associated configuration in $L$. Consider a disk $D\in\mathscr{D}$ with
boundary the lift of $\gamma\in\mathcal{C}(\Gamma)$. Then there is a natural
identification between the configuration $\mu_{D}(\Gamma)$ and the
$\gamma$-exchange of $\Gamma$.

###### Proof.

Lagrangian disk surgery occurs in a neighborhood of $D$ in
$(\mathbb{D}^{4},\lambda_{st})$. It is shown in [CZ22a, Section 4.8] that it
can be local modeled by the weave mutation in Figure 30. For this proof, we
assume familiarity with [CZ22a, Section 2], or [CW, Section 3]. It suffices to
understand how Lagrangian disk surgery along the disk $D$ bounding $\gamma$
affects the boundary of the other disks in $\mathscr{D}$. There are two cases:
positive and negative intersections, represented by the segments $\tau_{+}$
and $\tau_{-}$ in Figure 11 (left).

Figure 30. (Upper left) Lagrangian disk surgery in the 2-weave model along a
short $\sf I$-cycle $\gamma$. (Upper right) The short $\sf I$-cycle $\gamma$
depicted in pink, drawn before and after $\mu_{\gamma}$. (Lower left) A weave
line in yellow smoothly representing a relative homology class $\tau_{+}$ with
unique geometric intersection $+1$ with $\gamma$, drawn before and after
$\mu_{\gamma}$. (Lower right) A weave line in green smoothly representing a
relative homology class $\tau_{-}$ with a unique geometric intersection $-1$
with $\gamma$, also drawn before and after $\mu_{\gamma}$.

In the weave diagram, the segments $\tau_{+}$ and $\tau_{-}$ in Figure 11
(left) can be represented by the weave lines $\tau_{+}$ and $\tau_{-}$ in the
second row of Figure 30.262626The segment $\tau_{-}$ remains green, while
$\tau_{+}$ is now drawn in yellow because the weave is typically drawn in
blue. In this situation, before the mutation, $\tau_{+}\cap\gamma=+1$,
$\tau_{-}\cap\gamma=-1$ and $\tau_{+}\cap\tau_{-}=0$. Note that smooth curves
are encoded by their relative homology class in the context of curves in
surfaces, homology classes are entirely determined by their intersections and
thus we record those. The 2-weave in the left of Figure 30 (upper left)
represents a Lagrangian cylinder, and the left parts of the other three
figures indicate how to draw $\tau_{\pm}$ and $\gamma$ in that cylinder.

It suffices to understand how $\tau_{+},\tau_{-}$ and $\gamma$ in the weave
change under weave mutation. This is drawn in the right parts of the upper
right and the second row of Figure 30. The resulting curves
$\mu_{\gamma}(\tau_{+})$ and $\mu_{\gamma}(\tau_{-})$ are shown, also in
yellow and green respectively; the curve $\mu_{\gamma}(\gamma)$ is drawn in
pink. By using the intersection numbers on weaves, cf. [CZ22a, Section 2] or
[CGG+22, Section 4.4], we obtain that
$\mu_{\gamma}(\tau_{+})\cap\mu_{\gamma}(\gamma)=-1$,
$\mu_{\gamma}(\gamma)\cap\mu_{\gamma}(\tau_{-})=-1$ and
$\mu_{\gamma}(\tau_{+})\cap\mu_{\gamma}(\tau_{-})=1$. Therefore the curves
change exactly according to a $\gamma$-exchange. ∎

######  Remark 4.6.

Lemma 4.5 could be proven using other models, such as plabic fences and
Lagrangian conjugate surfaces, cf. [CL22, Section 3] or [STWZ19, Section 5.2].
We also refer the reader to [STW16, Section 2] for an explanation using a
conical model and the discussion in [PT, Section 4]. $\Box$

