid
$\mathscr{B}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ is the image of
$\mathscr{B}$ under the operation of satelliting the zero section
$S^{1}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ along the standard Legendrian
unknot.

Let $k\in\mathbb{N}$, a geometric braid
$\mathscr{B}\subseteq(J^{1}S^{1},\xi_{\text{st}})$, which is a Legendrian
link, can be encoded algebraically by a representation $\beta$ of an element
$[\beta]\in B_{k}$ of the $k$-stranded braid group

$B_{k}:=\langle\sigma_{1},\ldots,\sigma_{k-1}|\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}\mbox{
for }j\neq i\pm 1,1\leq i,j\leq k-1\rangle,$

where $\sigma_{i}$ are the standard Artin generators. The choice of
representation $\beta$, i.e. braid word, for the element $[\beta]\in B_{k}$ is
not unique, as one might use the word relations in $B_{k}$ to obtain different
representations $\beta_{1},\beta_{2}$ such that $[\beta_{1}]=[\beta_{2}]$ in
$B_{k}$. Given a braid word $\beta$, we denote by
$\mathscr{B}(\beta)\subseteq(J^{1}S^{1},\xi_{\text{st}})$ the Legendrian link
associated to $\beta$, and denote by
$\Lambda(\beta)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ the Legendrian link
$\Lambda(\mathscr{B}(\beta))$.

Notice that the geometric braid
$\mathscr{B}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ has a front in
$S^{1}\times\mathbb{R}$. Thus, we fix a basepoint $\theta_{0}\in S^{1}$ and
require that a braid word $\beta$ representing $\mathscr{B}$ has the form:

$\beta=\prod_{j=1}^{l}\sigma_{i_{j}},\quad 1\leq i_{j}\leq k-1,$

where $\sigma_{i_{1}}$ is the first crossing in the front diagram of
$\mathscr{B}$ on the right of the vertical line
$\\{\theta_{0}\\}\times\mathbb{R}\subseteq S^{1}\times\mathbb{R}$ and the
crossings are read from left to right.

In this article, we shall construct Legendrian loops by performing Legendrian
isotopies which primarily consist of Reidemeister moves in the front. In
particular, the three central operations that we use are:

  * (i)

Reidemeister III Moves. In terms of the given braid word presentation
$\mathscr{B}=\mathscr{B}(\beta)$, the Reidemeister III move consists in
applying the relation
$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$. We
shall refer to a Legendrian isotopy which implements the substitution

$\sigma_{i}\sigma_{i+1}\sigma_{i}\longmapsto\sigma_{i+1}\sigma_{i}\sigma_{i+1}$

as an ascending Reidemeister III Move, and denote it by R3a. Similarly, to a
Legendrian isotopy which implements the substitution

$\sigma_{i+1}\sigma_{i}\sigma_{i+1}\longmapsto\sigma_{i}\sigma_{i+1}\sigma_{i}$

as a descending Reidemeister III Move, and denote it by R3d. Thus, either R3
Reidemeister move is understood as a Legendrian isotopy.

  * (ii)

Cyclic Permutation. Consider a braid $\mathscr{B}(\beta)$ represented by

$\beta=\prod_{j=1}^{l}\sigma_{i_{j}},\quad 1\leq i_{j}\leq k-1.$

By definition, a cyclic shift $\delta$ is a Legendrian isotopy
$\\{\psi_{t}\\}_{t\in[0,1]}$ which brings the geometric braid
$\mathscr{B}(\beta)$ to $\psi_{1}(\mathscr{B}(\beta))$ such that the braid
word $\psi_{1}(\beta)$ for the latter is

$\psi_{1}(\beta)=\left(\prod_{j=2}^{l}\sigma_{i_{j}}\right)\sigma_{i_{1}},\quad
1\leq i_{j}\leq k-1.$

Note that this braid word for $\psi_{1}(\beta)$ is read with respect to the
fixed basepoint $\theta_{0}$. Explicitly, this Legendrian isotopy can be
geometrically visualized by rotating $\mathscr{B}(\beta)$ to the left by an
appropriate angle while keeping the zero section fixed.

Since we study Legendrian braids in $(J^{1}S^{1},\xi_{\text{st}})$, rather
than $(J^{1}[0,1],\xi_{\text{st}})$, two braid words $\beta_{1}=\beta_{2}$
which differ by a cyclic permutation yield Legendrian isotopic
$\Lambda(\beta_{1})$ and $\Lambda(\beta_{2})$. Hence, the operations above
produce Legendrian isotopies.

  * (iii)

Commutation. The third move $\gamma$ is just implementing the commutation
relation in the braid group $B_{k}$. It is described as follows:

$\left(\prod_{j=1}^{p-1}\sigma_{i_{j}}\right)\sigma_{i_{p}}\sigma_{i_{p+1}}\left(\prod_{j=p+2}^{l}\sigma_{i_{j}}\right)\longmapsto\left(\prod_{j=1}^{p-1}\sigma_{i_{j}}\right)\sigma_{i_{p+1}}\sigma_{i_{p}}\left(\prod_{j=p+2}^{l}\sigma_{i_{j}}\right),$

with indices $1\leq i_{j}\leq k-1,1\leq p\leq l-1$ and $i_{p+1}\neq i_{p}\pm
1$. This move can be realized by a compactly supported Legendrian isotopy in
$(J^{1}S^{1},\xi_{\text{st}})$ which we also refer to as $\gamma$, which is
the greek letter for $c$, standing for commutation.

######  Example 2.1.

Consider the braid word
$\beta=(\sigma_{1}\sigma_{2}\cdot\ldots\cdot\sigma_{n-1})^{m+n}$ which
geometrically represents the Legendrian torus link
$\Lambda(n,m)=\Lambda(\beta)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$. Then
the composition of the cyclic shift $\delta$ exactly $(n-1)$-times yields a
Legendrian loop $\delta^{n-1}$ for $\Lambda(n,m)$. This is the Legendrian loop
studied in [37] where it is shown to be a non-trivial Legendrian loop. We
shall provide our own alternative proof of this non-triviality. $\Box$

###  2.3. The $\Sigma_{1}$-Loop for $\Lambda(3,6)$

In this subsection we define a Legendrian loop $\Sigma_{1}$ for the maximal-tb
Legendrian links $\Lambda(3,3s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$,
$s\in\mathbb{N}$, represented by the positive braid $\mathscr{B}(\beta)$, with
braid word $\beta=(\sigma_{1}\sigma_{2})^{3(s+1)}$, in the front domain
$S^{1}\times\mathbb{R}$. The loop $\Sigma_{1}$ is defined as the composition
of Legendrian isotopies induced by the following sequence of moves:

$\displaystyle(\sigma_{1}\sigma_{2})^{3(s+1)}$
$\displaystyle=(\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2})^{s+1}\stackrel{{\scriptstyle\delta}}{{\approx}}(\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma_{1}})^{s+1}\stackrel{{\scriptstyle\mbox{\tiny
R3}^{d}}}{{\approx}}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma_{1}\sigma_{2}\sigma_{1}}\sigma_{1}\sigma_{2}\sigma_{1})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
R3}^{a}}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma_{2}\sigma_{1}\sigma_{2}})^{(s+1)}=(\sigma_{1}\sigma_{2})^{3(s+1)}.$

In the above sequence, the letters in blue indicate changes in the braid word.
In words, the first isotopy is a cyclic shift moving $\sigma_{1}$ to the end
of the braid by shifting left past $\\{\theta_{0}\\}\times\mathbb{R}\subseteq
S^{1}\times\mathbb{R}$. The second isotopy consists of $(3s+3)$ simultaneous
and commuting Reidemeister R3d moves, whereas the third isotopy consists of
$(3s+3)$ simultaneous and commuting Reidemeister R3a moves. The composition of
these isotopies yields the initial braid word
$(\sigma_{1}\sigma_{2})^{3(s+1)}$ and thus it generates a Legendrian loop.

######  Definition 2.2.

Consider $\Lambda(3,3s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$, the
Legendrian isotopy $\Sigma_{1}$ is the Legendrian loop of $\Lambda(3,3s)$
induced by the sequence of Legendrian isotopies

$(\sigma_{1}\sigma_{2})^{3(s+1)}\stackrel{{\scriptstyle\delta}}{{\approx}}\sigma_{2}(\sigma_{1}\sigma_{2})^{3s+2}\sigma_{1}\stackrel{{\scriptstyle\mbox{\tiny
R3}^{d}}}{{\approx}}((\sigma_{1}\sigma_{2}\sigma_{1})(\sigma_{1}\sigma_{2}\sigma_{1}))^{(s+1)}\stackrel{{\scriptstyle\mbox{\tiny
R3}^{a}}}{{\approx}}((\sigma_{1}\sigma_{2}\sigma_{1})(\sigma_{2}\sigma_{1}\sigma_{2}))^{(s+1)},$

once the zero section $S^{1}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ is
satellited to the standard unknot. $\Box$

Definition 2.2 yields Legendrian loops for $\Lambda(3,3s)$ for any
$s\in\mathbb{N}$. In this article it shall suffice to focus on the case $s=2$.
It might be relevant to notice that in Section 4 we shall prove that the loop
$\Sigma_{1}$ is non-trivial as a Legendrian loop and it is different from the
cyclic loop in Example 2.1. In fact, the $\Sigma_{1}$-loop and the cyclic
shift $\delta$ for the braid $\beta=(\sigma_{1}\sigma_{2})^{9}$ will suffice
in order to construct the representation in Theorem 1.1.

######  Remark 2.3.

The Legendrian loop $\Sigma_{1}$ is geometrically constructed in order to
algebraically act as the first Artin generator for a braid group action of
$B_{3}$ into $\mathbb{C}[\Lambda(3,6)]$. $\Box$

###  2.4. The $\Xi_{1}$-Loop for $\Lambda(4,4)$

Let us now define a Legendrian loop $\Xi_{1}$ for the maximal-tb Legendrian
links $\Lambda(4,4s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$,
$s\in\mathbb{N}$, represented by the 3-stranded positive braid
$\mathscr{B}(\beta)$, with braid word
$\beta=(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}$.

The Legendrian loop $\Xi_{1}$ is described by the cyclic shift

(2.1) $\displaystyle(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}$
$\displaystyle\stackrel{{\scriptstyle\delta}}{{\approx}}\sigma_{2}\sigma_{3}(\sigma_{1}\sigma_{2}\sigma_{3})^{4s+3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sigma_{1}}=(\sigma_{2}\sigma_{3}\sigma_{1})^{4(s+1)}$

followed by the sequence of moves:

$\displaystyle(\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1})^{(s+1)}\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}}}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}(\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{3}\sigma_{2}}}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1})^{(s+1)}\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{2}\sigma_{1}}}\sigma_{3}\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1})^{(s+1)}\quad(\Psi^{(1)}_{t}\textrm{
until here})$
$\displaystyle\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{1}}}\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1})^{(s+1)}\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}}})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{1}\sigma_{2}}}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{2}\sigma_{3}}}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{1}}}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}=(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}\quad(\Psi^{(2)}_{t}\textrm{
until here}).$

In each of the above rows, the letters emphasized in color blue represent
those braid generators, equivalently crossings of the front, which have been
affected at each step when performing the indicated Legendrian isotopy,
consisting either of a Reidemeister R3 move or a cyclic shift $\delta$. Note
that the sequence above ends with the braid word
$(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}$, and thus yields a Legendrian loop
when preconcatenated with the Legendrian isotopy in Equation 2.1.