Lemma 4.5 clarifies the combinatorics of $\Gamma$ that might lead to immersed
curves for $\mu_{D}(\Gamma)$. Indeed, the existence of a $2$-cycle in
$Q(\Gamma)$ is equivalent to the fact that $\mu_{D}(\Gamma)$, obtained by a
$\gamma$-exchange, has (at least) one immersed curve. Indeed, Proposition 2.22
implies that $\gamma$-exchange leads to a quiver mutation, thus
$\mu_{\gamma}Q(\Gamma)=Q(\mu_{D_{\gamma}}(\Gamma))$, and $\mu_{D}(\Gamma)$ has
an immersed curve if and only if $Q(\mu_{D}(\Gamma))$ contains a
loop.272727Technically, here $\mu_{v}Q$ refers to mutation of a quiver with
2-cycles. This is defined exactly the same as mutation of quivers, in the
sense that the two arrows $a,b\in Q_{1}$, $h(a)=t(b)=v$, in a 2-cycle lead to
the composed arrow $[ab]\in(\mu_{v}Q)_{1}$, which is a loop.

###  4.6. Iteration of Lagrangian disk surgeries and QP non-degeneracy

Let $L\subset(\mathbb{D}^{4},\lambda_{st})$ be an embedded exact Lagrangian
filling and $\Gamma$ an $\mathbb{L}$-compressing system for $L$. Suppose that
the configuration $\mathcal{C}(\Gamma)$ is reduced and
$(Q(\mathcal{C}(\Gamma)),W(\mathcal{C}(\Gamma)))$ is non-degenerate. In
particular, there are no 2-cycles in $Q(\mathcal{C}(\Gamma))$.

Consider an $\mathbb{L}$-compressing disk $D\in\mathscr{D}(\Gamma)$ with
boundary $\gamma\in\mathcal{C}(\Gamma)$ and perform Lagrangian disk surgery on
$(L,D)$. This produces a new Lagrangian filling $(L^{\prime},D^{\prime})$ with
an $\mathbb{L}$-compressing disk $D^{\prime}$. The $\mathbb{L}$-compressing
system $\Gamma$ for $L$ yields an $\mathbb{L}$-compressing system
$\Gamma^{\prime}$ for $L^{\prime}$, whose configuration of curves
$\mathcal{C}(\Gamma^{\prime})$ is as described by Lemma 4.5. We wish to be
able to iterate this procedure arbitrarily: given any curve
$\gamma_{1}\in\mathcal{C}(\Gamma^{\prime})$ with associated
$\mathbb{L}$-compressing disk $D_{1}$, we want to be able to perform
Lagrangian disk surgery to $L^{\prime}$ along $D_{1}$. The two problems at
this stage, which are well-known, are:

  * (i)

It might be that $\mathcal{C}(\Gamma^{\prime})$ does not bound any bigons but
there are curves $\gamma_{i},\gamma_{j}\in\mathcal{C}(\mathscr{D}_{1})$ with
two points of geometric intersection, one positive and one negative, and these
two intersection points do not bound a bigon.

  * (ii)

It might be that $\mathcal{C}(\Gamma^{\prime})$ is a non-reduced
configuration, bounding bigons. That is, there are curves
$\gamma_{i},\gamma_{j}\in\mathcal{C}(\mathscr{D}_{1})$ with two points of
geometric intersection, one positive and one negative, and these two
intersection points bound a bigon.

In either of these two cases, if the curve $\gamma_{1}\subset L^{\prime}$ at
which we must mutate was precisely one such $\gamma_{i}$, then Lagrangian disk
surgery along $D_{1}$ would make $\gamma_{j}$ immersed. It would produce a
Lagrangian filling $\mu_{D_{1}}(L^{\prime})$ but the resulting collection of
curves would have at least an immersed curve. In that case, we can no longer
iterate our sequence of Lagrangian disk surgeries. This is an obstruction that
we cannot bypass merely with the information of $Q(\mathcal{C}(\Gamma))$. In
fact, $Q(\mathcal{C}(\Gamma))$ is not able to distinguish the two cases $(i)$
and $(ii)$ above, as both are 2-cycles in $Q(\mathcal{C}(\Gamma))$.