######  Definition 2.4.

Consider $\Lambda(4,4s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$, the
Legendrian isotopy $\Xi_{1}$ is the Legendrian loop of $\Lambda(4,4s)$ given
by concatenating the two sequences above and satelliting the zero section
$S^{1}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ to the standard unknot. $\Box$

Let us now proceed with the construction of the second Legendrian loop
$\Xi_{2}$, also associated to the Legendrian links $\Lambda(4,4s)$. In
conjunction with $\Xi_{3}$, to be described momentarily, and the Legendrian
loop $\Xi_{1}$ above, $\Xi_{1},\Xi_{2},\Xi_{3}$ will be the geometric
ingredient for Theorem 1.3.

######  Remark 2.5.

The Legendrian loops $\Xi_{1},\Xi_{2},\Xi_{3}$ are geometrically constructed
to algebraically produce an action of the braid group $B_{4}$ into
$\mathbb{C}[\mathcal{M}(\Lambda(4,4))]$. Intuitively,
$\Xi_{1},\Xi_{2},\Xi_{3}$ act respectively as the three Artin generators for
$B_{4}$. $\Box$

###  2.5. The $\Xi_{2}$-Loop for $\Lambda(4,4)$

Let us now construct the Legendrian loop $\Xi_{2}$ for the maximal-tb
Legendrian links $\Lambda(4,4s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$,
$s\in\mathbb{N}$. We shall describe it using the same notation as in
Subsection 2.4 above. The Legendrian loop $\Xi_{2}$ starts with the braid word
$(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}$ and it is described by the
following sequence of Legendrian isotopies:

$\displaystyle(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}}}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{1}\sigma_{2}}}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle\delta}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}}})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}}}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2})^{(s+1)}\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{3}\sigma_{2}}}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2})^{(s+1)}\quad(\Psi^{(1)}_{t}\text{until
here})$ $\displaystyle\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{2}\sigma_{1}}}\sigma_{3}\sigma_{2}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2})^{(s+1)}\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{3}\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{2}\sigma_{3}}})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle\gamma^{2}}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{1}}}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{1}}}\sigma_{2}\sigma_{3})^{(s+1)}\quad(\Psi^{(2)}_{t}\text{until
here}).$

In each row, the letters emphasized in color blue represent those crossings
which have been affected when performing the indicated Legendrian isotopy,
consisting either of Reidemeister R3 move, a cyclic shift $\delta$ or a
commutation $\gamma$. In the above description, we denote by $\Psi^{(1)}_{t}$
the Legendrian isotopy consisting of the moves performed in the first six
equivalences, and we denote by $\Psi^{(2)}_{t}$ the Legendrian isotopy
consisting of the moves performed in the last four equivalences. The
decomposition into the two pieces $\Psi^{(1)}_{t}$ and $\Psi^{(2)}_{t}$ will
be used in Section 5. Finally, note that the sequence starts and ends with the
braid word $(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}$, and thus defines a
Legendrian loop for $\Lambda(4,4s)$ according to the fronts represented by
each braid word.

######  Definition 2.6.

Consider $\Lambda(4,4s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$, the
Legendrian isotopy $\Xi_{2}$ is the Legendrian loop of $\Lambda(4,4s)$ given
by the sequence of Legendrian isotopies above once the zero section
$S^{1}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ is satellited to the standard
unknot. $\Box$

###  2.6. The $\Xi_{3}$-Loop

We now construct the third Legendrian loop $\Xi_{3}$ for
$\Lambda(4,4s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$, $s\in\mathbb{N}$.
The Legendrian loop $\Xi_{3}$ starts with the braid word
$(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}$ and it is described by the
following sequence of Legendrian isotopies:

$\displaystyle(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}}}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{1}\sigma_{2}}}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}(\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{2}\sigma_{3}}}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\quad(\Psi^{(1)}_{t}\text{until
here})$
$\displaystyle\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{1}}}\sigma_{2}\sigma_{3}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle\gamma^{2}}}{{\approx}}(\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}\sigma_{3}}}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}(\sigma_{2}\sigma_{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{1}\sigma_{2}}}\sigma_{3}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle
R3^{a}}}{{\approx}}({\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{2}\sigma_{3}}}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\quad(\Psi^{(2)}_{t}\text{until
here}).$
$\displaystyle\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{3}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}}}\sigma_{2}\sigma_{3}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}(\sigma_{3}\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{3}\sigma_{2}}}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}(\sigma_{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{2}\sigma_{1}}}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{3}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{1}}}\sigma_{2}\sigma_{3}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{3}}}\sigma_{2}\sigma_{3})^{(s+1)}\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}(\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{2}\sigma_{1}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{2}\sigma_{3}\sigma_{2}}})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle
R3^{d}}}{{\approx}}(\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{1}\sigma_{2}\sigma_{1}}}\sigma_{3}\sigma_{2})^{(s+1)}\stackrel{{\scriptstyle\gamma}}{{\approx}}(\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}\sigma_{1}}}\sigma_{2})^{(s+1)}$
$\displaystyle\stackrel{{\scriptstyle\delta}}{{\approx}}(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{1}\sigma_{2}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\sigma_{3}}})^{(s+1)}=(\sigma_{1}\sigma_{2}\sigma_{3})^{4(s+1)}\quad(\Psi^{(3)}_{t}\text{until
here}).$

######  Definition 2.7.

Consider $\Lambda(4,4s)\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$, the
Legendrian isotopy $\Xi_{3}$ is the Legendrian loop of $\Lambda(4,4s)$ given
by concatenating the sequence of Legendrian isotopies above once the zero
section $S^{1}\subseteq(J^{1}S^{1},\xi_{\text{st}})$ is satellited to the
standard unknot. $\Box$

The loops $\Sigma_{1},\delta^{2},\Xi_{1},\Xi_{2}$ and $\Xi_{3}$ are the needed
geometric ingredients in our proof of Theorems 1.1 and Theorems 1.3. The
Legendrian loops $\Sigma_{1},\delta^{2}$ will give rise to the modular action,
and $\Xi_{1},\Xi_{2},\Xi_{3}$ to the faithful representation of $M_{0,4}$.
From a contact topology viewpoint, it is quite outstanding that the infinitely
many Lagrangian fillings in Corollary 1.5 can arise in this direct and
explicit manner. Let us now move to the algebraic invariants that we shall use
in order to build the representations of the modular group
$\operatorname{PSL}(2,\mathbb{Z})$ and the mapping class group $M_{0,4}$.

######  Remark 2.8.

The reader is invited to discover the analogue of $\Xi_{i}$,$1\leq i\leq 3$,
for the positive braid
$\beta=(\sigma_{1}\sigma_{2}\cdot\ldots\cdot\sigma_{n-1})^{n(s+1)}$. These are
Legendrian loops for the $n$-component Legendrian links $\Lambda(n,ns)$. We
shall nevertheless not need these loops in the present article and thus we do
not presently discuss them. $\Box$

##  3\. Microlocal Legendrian Invariants

In this section we introduce the algebraic invariants that we use in order to
construct the representations in Theorems 1.1 and 1.3. These are Legendrian
invariants arising from microlocal analysis and the study of constructible
sheaves on stratified spaces, as introduced by M. Kashiwara and P. Schapira in
the works [34, 38]. The articles [58, 59] have recently been developing these
Legendrian invariants. The present manuscript highlights a remarkable
application of these invariants to the study of Lagrangian fillings.

Let $\Lambda\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ be a Legendrian link
and identify the standard contact 3-space with the positive hemisphere bundle
$(T^{\infty,+}(\mathbb{R}^{2}),\xi_{\text{st}})$ of the real 2-plane. Let
$\operatorname{Sh}_{\Lambda}(\mathbb{R}^{2},\mathbb{C})$ be the derived dg-
category of constructible sheaves of $\mathbb{C}$-vector space on
$\mathbb{R}^{2}$ with singular support intersecting $T^{\infty}\mathbb{R}^{2}$
within the Legendrian $\Lambda$. Suppose that $\mbox{rot}(\Lambda)=0$ and
consider the microlocal monodromy functor
$\mu\mbox{mon}:\operatorname{Sh}_{\Lambda}(\mathbb{R}^{2},\mathbb{C})\longrightarrow\mbox{Loc}(\Lambda)$
to the category of local system of complexes of $\mathbb{C}$-vector spaces
[59, Section 5.1]. This allows us to consider the following moduli of objects

$\mathcal{M}^{\circ}(\Lambda):=\\{\mathscr{F}^{\bullet}\in\operatorname{Sh}_{\Lambda}(\mathbb{R}^{2},\mathbb{C}):\mbox{rk}(\mu\mbox{mon}(\mathscr{F}^{\bullet}))=1,\mu\mbox{mon}(\mathscr{F}^{\bullet})\mbox{
concentrated in degree }0\\}.$

It is shown in [34, 59] that the category
$\operatorname{Sh}_{\Lambda}(\mathbb{R}^{2},\mathbb{C})$, and in particular
$\mathcal{M}^{\circ}(\Lambda)$, is a Legendrian invariant of $\Lambda$. In the
present article, we restrict to Legendrian links
$\Lambda\subseteq(\mathbb{R}^{3},\xi_{\text{st}})$ which arise as
$\Lambda(\beta)$ for a positive braid $\beta$. For this class of Legendrian
links, $\mbox{rot}(\Lambda(\beta))=0$ and there exists a binary Maslov
potential. Indeed, the braid piece carries the zero Maslov potential and
satelliting to the standard Legendrian unknot - with its standard front -
increases the Maslov potential by exactly one.

###  3.1. The Broué-Deligne-Michel Description

In order to directly compute with the moduli spaces
$\mathcal{M}^{\circ}(\Lambda(\beta))$ and construct the representations in
Theorems 1.1 and 1.3, we require a more explicit description of the moduli
space $\mathcal{M}^{\circ}(\Lambda(\beta))$. This description is available due
to the work [59], which proves that $\mathcal{M}^{\circ}(\Lambda(\beta))$ is
isomorphic to a classical moduli BS($\beta$), modulo the gauge action,
associated to a braid $\beta$ by M. Broué-J. Michel [10] and Deligne [16].