The core contribution of this manuscript is that we can solve this problem by
using the curve potential $W(\mathcal{C}(\Gamma))$. This potential
$W(\mathcal{C}(\Gamma))$ must be precisely the one we constructed and studied
in Sections 2 and 3. Another choice of potential $W$ for
$Q(\mathcal{C}(\Gamma))$, unrelated to the specific geometry of polygons for
curve configurations $\mathcal{C}(\Gamma)$ would be of no use. Indeed, by
construction, $W(\mathcal{C}(\beta))$ keeps track of polygons and, in
particular, bigons. This is used when solving problem $(ii)$ above. By
Proposition 3.12, $(Q(\mathcal{C}(\Gamma)),W(\mathcal{C}(\Gamma)))$ is non-
degenerate if $\Gamma$ is an $\mathbb{L}$-compressing system of the form
$\Gamma=\Gamma(\beta)$. This is used for solving problem $(i)$ above.

Consider the curve quiver with potential
$(Q(\mathcal{C}(\beta)),W(\mathcal{C}(\beta)))$ constructed in Section 3. We
have the following two properties:

  * (P1)

The QP $(Q(\mathcal{C}(\beta)),W(\mathcal{C}(\beta)))$ is reduced, as defined
in Section 2.3.2. Indeed, the construction of the QP
$(Q(\mathbb{G}),W(\mathbb{G}))$ in Section 3.2 implies that is reduced for any
plabic fence $\mathbb{G}$. By Proposition 3.8,
$(Q(\mathcal{C}(\beta)),W(\mathcal{C}(\beta)))$ coincides with
$(Q(\mathbb{G}(\beta)),W(\mathbb{G}(\beta)))$, where $\mathbb{G}(\beta)$ is
the plabic fence associated to $\beta$. Thus
$(Q(\mathcal{C}(\beta)),W(\mathcal{C}(\beta)))$ is reduced as well.

  * (P2)

The QP $(Q(\mathcal{C}(\beta)),W(\mathcal{C}(\beta)))$ is a curve QP, as in
Definition 2.4. By Proposition 3.12, it is a non-degenerate QP. In particular,
there are never 2-cycles in any QP $(Q,W)$ which is related to
$(Q(\mathcal{C}(\beta)),W(\mathcal{C}(\beta)))$ by a sequence of QP mutations.

Let us use these properties, along with the results in Sections 2 and 3, and
prove the following result.

######  Proposition 4.7.

Let $L\subset(\mathbb{D}^{4},\lambda_{st})$ be an embedded exact Lagrangian
filling and $\Gamma$ an $\mathbb{L}$-compressing system for $L$. Suppose that
the configuration $\mathcal{C}(\Gamma)$ is reduced and
$(Q(\mathcal{C}(\Gamma)),W(\mathcal{C}(\Gamma)))$ is non-degenerate.

If $D\in\mathscr{D}(\Gamma)$ is an $\mathbb{L}$-compressing disk with boundary
$\gamma\in\mathcal{C}(\Gamma)$ then:

  * (i)

The Lagrangian filling $\mu_{D}(L)$ obtained from $L$ by Lagrangian disk
surgery on $D$ admits an $\mathbb{L}$-compressing system $\mu_{D}(\Gamma)$
such that its associated QP $(Q(\mu_{D}(\Gamma)),W(\mu_{D}(\Gamma)))$ is
reduced.

  * (ii)

The quiver with potential associated to $\mu_{D}(\Gamma)$ satisfies

$(Q(\mathcal{C}(\mu_{D}(\Gamma))),W(\mathcal{C}(\mu_{D}(\Gamma)))=(\mu_{\gamma}Q(\mathcal{C}(\Gamma)),\mu_{\gamma}W(\mathcal{C}(\Gamma)))$

and the quiver $Q(\mathcal{C}(\mu_{D}(\Gamma)))$ contains no 2-cycles.