Let $G=\operatorname{GL}_{k}(\mathbb{C})$ and $B\subseteq G$ the Borel
subgroup of upper triangular matrices. The quotient $G/B$ is the flag variety,
whose points parametrize complete flags $V^{\bullet}$ of vector subspaces of
$\mathbb{C}^{k}$. The Bruhat decomposition

$\displaystyle G/B=\bigsqcup_{w\in W}BwB/B$

implies that the relative position of a pair of flags
$(V_{1}^{\bullet},V_{2}^{\bullet})$ is determined by an element $s\in
S_{k}=\mbox{Weyl}{(G)}$ of the Weyl group, in this case a permutation in the
symmetric group. Consider the Artin generators $\sigma_{i}\in B_{k}$, $1\leq
i\leq k-1$, and denote by $\overline{\sigma_{i}}$ the image of $\sigma_{i}$
under the projection $B_{k}\longrightarrow S_{k}$ from the braid group to the
$A_{k-1}$-Coxeter group $S_{k}$. Given a flag $V^{\bullet}$ and a permutation
$s\in S_{k}$, let $S_{s}(V^{\bullet})$ be the associated Schubert cell, i.e.
the set of flags in relative $s$-positive with respect to $V^{\bullet}$.

Let $\beta$ be a positive braid word

$\beta=\prod_{j=1}^{l(\beta)}\sigma_{i_{j}},\quad 1\leq i_{j}\leq k-1,$

and consider the subset

$C(\beta)=\\{(V^{\bullet}_{1},\ldots,V^{\bullet}_{l(\beta)})\in(G/B)^{l(\beta)}:V^{\bullet}_{m+1}\in
S_{\overline{\sigma}_{i_{m}}}(V^{\bullet}_{m}),1\leq m\leq
l(\beta)\\}\subseteq(G/B)^{l(\beta)},$

where the index $1\leq m\leq l(\beta)$ is understood cyclically modulo
$l(\beta)$, i.e. the condition for $m=l(\beta)$ reads $V^{\bullet}_{1}\in
S_{\overline{\sigma}_{i_{l(\beta)}}}(V^{\bullet}_{l(\beta)})$.

######  Definition 3.1.

The open Bott-Samelson cell BS($\beta$) associated to $\beta$ is the space

$\mbox{BS}(\beta)=C(\beta)\setminus(\widetilde{\Delta}\cap C(\beta))$

where $\widetilde{\Delta}\subseteq(G/B)^{l(\beta)}$ denotes the fat diagonal.
$\Box$

For each $\beta$, the group $G$ acts on the open Bott-Samelson cell
BS($\beta$) diagonally on the left, given that the flag variety $G/B$ is given
by the $B$-action on the right. The article [59, Section 6] identifies
$\mathcal{M}^{\circ}(\Lambda(\beta))$ with the quotient
$G\backslash$BS($\beta$). It is a consequence of this identification that our
moduli space $\mathcal{M}^{\circ}(\Lambda(\beta))$ can be described as
follows.

Choose a set of points
$\\{\theta_{0},\theta_{1},\ldots,\theta_{l(\beta)}\\}\in S^{1}$ such that the
vertical lines $\\{\theta_{m}\\}\times\mathbb{R}$, $0\leq m\leq l(\beta)$ do
not intersect the front $\beta\subseteq S^{1}\times\mathbb{R}$ at a crossing
and there exists a unique crossing of $\beta$ between
$\\{\theta_{m}\\}\times\mathbb{R}$ and $\\{\theta_{m+1}\\}\times\mathbb{R}$,
$0\leq i\leq l(\beta)$. Then $\mathcal{M}^{\circ}(\Lambda(\beta))$ is the
moduli space given by associating a complete flag $V^{\bullet}_{m}$ along each
vertical line $\\{\theta_{m}\\}\times\mathbb{R}$ such that
$V^{\bullet}_{0}=V^{\bullet}_{l(\beta)}$ and two flags $V^{\bullet}_{m}$ and
$V^{\bullet}_{m+1}$ differ only and exactly in their $i_{m+1}$-dimensional
subspaces for all $1\leq m\leq l(\beta)$, modulo the gauge group action of
$\operatorname{GL}_{k}(\mathbb{C})$. This description in terms of BS($\beta$)
will be used in Sections 4 and 5.

###  3.2. Moduli of Framed Sheaves

In the proof of Theorems 1.1 and 1.3 we shall need a framed enhancement
$\mathcal{M}(\Lambda(\beta),\tau)$ of the Bott-Samelson cells
$\mathcal{M}^{\circ}(\Lambda(\beta))$. In precise terms, the points of
$\mathcal{M}(\Lambda(\beta),\tau)$ are given by the $l(\beta)$-tuples of flags
$[(V^{\bullet}_{1},\ldots,V^{\bullet}_{l(\beta)})]\in\mathcal{M}^{\circ}(\Lambda(\beta))$
equipped with trivializations $\tau$ for the stalks at a specified set of
points. In this case, we choose the set of points such that the set contains
exactly one point for each region where the constructible sheaf has a
1-dimensional stalk. Given that they are in bijection, we will interchangeably
speak of these points or the open strata in the front diagram that contain
them, these open strata shall also be referred to as regions. Hence, in the
language of Bott-Samelson cells, the trivialization $\tau$ consists of a
series of isomorphisms

$V^{(1)}_{m}\cong\mathbb{C},\quad 1\leq m\leq l(\beta).$

In our context, the moduli spaces of framed sheaves
$\mathcal{M}(\Lambda(\beta),\tau)$ are algebraic varieties [59]. It should be
emphasized that the moduli space $\mathcal{M}(\Lambda(\beta),\tau)$ depends on
the choice of trivialization $\tau$. In our choice above,
$\mathcal{M}(\Lambda(\beta),\tau)$ shall depend on the choice of braid word
$\beta$. Indeed, the length of the tuple is precisely $l(\beta)$.
Nevertheless, the article [58] shows that a Legendrian isotopy generates an
equivalence of moduli space $\mathcal{M}(\Lambda(\beta),\tau)$ of framed
sheaves, with the trivialization, and its region, being pushed forward under
the isotopy. Thus, in studying the action of a Legendrian loop on
$\mathcal{M}(\Lambda(\beta),\tau)$ we identify the moduli spaces of framed
sheaves along the Legendrian isotopy and compare the action at the canonically
identified endpoints of the Legendrian loop.

Explicitly, let $\\{\Psi_{t}\\}_{t\in[0,1]}$ be a Legendrian loop based at the
identity, i.e. $\Psi_{0}=\mbox{Id}$. By [58, Section 2], there is a canonical
isomorphism between the moduli spaces $\mathcal{M}(\Lambda,\tau)$ and
$\mathcal{M}(\Psi_{t}(\Lambda),(\Psi_{t})_{*}\tau)$ for all $t\in[0,1]$. By
virtue of being a Legendrian loop, $\Psi_{1}=\mbox{Id}$ and thus we obtain an
algebraic automorphism $\Psi\in\operatorname{Aut}(\mathcal{M}(\Lambda,\tau))$
of the moduli space $\mathcal{M}(\Lambda,\tau)$. This automorphism is to be
understood as the monodromy of the Legendrian loop
$\\{\Psi_{t}\\}_{t\in[0,1]}$, in line with T. Kálmán’s [37, Section 3]
monodromy invariant. The automorphism
$\Psi\in\operatorname{Aut}(\mathcal{M}(\Lambda,\tau))$ in turn induces an
automorphism
$\Psi^{*}\in\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda,\tau)])$ in the
coordinate ring of regular functions on $\mathcal{M}(\Lambda,\tau)$.

In Theorems 1.1 and 1.3 the focus will be on two moduli spaces
$\mathcal{M}(\Lambda(\beta),\tau)$ for the two braid words
$\beta=(\sigma_{1}\sigma_{2})^{9}$ and
$\beta=(\sigma_{1}\sigma_{2}\sigma_{3})^{8}$ and a chosen trivialization
$\tau$.

In the next two sections we prove Theorem 1.1 and 1.3. These two proofs are
independent of each other. The reader is nevertheless encouraged to read the
proof of Theorem 1.1 first, as it also sets the main techniques and notations
for the proof of Theorem 1.3.

##  4\. The $\operatorname{PSL}(2,\mathbb{Z})$ representation for
$\Lambda(3,6)$

Let us prove Theorem 1.1. For that, we shall compute the action of the two
Legendrian loops $\Sigma_{1},\delta^{2}$ constructed in Section 2 into the
coordinate ring $\mathbb{C}[\mathcal{M}(\Lambda(\beta),\tau)]$ of the framed
Bott-Samelson cell $\mathcal{M}(\Lambda(\beta),\tau)$, where the braid is
fixed to be $\beta=(\sigma_{1}\sigma_{2})^{9}$ and the trivialization $\tau$
is given at the 1-dimensional stalks depicted as dots Figure 2, where the
braid $\beta$ for the Legendrian link $\Lambda(3,6)$ is also depicted. These
monodromy invariants
$\Sigma_{1}^{*},(\delta^{2})^{*}\in\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda(\beta),\tau)])$
will be shown to be non-trivial and generate an action of an infinite group on
the coordinate ring $\mathbb{C}[\mathcal{M}(\Lambda(\beta),\tau)]$.

Figure 2. The identification of $\mathcal{M}(\Lambda(3,6))$ with the positroid
cell in $\operatorname{Gr}(3,9)$.

###  4.1. The Monodromy Effect within $\operatorname{Gr}(3,9)$

The first step of the argument is to identify the moduli
$\mathcal{M}(\Lambda(\beta),\tau)$ with the positroid stratum
$\Pi_{r}\subseteq\operatorname{Gr}(3,9)$ in the projective Grassmannian
$\operatorname{Gr}(3,9)$, where $r$ is the cyclic rank matrix associated to
the positive braid $(\sigma_{1}\sigma_{2})^{9}$. The canonical embedding of
$\mathcal{M}(\Lambda(\beta),\tau)$ into $\operatorname{Gr}(3,9)$, with image
$\Pi_{r}$, is obtained as follows [58, Section 3.2]. Given a point
$(V^{\bullet}_{1},\ldots,V^{\bullet}_{18})\in\mathcal{M}(\Lambda(\beta),\tau)$,
the $9$-tuple of vectors

$(v_{1},v_{2},\ldots,v_{9})\in(V^{(1)}_{1},V^{(1)}_{3},V^{(1)}_{5},\ldots,V^{(1)}_{17}),$

modulo the $\operatorname{GL}_{9}(\mathbb{C})$-action, defines a point in
$\operatorname{Gr}(3,9)$, where the choice of vectors is given by the framing.
These vectors $(v_{1},v_{2},\ldots,v_{9})$ are depicted in Figure 2. The
advantage of this algebraic embedding
$\mathcal{M}(\Lambda(\beta),\tau)\longrightarrow\operatorname{Gr}(3,9)$ is
that it allows us to use elements in the homogeneous coordinate ring of
$\operatorname{Gr}(3,9)$ restricted to $\mathcal{M}(\Lambda(\beta),\tau)$ in
order to study the effect of the monodromies
$\Sigma_{1}^{*},(\delta^{2})^{*}\in\operatorname{Aut}(\mathbb{C}[\mathcal{M}(\Lambda(\beta),\tau)])$.
We shall henceforth denote the framed moduli space by
$\mathcal{M}(\Lambda(\beta))$, where the trivialization $\tau$ is implicitly
chosen to be as above.

######  Remark 4.1.

Consider three vectors spaces $U,W,V$ of dimensions $\dim(U)=1$, $\dim(W)=2$
and $\dim(V)=3$. A framed constructible sheaf
$\mathscr{F}\in\mathcal{M}(\Lambda(\beta),\tau)$ has stalks isomorphic to $U$,
$W$ and $V$ as depicted in Figure 2. The $9$-tuple of vectors described above
can also be obtained by parallel transport of the stalk of
$\mathscr{F}\in\mathcal{M}(\Lambda(\beta))$ in the $U$-region to the
$V$-region along the dashed paths depicted in Figure 2. Note that it does not
matter whether a dashed arrow passes a crossing from its left or its right.
$\Box$

Let us now analyze the action of the Legendrian loops $\Sigma_{1}$ and
$\delta^{2}$ on the coordinate ring of $\mathcal{M}(\Lambda(\beta))$ by
studying their action on the $9$-tuples of vectors
$(v_{1},v_{2},\ldots,v_{9})$. For that, we must identify the explicit effect
of each of the Legendrian isotopies constituting $\Sigma_{1}$ and
$\delta^{2}$. These consist of cyclic shifts and Reidemeister III moves.

The effect of the Legendrian isotopy $\delta^{2}$ described in Subsection
2.2.(ii) above is precisely the cyclic shift on the $9$-tuple of vectors:

$\delta^{2}(v_{1},v_{2},\ldots,v_{9})=(v_{9},v_{2},\ldots,v_{1}).$

The effect of Reidemeister III moves is more interesting. Indeed, the
Reidemeister R3d introduces a $U$-region and thus contributes to a vector
$u_{1}$, whereas the Reidemeister R3a, conversely, reduces the number of
$U$-regions by exactly one, thus making a vector disappear. Figure 3 depicts
the case where the $3$-tuple $(v_{1},v_{2},v_{3})$, in the region given by the
braid $\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}$ becomes the $4$-tuple
$(v_{1},v_{2},u_{1},v_{3})$ for the braid
$\sigma_{1}\sigma_{1}\sigma_{2}\sigma_{1}$.

Figure 3. The effect of a Reidemeister R3d move on
$\mathcal{M}(\Lambda(\beta))$.

In terms of the $5$-tuple of flags $(V^{\bullet}_{1},\ldots,V^{\bullet}_{5})$,
associated to the braid $\sigma_{1}\sigma_{2}\sigma_{1}\sigma_{2}$, as
described in Section 3, the vectors are $v_{1}=V^{(1)}_{1}$,
$v_{2}=V^{(1)}_{2}=V^{(1)}_{3}$ and $v_{3}=V^{(1)}_{4}=V^{(1)}_{5}$.
Performing the descending Reidemeister III in Figure 3 yields the new vector
$u_{1}$, whose direction is uniquely defined by $V^{(2)}_{1}\cap V^{(2)}_{5}$
and the normalization is given by the framing. The following proposition
describes the algebraic effect of $\Sigma_{1}$:

######  Proposition 4.2.

The Legendrian loop $\Sigma_{1}$ induces the morphism

$(v_{1},v_{2},v_{3};v_{4},v_{5},v_{6};v_{7},v_{8},v_{9})\longmapsto(v_{2},u_{1},v_{3};v_{5},u_{2},v_{6};v_{8},u_{3},v_{9}),$

where $u_{1},u_{2}$ and $u_{3}$ are given by the intersections

$u_{1}\in\langle v_{1},v_{2}\rangle\cap\langle v_{3},v_{4}\rangle,\quad
u_{2}\in\langle v_{4},v_{5}\rangle\cap\langle v_{6},v_{7}\rangle,\quad
u_{3}\in\langle v_{7},v_{8}\rangle\cap\langle v_{9},v_{1}\rangle,$

and the three normalizing conditions $v_{1}v_{2}=v_{2}u_{1}$,
$v_{4}v_{5}=v_{5}u_{2}$ and $v_{7}v_{8}=v_{8}u_{3}$ in $V\wedge V$. $\Box$

Figure 4 depicts instances of the conclusion of Proposition 4.2 in terms of G.
Kuperberg’s $\operatorname{SL}_{3}$-web combinatorics [41, Section 4]. The
reader is also referred to [28, 40] for the basics of planar tensor diagrams
and $\operatorname{SL}_{3}$-webs, which we shall use in Subsection 4.2. In
particular, Figure 4 displays the pull-backs $(\Sigma_{1})^{*}P_{147}$,
$(\Sigma_{1})^{*}P_{258}$ and $(\Sigma_{1})^{*}P_{369}$ of three Plücker
coordinates $P_{ijk}\in\mathbb{C}[\operatorname{Gr}(3,9)]$, $1\leq i<j<k\leq
9$, where $P_{ijk}=v_{i}\wedge v_{j}\wedge v_{k}$. In particular, Proposition
4.2 implies $(\Sigma_{1})^{*}P_{147}=P_{258}$ and
$(\Sigma_{1})^{*}P_{369}=P_{369}$.

Figure 4. The webs associated to $(\Sigma_{1})^{*}P_{147}$, on the left,
$(\Sigma_{1})^{*}P_{258}$, in the center, and $(\Sigma_{1})^{*}P_{369}$, on
the right, according to Proposition 4.2. In the web for
$(\Sigma_{1})^{*}P_{258}$ we have depicted $u_{1}$ in green, $u_{2}$ in red
and $u_{3}$ in blue.

We shall provide the proof of Proposition 4.2 momentarily. Let us however
first conclude the proof of Theorem 1.1 assuming Proposition 4.2.

###  4.2. The Faithful $\operatorname{PSL}(2,\mathbb{Z})$-Action

For that, we study the monodromy action of the subgroup
$\Gamma=\langle[\Sigma_{1}],[\delta^{2}]\rangle\subseteq\pi_{1}(\mathcal{L}(3,6))$
generated by the homotopy classes of the two Legendrian loops
$\Sigma_{1},\delta^{2}$ into the set of $9$-tuples of vectors in
$\mathbb{C}^{3}$. In order to show that this action is indeed non-trivial we
choose a function $\Delta\in\mathbb{C}[\operatorname{Gr}(3,9)]$ and ensure
that the pull-backs of this function are distinct. For our braid
$\beta=(\sigma_{1}\sigma_{2})^{9}$, let us choose the Plücker coordinate
$\Delta=P_{147}$ in $\mathbb{C}[\operatorname{Gr}(3,9)]$, given by
$P_{147}(v_{1},\ldots,v_{9})=v_{1}\wedge v_{4}\wedge v_{7}$. The algebraic
claim that needs to be proven is that the monodromy of $\Sigma_{1},\delta^{2}$
induces a faithful $\operatorname{PSL}(2,\mathbb{Z})$-action on the orbit
$\mathcal{O}(P_{147})$.

First, let $A=\delta^{2}$ and $B=\Sigma_{1}\circ\delta^{2}$. We have that
$A^{*}(P_{147})=P_{258}$, $(A^{2})^{*}(P_{147})=P_{369}$ and
$(A^{3})^{*}(P_{147})=P_{471}=P_{147}$ and thus $A$ generates a
$\mathbb{Z}_{3}$-action on the orbit $\mathcal{O}(P_{147})$. In general, the
action of $A$ and $B$ cannot be exclusively written in terms of Plücker
coordinates, and in order to study our monodromy action we shall be using G.
Kuperberg’s web $\operatorname{SL}_{3}$-invariants [41, Section 4]. In terms
of $\operatorname{SL}_{3}$-webs, the diagrams associated to the Plücker
coordinates $P_{147}$ and $A^{*}(P_{147})=P_{258}$ are depicted in Figure 5.

Figure 5. The webs associated to the Plücker coordinates $P_{147}$, on the
left, and $P_{258}$, on the right. In general, $\delta^{2}$ acts by clockwise
rotation on webs.

The monodromy of the Legendrian loop $B$ generates a $\mathbb{Z}_{2}$-action
on the orbit $\mathcal{O}(P_{147})$. Indeed, the square
$B^{2}=\Sigma_{1}\circ\delta^{2}\circ\Sigma_{1}\circ\delta^{2}$ pull-backs
$P_{147}$ as follows

$\displaystyle(B^{*})^{2}(P_{147})$
$\displaystyle=(\delta^{2})^{*}\circ(\Sigma_{1})^{*}\circ(\delta^{2})^{*}\circ\Sigma_{1}^{*}(P_{147})=(\delta^{2})^{*}\circ(\Sigma_{1})^{*}\circ(\delta^{2})^{*}(P_{258})=$
$\displaystyle=(\delta^{2})^{*}\circ(\Sigma_{1})^{*}(P_{369})=(\delta^{2})^{*}(P_{369})=P_{147},$

where we have used $(\Sigma_{1})^{*}(P_{369})=P_{369}$, as implied by
Proposition 4.2. Thus, $A^{*}$ generates a $\mathbb{Z}_{3}$-action and $B^{*}$
generates a $\mathbb{Z}_{2}$-action. Since the modular group
$\operatorname{PSL}(2,\mathbb{Z})\cong\mathbb{Z}_{3}*\mathbb{Z}_{2}$ is a free
product, it suffices to show that $A$ and $B$ generate a faithful action with
no relations in the subgroup $\langle[A],[B]\rangle$. Following [30, Section
10], we will prove this by using the Ping-Pong Lemma [15, Section II.B]:

######  Lemma 4.3 ([15, 45]).

Let $\Gamma$ be a group acting on a set $X$, let $\Gamma_{1},\Gamma_{2}$ be
two subgroups of $\Gamma$, and let $G$ be the subgroup of $\Gamma$ generated
by $\Gamma_{1}$ and $\Gamma_{2}$. Suppose that $|\Gamma_{1}|\geq 3$ and
$|\Gamma_{2}|\geq 2$.

Assume that there exist two non-empty subsets $X_{1},X_{2}$ in $X$, with
$X_{2}$ not included in $X_{1}$, such that

$\gamma(X_{2})\subseteq X_{1},\quad\forall\gamma\in\Gamma_{1},\gamma\neq 1$
$\gamma(X_{1})\subseteq X_{2},\quad\forall\gamma\in\Gamma_{2},\gamma\neq 1.$

Then $G$ is isomorphic to the free product $\Gamma_{1}*\Gamma_{2}$. $\Box$

We apply Lemma 4.3 for $\Gamma=\pi_{1}(\mathcal{L}(3,6))$,
$G=\langle[A],[B]\rangle$, $\Gamma_{1}=\langle[A]\rangle$ and
$\Gamma_{2}=\langle[B]\rangle$, which indeed satisfy $|\Gamma_{1}|\geq 3$ and
$|\Gamma_{2}|\geq 2$. The action of $G$ in $X$ is given by the induced
monodromy, as described in Section 3. Consider $X=\mathcal{O}(P_{147})$ to be
the orbit of $P_{147}$. Let us now define the Ping-Pong sets $X_{1}$ and
$X_{2}$. This shall be done in terms of their web diagrams, as follows.

######  Definition 4.4.

The set $X_{1}\subseteq\mathcal{O}(P_{147})$ is the set of all (non-elliptic)
webs in $\mathbb{C}[\operatorname{Gr}(3,9)]$ which do not contain any of the
pieces in Figure 6. That is, a web is in $X\setminus X_{1}$ if it contains at
least one of the pieces in Figure 6.

Figure 6. The webs in the set $X_{1}$ do not contain any of these six pieces.

Similarly, the set $X_{2}\subseteq\mathcal{O}(P_{147})$ is the set of all
(non-elliptic) webs in $\mathbb{C}[\operatorname{Gr}(3,9)]$ which do not
contain any of the pieces in Figure 7.

Figure 7. The webs in the set $X_{2}$ do not contain any of these six pieces.

It suffices to prove that $X_{1},X_{2}$ in Definition 4.4 are Ping-Pong sets
for the monodromy action. It is useful to remind ourselves that the pull-back
$A^{*}=(\delta^{2})^{*}$ acts by clockwise rotation by $2\pi/9$-radians on the
web diagram.

First, let us prove the inclusion $A(X_{2})\subseteq X_{1}$. Suppose that we
have a web $W_{2}\in X_{2}$, we need to argue that $A(W_{2})$ contains none of
the six patterns displayed in Figure 6. Suppose $A(W_{2})$ contained any of
these six, then the clockwise rotation by $4\pi/9$-radians of the diagram
$A(W_{2})$ will contain a spike at one of the boundary vertices $2$,$5$ or
$8$, and thus not be in $X_{2}$. This rotation by $4\pi/9$-radians of the
diagram $A(W_{2})$ represents $A^{2}(A(W_{2}))=W_{2}$ since $A^{3}=\mbox{id}$
in $\mathcal{O}(P_{147})$, which contradicts $W_{2}\in X_{2}$. This shows
$A(X_{2})\subseteq X_{1}$.

Second, let us prove the inclusion $B(X_{1})\subseteq X_{2}$. Consider a web
$W_{1}\in X_{1}$, we need to argue that $B(W_{1})$ contains none of the six
patterns displayed in Figure 7. Suppose $B(W_{1})$ contained any of the three
patterns displayed in the first row of Figure 7, i.e. a spike at either one of
the boundary vertices $2$,$5$ or $8$. By Proposition 4.2, the web
$\Sigma_{1}^{*}(B(W_{1}))$ contains one of the three patterns in the first row
Figure 6 rotated counter-clockwise by an angle of $2\pi/9$-radians. In
consequence, the $2\pi/9$-clockwise rotation

$(\delta^{2})^{*}\circ\Sigma_{1}^{*}(B(W_{1}))=B(B(W_{1}))=W_{1}\not\in
X_{1},$

of $\Sigma_{1}^{*}(B(W_{1}))$ does not belong to $X_{1}$, which contradicts
$W_{1}\in X_{1}$. Thus $B(W_{1})$ does not contain any of the three patterns
displayed in the first row of Figure 7.

Now, suppose that $B(W_{1})$ contained any of the three patterns displayed in
the second row of Figure 7. Proposition 4.2 implies that the web
$\Sigma_{1}^{*}(B(W_{1}))$ contains a counter-clockwise rotated copy, by an
angle of $2\pi/9$-radians, of one of the three patterns in the second row of
Figure 6. Thus, the $2\pi/9$-clockwise rotation
$(\delta^{2})^{*}\circ\Sigma_{1}^{*}(B(W_{1}))$ does not belong to $X_{2}$.
This is a contradiction with

$(\delta^{2})^{*}\circ\Sigma_{1}^{*}(B(W_{1}))=B(B(W_{1}))=W_{1}\not\in
X_{1}.$

Hence $B(W_{1})$ cannot contain any of the three patterns displayed in the
second row of Figure 7. This shows $B(X_{1})\subseteq X_{2}$, as desired. In
conclusion, $X_{1}$ and $X_{2}$ are Ping-Pong sets and Lemma 4.3 implies that
$G=\langle[A],[B]\rangle$ is isomorphic to $\operatorname{PSL}(2,\mathbb{Z})$
and thus the restriction of the monodromy action to this subgroup is a
faithful $\operatorname{PSL}(2,\mathbb{Z})$-representation along the orbit
$X=\mathcal{O}(P_{147})$. This concludes the proof of Theorem 1.1 once
Proposition 4.2 has been proven. $\Box$

###  4.3. Proof of Proposition 4.2

Let us consider the braid word $\beta=(\sigma_{1}\sigma_{2})^{9}$ and consider
the braid word given by the piece $\beta_{0}=(\sigma_{1}\sigma_{2})^{3}$, such
that $\beta=\beta_{0}^{3}$ is a concatenation of $\beta_{0}$ three times. We
refer to the piece $\beta_{0}$ as a window for the braid $\beta$, such that
$\beta$ consists of three windows. The Legendrian loop $\Sigma_{1}$ consists
of a cyclic permutation and a sequence of braid equivalences given by the
Reidemeister III moves. The braid equivalence can be performed equivariantly
over each of the three windows, and hence the morphism induced from
$\Sigma_{1}$ is periodic with respect to this prescribed window decomposition
once the shift is applied. It thus suffices to work with one window to
describe the morphism. Figure 8 depicts the window before a cylic shift,
bounded by the vertical grey boundaries, and after a cyclic shift, which is
bounded by the vertical blue boundaries.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{2}v_{3}$$v_{3}v_{4}$$v_{4}v_{5}$$V=v_{1}v_{2}v_{3}=v_{2}v_{3}v_{4}=v_{3}v_{4}v_{5}$$v_{3}v_{5}$$u_{1}$$v_{2}u_{1}$$u_{1}v_{3}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{3}v_{4}$$v_{4}v_{5}$$V$$u_{1}$$v_{2}u_{1}$$u_{1}v_{3}$$v_{1}$$v_{2}$$v_{3}$$v_{5}$$v_{1}v_{2}$$v_{3}v_{4}$$v_{4}v_{5}$$V$
Figure 8. First window $\beta_{0}$ of the Legendrian braid
$\beta=(\sigma_{1}\sigma_{2})^{9}$ associated to the Legendrian link
$\Lambda(3,6)$. There are total of three windows.

Consider the union of the first window with its cyclic shift, as depicted in
Figure 8. A framed sheaf restricted to this union is determined by vectors
$\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$, which are placed at the regions bounded
by the first and second strands. In the diagrams in Figure 8 the (stalk of)
the sheaf is specified in each open region given by the stratification of the
front diagram, by associating the vector space spanned by the vectors written
in the region. The volume form in each region is given by the ordered wedge
product of vectors in that region.

Now we focus on the grey window. Its boundary underlines the two complete
flags

$0\subset\langle v_{1}\rangle\subset\langle v_{1},v_{2}\rangle\subset V,\quad
0\subset\langle v_{4}\rangle\subset\langle v_{4},v_{5}\rangle\subset V.$

Each flag is shared by two nearby windows. To reduce this replication, one can
break the symmetry by choosing one flag for each window. Without loss of
generality, we choose the flag on the left boundary of each window. In
particular, the sheaf restricted to the grey window is reduced to the data of
three vectors $\\{v_{1},v_{2},v_{3}\\}$ in $V$.

Note that even though the subspaces $\langle v_{4}\rangle$ and $\langle
v_{4},v_{5}\rangle$ cannot be computed from $\\{v_{1},v_{2},v_{3}\\}$, they
are uniquely determined by the next window, and the sheaf is still well-
defined over the grey window. We now perform the descending Reidemeister move
III depicted in the middle of Figure 8. This R3d move creates a region and a
new vector $u_{1}$, as we described in the discussion preceding Proposition
4.2. From the front, the microlocal support condition for our constructible
sheaf implies that

$\langle v_{1},v_{2}\rangle=\langle v_{2},u_{1}\rangle,\qquad\langle
v_{3},v_{4}\rangle=\langle u_{1},v_{2}\rangle.$

Hence $u_{1}$ lies in both $\langle v_{1},v_{2}\rangle$ and $\langle
v_{1},v_{2}\rangle$. Moreover, the crossing condition at the crossing depicted
in red in Figure 8 yields that the complex

$0\longrightarrow\langle u_{1}\rangle\longrightarrow\langle
v_{1},v_{2}\rangle\oplus\langle v_{3},v_{4}\rangle\longrightarrow V\rightarrow
0$

is a short exact sequence of $\mathbb{C}$-vector spaces. Therefore

$\langle u_{1}\rangle=\langle v_{1},v_{2}\rangle\cap\langle
v_{3},v_{4}\rangle,$

and $u_{1}$ is the unique vector such that

$v_{1}v_{2}=v_{2}u_{1}.$

This establishes the description of $u_{1}$ in the statement of Proposition 8.
Let us now shift to the blue window. The constructible sheaf restricted to
this window is determined by $\\{v_{2},u_{2},v_{3},v_{4}\\}$. The fourth
vector $v_{4}$ disappears upon performing the ascending Reidemeister III move,
as depicted in the bottom of Figure 8. After this R3a move, the sheaf is
uniquely determined by $\\{v_{2},u_{2},v_{3}\\}$. The morphism induced by
$\Sigma_{1}$ thus starts with

$(v_{1},v_{2},v_{3})\mapsto(v_{2},u_{1},v_{3}),$

where $u_{1}\in\langle v_{1},v_{2}\rangle\cap\langle v_{3},v_{4}\rangle$ and
$v_{1}v_{2}=v_{2}u_{1}$, and continues to remove $v_{4}$. These two moves are
preceded by the cyclic shift, and their composition yields the expression in
the first, and thus any, window in the statement of Proposition 4.2, as
required. $\Box$

###  4.4. Comments on the Proof

This concludes the proof of Theorem 1.1. Before proceeding with Theorem 1.3,
the following comments might be clarifying. The geometric loops
$\Sigma_{1},\delta^{2}$ are studied in the above proof of Theorem 1.1 by
analyzing their action on the ring of functions
$\mathbb{C}[\mathcal{M}(\Lambda(\beta),\tau)]$ of the framed moduli space
$\mathcal{M}(\Lambda(\beta),\tau)$. It should be equally possible to deduce
Theorem 1.1 by studying their monodromy invariants in the ring of regular
functions $\mathbb{C}[\mathcal{M}(\Lambda(\beta))]$ of the moduli spaces of
sheaves, with no frame $\tau$ chosen, with corresponding
$(\mathbb{C}^{*})^{9}$-equivariant condition added. Indeed, the positroid
embedding of $\mathcal{M}(\Lambda(\beta),\tau)$ inside the Grassmannian
$\operatorname{Gr}(3,9)$ yields an embedding of the moduli of sheaves
$\mathcal{M}(\Lambda(\beta))$ into the quotient
$\operatorname{Gr}(3,9)/(\mathbb{C}^{*})^{9}$ of the Grassmannian
$\operatorname{Gr}(3,9)$ by the diagonal subgroup of
$\operatorname{GL}_{9}(\mathbb{C})$ acting on the right, i.e. by column
$\mathbb{C}^{*}$-rescaling.

It is our aesthetic opinion that working directly in the unquotiented
Grassmannian $\operatorname{Gr}(3,9)$ yields a clearer understanding of the
geometry, thus our choice of using the moduli space of framed sheaves. In
terms of cluster algebras, the quotient
$\operatorname{Gr}(3,9)/(\mathbb{C}^{*})^{9}$ has no frozen cluster variables,
whereas the Grassmannian $\operatorname{Gr}(3,9)$ [29, 53] has the cyclically
consecutive Plücker coordinates as frozen cluster variables.

######  Remark 4.5.

The articles [30, 56] respectively use the affine cone on the projective
Grassmannian $\operatorname{Gr}(3,9)$ [30, Section 3] and the decorated
Grassmannian $\mathscr{G}r(3,9)$ [56, Section 2.1]. These can be equivalently
considered [56, Lemma 2.6] and correspond to matrices $\mbox{Mat}_{3,9}$ up to
the left action of $\operatorname{SL}_{3}(\mathbb{C})$, rather than
$\operatorname{GL}_{3}(\mathbb{C})$, which would yield the projective
Grassmannian $\operatorname{Gr}(3,9)$. In terms of the moduli space of framed
sheaves $\mathcal{M}(\Lambda(\beta),\tau)$ used in our proof of Theorem 1.1,
we should require the additional data of a trivialization of the microlocal
monodromy along $\Lambda(\beta)$ itself [59, Section 5.1]. By context, it
seems appropriate to refer to this space as the moduli space of decorated
sheaves. The line of argument above should also work by using the decorated
positroid embedding of the space of decorated sheaves into the decorated
Grassmannian. $\Box$

Let us now move forward with Theorem 1.3. Note that Theorem 1.1 on its own
allows us to conclude Corollaries 1.5 and 1.10 in the cases
$(n,m)\in\mathcal{H}\setminus\\{(4,4),(4,5)\\}$, and Corollaries 1.7 and 1.8
for $\Lambda(3,6)$. In order to cover the Legendrian links $\Lambda(4,4)$ and
$\Lambda(4,5)$, and for completeness, we now include the proof of Theorem 1.3,
which is in line with that of Theorem 1.1 above.

##  5\. The $M_{0,4}$ representation for $\Lambda(4,4)$

In this section we prove Theorem 1.3. The argument reproduces the strategy for
Theorem 1.1 above. In this case, the braid is
$\beta=(\sigma_{1}\sigma_{2}\sigma_{3})^{8}$ and the moduli space
$\mathcal{M}(\Lambda(\beta))$ is identified with a positroid cell
$\Pi_{r(\beta)}\subseteq\operatorname{Gr}(4,8)$ by the same procedure. The
action of the Legendrian loops $\Xi_{1},\Xi_{2},\Xi_{3}$ is described by the
following three crucial Propositions:

######  Proposition 5.1.

The Legendrian loop $\Xi_{1}$ induces the morphism

$(v_{1},v_{2},v_{3},v_{4};v_{5},v_{6},v_{7},v_{8})\longmapsto(v_{2},u_{1},v_{3},v_{4};v_{6},u_{2},v_{7},v_{8}),$

where $u_{1},u_{2}$ are given by the intersections

$u_{1}\in\langle v_{1},v_{2}\rangle\cap\langle v_{3},v_{4},v_{5}\rangle,\quad
u_{2}\in\langle v_{5},v_{6}\rangle\cap\langle v_{7},v_{8},v_{1}\rangle,$

and the normalizing conditions $v_{1}v_{2}=v_{2}u_{1}$ and
$v_{5}v_{6}=v_{6}u_{2}$ in $V\wedge V$.

######  Proposition 5.2.

The Legendrian loop $\Xi_{2}$ induces the morphism

$(v_{1},v_{2},v_{3},v_{4};v_{5},v_{6},v_{7},v_{8})\longmapsto(v_{1},v_{3},u_{1},v_{4};v_{5},v_{7},u_{2},v_{8}),$

where $u_{1},u_{2}$ are given by the intersections

$u_{1}\in\langle v_{2},v_{3}\rangle\cap\langle v_{4},v_{5},v_{6}\rangle,\quad
u_{2}\in\langle v_{6},v_{7}\rangle\cap\langle v_{8},v_{1},v_{2}\rangle,$

and the normalizing conditions $v_{2}v_{3}=v_{3}u_{1}$ and
$v_{6}v_{7}=v_{7}u_{2}$ in $V\wedge V$.

######  Proposition 5.3.

The Legendrian loop $\Xi_{3}$ induces a morphism

$(v_{1},v_{2},v_{3},v_{4};v_{5},v_{6},v_{7},v_{8})\longmapsto(v_{1},v_{2},v_{4},u_{1};v_{5},v_{6},v_{8},u_{2}),$

where $u_{1},u_{2}$ are given by the intersections

$u_{1}\in\langle v_{3},v_{4}\rangle\cap\langle v_{5},v_{6},v_{7}\rangle,\quad
u_{2}\in\langle v_{7},v_{8}\rangle\cap\langle v_{1},v_{2},v_{3}\rangle,$

and the normalizing conditions $v_{3}v_{4}=v_{4}u_{1}$ and
$v_{7}v_{8}=v_{8}u_{2}$ in $V\wedge V$.

Propositions 5.1, 5.2 and 5.3 are proven at the end of this section. The
action of the group $\Gamma_{2}=\langle\Xi_{1},\Xi_{2},\Xi_{3}\rangle$ in the
set of 8-tuples of vectors, representing a point in $\operatorname{Gr}(4,8)$,
yields via pull-back an action on a subset of the homogeneous coordinate ring
$\mathbb{C}[\operatorname{Gr}(4,8)]$. For the braid $\Lambda(4,4)$ it does not
suffice to study the $\Gamma_{2}$-orbit of a Plücker coordinate, as we
directly did for Theorem 1.1, but rather a set of Plücker coordinates. In this
proof for Theorem 1.3, we directly refer to known algebraic arguments whose
nature is on par with Subsection 4.2, as follows. Indeed, [30, Lemma 10.8]
proves that the group $\langle\Xi_{1},\Xi_{2},\Xi_{3}\rangle$ generated by the
monodromies of the three Legendrian loops generates a faithful action of
$\operatorname{PSL}(2,\mathbb{Z})\cong\mathbb{Z}_{2}*\mathbb{Z}_{3}$ on the
(cluster) automorphism group of the coordinate ring
$\mathbb{C}[\operatorname{Gr}(4,8)]$. This is achieved by studying the orbit
of the Plücker set:

$\mathscr{P}=\\{P_{1378},P_{2348},P_{2367},P_{4678},P_{3457},P_{2347},P_{2378},P_{3678},P_{3467}\\},$

which is a cluster seed for a triangulation of the annulus with four boundary
marked points. By [26, Proposition 2.7], the mapping class group of the four-
punctured sphere $M_{0,4}$ is isomorphic to the semidirect product
$\operatorname{PSL}(2,\mathbb{Z})\ltimes(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$.
The article [30, Theorem 9.14] also shows that this faithful action of
$\operatorname{PSL}(2,\mathbb{Z})$ extends to the mapping class group
$M_{0,4}$ as required. This is achieved explicitly by studying the four cosets
of $\operatorname{PSL}(2,\mathbb{Z})$ into $M_{0,4}$. In the algebraic
argument the set $F$ can be chosen to be the union of four sets, as follows.
The first set $S$ is the union of a finite number of cluster charts [30,
Section 10.2] containing the set $\mathscr{P}$ of Plücker coordinates above,
and the remaining three sets are the coset translates $\Xi_{3}(S)$,
$\Xi_{3}\Xi_{2}(S)$, and $\Xi_{3}\Xi_{2}\Xi_{1}(S)$. Here
$\Xi_{3},\Xi_{3}\Xi_{2}$ and $\Xi_{3}\Xi_{2}\Xi_{1}$ are each a right coset
representative for each of the three non-trivial cosets of the inclusion of
$\operatorname{PSL}(2,\mathbb{Z})$ into $M_{0,4}$ above. $\Box$

The crucial ingredient for the proof of Theorem 1.3 above is the statement
that the Legendrian loops we constructed in Section 2 indeed induce an action
of the (spherical) braid group $B_{4}$. This is precisely the content of
Propositions 5.1, 5.2 and 5.3, which describe the algebraic effect of the
Legendrian loops $\Xi_{1},\Xi_{2}$ and $\Xi_{3}$. Let us now prove these three
propositions.

###  5.1. Proof of Proposition 5.1

Let us consider the braid words $\beta=(\sigma_{1}\sigma_{2}\sigma_{3})^{8}$
and $\beta_{0}=(\sigma_{1}\sigma_{2}\sigma_{3})^{4}$. Following the notation
in the proof of Proposition 4.2 above, each $\beta_{0}$ is a window and
$\beta=\beta_{0}^{2}$ is the concatenation of two windows. Similar to the
proof of Proposition 4.2, it suffices to compute the induced morphism in a
window.

Consider the union of the first window and its one-term cyclic shift, depicted
in the top diagram of Figure 9. The window before the shift has grey
boundaries. We choose to include the sheaf data on the left boundary in this
window, and leave the sheaf data on the right boundary to the next window.
With this choice, a framed constructible sheaf in the grey window is
determined by four vectors $\\{v_{1},v_{2},v_{3},v_{4}\\}$ in $V$.

Now we study the morphism induced by the Legendrian loop $\Xi_{1}$. The
sequence of braid moves can be carried out as the concatenation of two
Legendrian isotopies $\Psi_{1}^{(2)}\circ\Psi_{1}^{(1)}$. These two Legendrian
isotopies $\Psi_{t}^{(1)}$ and $\Psi_{t}^{(2)}$, $t\in[0,1]$, are defined in
Section 2. In Figure 9, $\Psi^{(1)}_{t}$ corresponds to the Legendrian isotopy
from the top diagram to the middle diagram, and $\Psi^{(2)}_{t}$ is depicted
from the middle diagram to the bottom diagram.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{2}v_{3}$$v_{3}v_{4}$$v_{4}v_{5}$$v_{6}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{3}v_{4}v_{5}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$v_{2}u_{1}v_{3}$$u_{1}$$u_{1}v_{3}$$v_{2}u_{1}$$u_{1}v_{3}v_{4}$$v_{3}v_{4}v_{6}$

$\Psi_{t}^{(1)}$

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{3}v_{4}$$v_{4}v_{5}$$v_{6}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{3}v_{4}v_{5}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$v_{2}u_{1}v_{3}$$u_{1}$$u_{1}v_{3}$$v_{2}u_{1}$$u_{1}v_{3}v_{4}$

$\Psi_{t}^{(2)}$

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{1}v_{2}$$v_{3}v_{4}$$v_{6}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{3}v_{4}v_{5}$$v_{4}v_{6}$$v_{5}v_{6}v_{7}$$V$$v_{4}v_{5}v_{6}$
Figure 9. First window $\beta_{0}$ of the Legendrian braid
$\beta=(\sigma_{1}\sigma_{2}\sigma_{3})^{8}$ associated to the Legendrian loop
$\Xi_{1}$ for the Legendrian link $\Lambda(4,4)$. There are a total of two
windows.

After performing the Legendrian isotopy $\Psi_{t}^{(1)}$, $t\in[0,1]$, the
diagram introduces a new vector $u_{1}$. From the diagram, we see that
$\langle v_{1},v_{2}\rangle=\langle v_{2},u_{1}\rangle$ and $\langle
u_{1},v_{3},v_{4}\rangle=\langle v_{3},v_{4},v_{5}\rangle$. Hence
$u_{1}\in\langle v_{1},v_{2}\rangle\cap\langle v_{3},v_{4},v_{5}\rangle$. To
argue that the intersection is a one-dimensional subspace, we should discard
the case that $\langle v_{1},v_{2}\rangle\subset\langle
v_{3},v_{4},v_{5}\rangle$. Inside the middle figure, the condition at the red
crossing yields a short exact sequence of complex vector spaces:

$0\rightarrow\langle u_{1},v_{3}\rangle\rightarrow\langle
v_{1},v_{2},v_{3}\rangle\oplus\langle v_{3},v_{4},v_{5}\rangle\rightarrow
V\rightarrow 0.$

If $\langle v_{1},v_{2}\rangle$ is contained in $\langle
v_{3},v_{4},v_{5}\rangle$, so is $\langle v_{1},v_{2},v_{3}\rangle$. Then it
is impossible to map the direct sum onto $V$, which is a contradiction.
Therefore

$\langle u_{1}\rangle=\langle v_{1},v_{2}\rangle\cap\langle
v_{3},v_{4},v_{5}\rangle,$

and the vector $u_{1}$ can be uniquely determined by

$v_{1}v_{2}=v_{2}u_{1}.$

At this stage, there are five vectors $\\{v_{2},u_{1},v_{3},v_{4},v_{5}\\}$
inside the (blue) shifted window. There is a redundancy which is removed via
the Legendrian isotopy $\Psi^{(2)}_{t}$. The bottom diagram in Figure 9
specifies how to determine the constructible sheaf using the four vectors
$\\{v_{2},u_{1},v_{3},v_{4}\\}$. In the end, the only regions including
$v_{5}$ are connected to the right blue boundary, which is determined by the
next window. Iterating this procedure in each window, we obtain that the
morphism determined by $\Xi_{1}$ is indeed that of the statement of
Proposition 5.1. $\Box$

###  5.2. Proof of Proposition 5.2

Let us consider the Legendrian isotopy $\Psi^{(1)}_{t}$, as defined in Section
2. This is the first of two pieces which constitute the Legendrian loop
$\Xi_{2}$. This Legendrian isotopy $\Psi^{(1)}_{t}$ is depicted from the top
to the middle in Figure 10. In this Legendrian isotopy, a $2$-dimensional
vector space $W$ appears in the front diagram. Let us show that $W=\langle
v_{2},v_{3}\rangle$.

$u_{1}$$v_{3}u_{1}$$u_{1}v_{4}$$u_{1}v_{4}v_{5}$$\Psi_{t}^{(1)}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{2}v_{3}$$v_{3}v_{4}$$v_{4}v_{5}$$v_{6}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{3}v_{4}v_{5}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$v_{1}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$W$$v_{3}v_{4}$$v_{4}v_{5}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$\Psi_{t}^{(2)}$$v_{1}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{2}v_{3}$$v_{4}v_{5}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$v_{1}v_{3}$$v_{1}v_{3}$
Figure 10. Window for the Legendrian loop $\Xi_{2}$ on $\Lambda(4,4)$ with the
needed information for the proof of Proposition 5.2.

Indeed, consider the crossing labeled in red. The crossing condition for a
microlocal sheaf supported in the Legendrian braid yields a short exact
sequence of complex vector spaces:

$0\rightarrow\langle v_{3}\rangle\rightarrow\langle v_{1},v_{3}\rangle\oplus
W\rightarrow\langle v_{1},v_{2},v_{3}\rangle\rightarrow 0.$

Therefore $W$ is a subspace of $\langle v_{1},v_{2},v_{3}\rangle$ and it
contains $\langle v_{3}\rangle$. Since $0\subset\langle
v_{1}\rangle\subset\langle v_{1},v_{2}\rangle\subset\langle
v_{1},v_{2},v_{3}\rangle$ is a complete flag, $v_{1},v_{2},v_{3}$ are three
distinct and non-planar vectors. Thus, we can assume the vector space $W$ is
spanned by $v_{3}$ and another vector $v=av_{1}+bv_{2}$ for some constants
$a,b\in\mathbb{C}$.

Now, consider the blue crossing and, by a similar argument, we obtain
$v=cv_{2}+dv_{4}$ for some other constants $c,d\in\mathbb{C}$. Finally,
consider the crossing condition at the purple crossing, there is a short exact
sequnce:

$0\rightarrow W\rightarrow\langle v_{1},v_{2},v_{3}\rangle\oplus\langle
v_{2},v_{3},v_{4}\rangle\rightarrow V\rightarrow 0.$

By the last map being surjective, $v_{4}$ is not contained $\langle
v_{1},v_{2},v_{3}\rangle$. In particular, it is not a linear combination of
$v_{1}$ and $v_{2}$ and there must be $d=0$. By a similar argument $a=0$ as
well. Hence $v$ is a multiple of $v_{2}$ and $W=\langle v_{2},v_{3}\rangle$.
This determines the algebraic effect of the Legendrian isotopy
$\Psi^{(1)}_{t}$.

At this stage, we continue with the second Legendrian isotopy
$\Psi^{(2)}_{t}$. This Legendrian isotopy creates a new region with a vector
$u_{1}$, as depicted in Figure 10. An argument in line with that of the proof
of Proposition 5.1 concludes that $u_{1}\in\langle
v_{2},v_{3}\rangle\cap\langle v_{4},v_{5},v_{6}\rangle$ and it is uniquely
determined by $v_{2}v_{3}=v_{3}u_{1}$, as required. This concludes the desired
transformation for the first window, with the transformations for the
remaining windows concluded similarly. $\Box$

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{2}v_{3}$$v_{3}v_{4}$$v_{4}v_{5}$$v_{6}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{3}v_{4}v_{5}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$\Psi_{t}^{(1)}$$v_{1}$$v_{3}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{1}v_{3}$$v_{3}v_{4}$$v_{4}v_{5}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{3}v_{4}v_{5}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$v_{1}v_{3}v_{4}$$\Psi_{t}^{(2)}$$v_{1}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{1}v_{4}$$v_{3}v_{4}$$v_{4}v_{5}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{3}v_{4}v_{5}$$v_{4}v_{5}v_{6}$$v_{5}v_{6}v_{7}$$V$$v_{1}v_{3}v_{4}$$v_{1}v_{2}v_{4}$$\Psi_{t}^{(3)}$$v_{1}$$v_{4}$$v_{5}$$v_{1}v_{2}$$v_{2}v_{4}$$v_{3}v_{4}$$v_{4}u_{1}$$v_{5}v_{6}$$v_{1}v_{2}v_{3}$$v_{2}v_{3}v_{4}$$v_{3}v_{4}v_{5}$$v_{5}v_{6}v_{7}$$V$$v_{1}v_{2}v_{4}$$v_{2}$$u_{1}$$u_{1}v_{5}$$v_{4}u_{1}v_{5}$$u_{1}v_{5}v_{6}$
Figure 11. Windows for the Legendrian loop $\Xi_{3}$ on $\Lambda(4,4)$
containing the required information for the proof of Proposition 5.3.

###  5.3. Proof of Proposition 5.3

The argument is identical to that in Propositions 5.1 and 5.2, and thus we
only provide the core steps. In particular, we have depicted the Legendrian
loop $\Xi_{3}$ in Figure 11 as well as its effect in three different pieces
$\Xi_{3}=\Psi_{1}^{(3)}\circ\Psi_{1}^{(2)}\circ\Psi_{1}^{(1)}$, as recorded in
Section 2. In short, the core information in studying the effect of $\Xi_{3}$
can be described as follows:

  * -

After $\Psi_{1}^{(t)}$, the subspaces are uniquely determined as indicated in
the figure. The vector space spanned by $v_{2}$ disappears but the vector
$v_{2}$ can be recovered from the new data. Namely, it is determined by the
intersection of $\langle v_{1},v_{2}\rangle$ and $\langle
v_{2},v_{3},v_{4}\rangle$, both of which are stalks of some regions in the
front diagram, and the volume form in either one of these vector spaces.

  * -

After $\Psi_{2}^{(t)}$, the subspaces are also uniquely determined as
indicated. The data of $v_{2}$ remains in the diagram implicitly.

  * -

The Legendrian isotopy $\Psi_{3}^{(t)}$, pulls down two strands in Figure 11
which are colored in red. The red strand on the left recovers the vector
$v_{2}$. The red strand on the right introduces a new vector $u_{1}$, which
satisfies $v_{3}v_{4}=v_{4}u_{1}$, and $u_{1}v_{5}v_{6}=v_{5}v_{6}v_{7}$. By a
similar argument with that for $\Xi_{1}$ and $\Xi_{2}$, we see that
$u_{1}\in\langle v_{3},v_{4}\rangle\cap\langle v_{5},v_{6},v_{7}\rangle$ and
that $v_{3}v_{4}=v_{4}u_{1}$.

In conclusion, the morphism sends the first window from the 4-tuple
$(v_{1},v_{2},v_{3},v_{4})$ to the 4-tuple $(v_{1},v_{2},v_{4},u_{1})$ as
required. The second window is concluded in similar manner. $\Box$

##  6\. Corollaries and Applications

In this section we prove Corollaries 1.5,1.6, 1.7, 1.8 and 1.10.

First, Corollary 1.7 follows by observing that a trivial concordance in the
Lagrangian concordance monoid $\mathbb{L}(3,6)$, and $\mathbb{L}(4,4)$, would
induce a trivial map on $\mathcal{M}(\Lambda(3,6))$, and
$\mathcal{M}(\Lambda(4,4))$ respectively. Theorems 1.1 and 1.3 imply that the
loops $\Sigma_{1},\delta^{2}$, for $\Lambda(3,6)$ and
$\Xi_{1},\Xi_{2},\Xi_{3}$, for $\Lambda(4,4)$, induce Legendrian loops which
act non-trivially on $\mathcal{M}(\Lambda(3,6))$, and
$\mathcal{M}(\Lambda(4,4))$ respectively. Hence the concordances induced by
graphing these Legendrian loops are themselves non-trivial. The same argument
concludes Corollary 1.8.

Let us now address Corollary 1.5 and Corollary 1.10, which shall follow from
Theorems 1.1 and 1.3, with the addition of the upcoming Proposition 6.1. For
that, let us consider the two-sided closure of the braid word
$\beta=(\sigma_{1}\cdot\sigma_{2}\cdot\ldots\cdot\sigma_{n-1})^{m}$ as
depicted in the upper leftmost diagram in Figure 12. Let us denote the
Legendrian associated to this front $\Lambda(\beta)$. Corollary 1.5 is proven
with the following geometric construction:

######  Proposition 6.1.

Let $\Lambda(n,m)=\Lambda(\beta)$ be the Legendrian torus link given by the
braid

$\beta=(\sigma_{1}\cdot\ldots\cdot\sigma_{n-1})^{m}.$

There exists a decomposable Lagrangian cobordism from $\Lambda(n,m)$ to
$\Lambda(n,m+1)$ whose Lagrangian handles have isotropic spheres away from the
region with the $\beta$-braiding.

Figure 12. Exact Lagrangian Cobordism from $\beta$ to
$\beta\sigma_{1}\sigma_{2}$.

###### Proof.

For any given $\sigma_{i}$, $1\leq i\leq n-1$, it suffices to construct a
decomposable Lagrangian cobordism with concave end $\Lambda(\beta)$ and convex
end $\Lambda(\beta\sigma_{i})$. For that, we first perform an upwards
Reidemeister I move on the right lower strand for the $i$th rightmost cusp.
Then, the left cusp created in this Reidemeister I move can be isotoped,
without introducing crossings in the front, to the same level as the rightmost
cusp for the $(i+1)$th strand. This is depicted in the second and fifth
diagrams of Figure 12 in the cases of $\sigma_{1}$ and $\sigma_{2}$. Once
these two cusps are aligned, we perform a reverse pinched-move [8, 47]
allowing this pair of opposite cusps to become two parallel strands. This
corresponds to a Lagrangian $1$-handle attachment $h_{1}$, and it is depicted
in the second to third, and fifth to sixth diagrams in Figure 12. The
decomposable Lagrangian cobordisms just described can be independently and
repeatedly performed for different $\sigma_{i}$, $1\leq i\leq n-1$. In
particular, by applying this cobordism for the Artin generators
$\sigma_{1},\sigma_{2}$ through $\sigma_{n-1}$, we obtain a decomposable
Lagrangian cobordism from $\Lambda(\beta)$ to
$\Lambda(\beta(\sigma_{1}\sigma_{2}\cdot\ldots\cdot\sigma_{n-1}))$, which
implies the statement in the proposition when applied to the braid
$\beta=(\sigma_{1}\cdot\ldots\cdot\sigma_{n-1})^{m}$. ∎

Note that Proposition 6.1 holds for any pair $(n,m)$ with no constraint $n\leq
m$ nor $(n,m)\in\mathcal{H}$.

###  6.1. Proof of Corollary 1.5

Let us first prove that $\Lambda(3,6)$ has infinitely many Lagrangian
fillings. Fix an exact Lagrangian filling $L\subseteq(\mathbb{R}^{4},\omega)$
for $\Lambda(3,6)$ obtained via a pinching sequence from the front diagram on
the left of Figure 1. Smoothly, this must be a third punctured genus-3
surface. By [58, Proposition 2.15], this exact Lagrangian filling yields on
open inclusion $\mbox{Loc}(L)\subseteq\mathcal{M}(\Lambda(3,6))$, where
Loc$(L)$ denotes the space of framed local systems in $L$. Now, given a
Legendrian loop $\vartheta\in\Gamma=\langle[A],[B]\rangle$ in the group
generated by the Legendrian loops $A,B$ (or equivalently
$\Sigma_{1},\delta^{2}$), we consider the Lagrangian filling $L_{\vartheta}$
obtained by applying the Legendrian loop $\vartheta$ to the Legendrian
$\Lambda(3,6)$ and then performing the fixed pinching sequence for the
Lagrangian filling fixed above. Choose an infinite sequence of distinct
elements
$(\vartheta_{i})_{i\in\mathbb{N}}\in\operatorname{PSL}(2,\mathbb{Z})$, since
$\vartheta_{i}$ are distinguished by their action on the infinite cluster
charts of $\operatorname{Gr}(3,9)$, the inclusions
$\mbox{Loc}(L_{\vartheta_{i}})\subseteq\mathcal{M}(\Lambda(3,6))$ yield
infinitely many distinct cluster charts. In consequence, the Lagrangian
fillings $L_{\vartheta_{i}}$ are not Hamiltonian isotopic [58, Proposition
6.1]. The same argument holds for the Legendrian link $\Lambda(4,4)$ once we
use the representation in Theorem 1.3 and the mapping class group $M_{0,4}$.

Let $\Lambda(n,m)$ be given with $(n,m)\in\mathcal{H}$ different from $(4,4)$
and $(4,5)$. The construction in the proof of Proposition 6.1 yields a
decomposable Lagrangian cobordism from $\Lambda(3,6)$ to $\Lambda(n,m)$. This
decomposable cobordism yields an injective map [46, Theorem 1.6] between the
augmentation varieties of $\Lambda(3,6)$ and $\Lambda(n,m)$. Since there are
infinitely many Lagrangian fillings for $\Lambda(3,6)$ distinguished by their
sheaves, the correspondence between augmentations and sheaves [43, Theorem
1.3] implies that these Lagrangian fillings are distinguished by their
augmentations on their Chekanov-Eliashberg algebra [13]. Thus, the infinitely
many Lagrangian fillings of $\Lambda(3,6)$ concatenated with the Lagrangian
cobordism in Proposition 6.1 induce different augmentations for
$\Lambda(n,m)$. In consequence, the infinitely many Lagrangian fillings of
$\Lambda(3,6)$ yield infinitely many Lagrangian fillings of $\Lambda(n,m)$.
For the remaining case of $(n,m)=(4,5)$ we apply Proposition 6.1 to obtain a
cobordism from $\Lambda(4,4)$ to $\Lambda(4,5)$ and proceed identically.
$\Box$

###  6.2. Proof of Corollary 1.6

Let $\Lambda\subseteq(S^{3},\xi_{\text{st}})$ be any Legendrian link with an
exact Lagrangian cobordism $\Lambda(3,6)\preceq\Lambda$, or
$\Lambda(4,4)\preceq\Lambda$. The argument for Corollary 1.5 implies that
$\Lambda$ itself will have infinitely many exact Lagrangian fillings. This
readily implies Corollary 1.6. Indeed, by [9, Theorem 1.1] the twisted torus
knots $K_{p,q,r,s}=T(p,q,r,s)$ with $1<r<p<q$ and $18p<s$ are hyperbolic
knots. Let $\Lambda_{K_{3,7,2,s}}$ be the maximal-tb Legendrian representative
obtained from the positive braid associated to the $T$-knot $T(3,7,2,s)$, with
$54<s$ and $s$ even, as described in [7, Section 1]. Then there exists an
exact Lagrangian cobordism $\Lambda(3,6)\preceq\Lambda_{K_{3,7,2,s}}$, and
hence $\Lambda_{K_{3,7,2,s}}$ is a hyperbolic knot which admits infinitely
many exact Lagrangian fillings. The same argument applies to the twisted torus
links $T(p,q,kq,s)$, $p,q,k,s\in\mathbb{N}$, which are proven to be
$(q,p+k^{2}qs)$-cables of the torus knot $T(k,ks+1)$ in [42]. $\Box$

###  6.3. Proof of Corollary 1.10

Consider an infinite collection of the exact Lagrangian fillings
$\\{L_{i}\\}_{i\in\mathbb{N}}$ constructed in Corollary 1.5, and denote by
$\overline{L}_{i}\subseteq M(n,m)$ the exact Lagrangian surfaces obtained by
capping $L_{i}$ with the unique defining $2$-handle of $M(n,m)$,
$i\in\mathbb{N}$. By the equivalences between sheaves and augmentations [43,
Theorem 1.3], these Lagrangian fillings $\\{L_{i}\\}_{i\in\mathbb{N}}$ are
distinguished by the augmentations they induce in the Chekanov-Eliashberg
differential graded algebra $A_{n,m}$ of $\Lambda(n,m)$. The wrapped Fukaya
categories of the Weinstein manifolds $M(n,m)$ are generated by their
respective unique cocore $C$ of their defining $2$-handle [1, 12], i.e. the
wrapped Fukaya category is identified with the category of dg-modules over
$\mbox{End}(C,C)\cong A_{n,m}$. Hence, the Lagrangian surfaces
$\overline{L}_{i}$, whose wrapped Floer complex $WF(C,\overline{L}_{i})$ has a
unique generator, yield distinct 1-dimensional $A_{n,m}$-modules. Thus
$\\{\overline{L}_{i}\\}_{i\in\mathbb{N}}$ represent distinct objects in the
wrapped Fukaya category and $\\{\overline{L}_{i}\\}_{i\in\mathbb{N}}$ are an
infinite collection of pairwise non-Hamiltonian isotopic exact Lagrangians.
$\Box$

The cluster modular groups of the remaining Grassmannians
$\operatorname{Gr}(n,m+n)$, with the pair
$(n,m)\in(\mathbb{N}\times\mathbb{N})\setminus\mathcal{H}$, are known to be
finite [5, 30]. Thus, for these remaining Legendrian links $\Lambda(n,m)$,
$(n,m)\in(\mathbb{N}\times\mathbb{N})\setminus\mathcal{H}$, our arguments will
only yield a representation of a finite group.

######  Remark 6.2.

In particular, we are almost certain that our results are sharp, i.e. we
conjecture that the Legendrian torus links $\Lambda(n,m)$ have finitely many
Lagrangian fillings if $(n,m)\not\in\mathcal{H}$. In fact, we believe that the
Legendrian torus links $\Lambda(2,n)$ must have exactly
$\frac{1}{n+1}{2n\choose n}$ Lagrangian fillings, $\Lambda(3,3)$ should have
exactly $50$ Lagrangian fillings, and $\Lambda(3,4)$ and $\Lambda(3,5)$ will
have exactly 883 and 25080 Lagrangian fillings respectively. $\Box$

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