In addition, the curve configuration associated to $\mu_{D}(\Gamma)$ is a
reduction of the $\gamma$-exchange of $\mathcal{C}(\Gamma)$.

###### Proof.

Let us prove $(i)$. First, the quiver $Q(\mathcal{C}(\Gamma))$ has no
2-cycles. Indeed, $\mathcal{C}(\Gamma)$ is reduced by hypothesis, and thus
there are no 2-cycles coming from the trivial part of
$(Q(\mathcal{C}(\Gamma)),W(\mathcal{C}(\Gamma)))$. By hypothesis as well,
$(Q(\mathcal{C}(\Gamma)),W(\mathcal{C}(\Gamma)))$ is non-degenerate, and thus
Definition 3.11 implies that there are no empty 2-cycles either. By Lemma 4.5,
the configuration of curves $\mathcal{C}(\Gamma)$ undergoes a
$\gamma$-exchange under Lagrangian disk surgery. Since
$Q(\mathcal{C}(\Gamma))$ has no 2-cycles, all curves in the resulting
configuration are embedded. None of them is immersed. Therefore, Lagrangian
disk surgery along $D$ produces a new $\mathbb{L}$-compressing system
$\Gamma^{\prime}$, such that $\mathcal{C}(\Gamma^{\prime})$ is the
$\gamma$-exchange of $\mathcal{C}(\Gamma)$.

Second, at this stage, $\mathcal{C}(\Gamma^{\prime})$ might not be a priori
reduced. By Theorem 2.9, there exists a reduction $\mu_{D}(\Gamma)$ of
$\mathcal{C}(\Gamma^{\prime})$. Since
$(Q(\mathcal{C}(\Gamma)),W(\mathcal{C}(\Gamma)))$ is non-degenerate, Lemma
2.21 applies and shows that

$(Q(\mu_{D}(\Gamma)),W(\mu_{D}(\Gamma)))=(Q(\Gamma)_{red},W(\Gamma)_{red}).$

This implies that $(Q(\mu_{D}(\Gamma)),W(\mu_{D}(\Gamma)))$ is reduced. This
implies $(i)$. By construction, $\mu_{D}(\Gamma)$ is a reduction of
$\mathcal{C}(\Gamma^{\prime})$, which itself is the $\gamma$-exchange of
$\mathcal{C}(\Gamma)$. The last sentence of the statement is thus proven as
well.

For item $(ii)$, Proposition 2.22 implies the equality between the QPs. By
item $(i)$, the quiver with potential
$(Q(\mathcal{C}(\mu_{D}(\Gamma))),W(\mathcal{C}(\mu_{D}(\Gamma)))$ is reduced
and thus the non-degeneracy of
$(Q(\mathcal{C}(\Gamma)),W(\mathcal{C}(\Gamma)))$ implies that
$Q(\mathcal{C}(\mu_{D}(\Gamma)))$ has no 2-cycles. ∎

###  4.7. The ring of regular functions $\mathbb{C}[X(\Lambda_{\beta},T)]$

Let $\beta$ be a positive braid word on $n$-strands. Consider the Legendrian
link $\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ and a set of marked
points $T\subset\Lambda_{\beta}$, one per component. Let $A_{\Lambda_{\beta}}$
be Legendrian contact dg-algebra of $(\Lambda_{\beta},T)$ with
$\mathbb{Z}$-coefficients. We refer to [CN22, Section 5.1] or the survey
[EN18] for details on $A_{\Lambda_{\beta}}$. Consider its augmentation variety
$X(\Lambda_{\beta},T)$, which is the space of all dg-algebra maps from
$A_{\Lambda_{\beta}}$ to $\mathbb{C}$, the latter considered as a dg-algebra
in grading $0$ and zero differential. Following [CN22, Section 5], the space
$X(\Lambda_{\beta},T)$, which is naturally an affine variety, can be
explicitly described as follows. For $k=[n-1]$, define the $n\times n$ matrix
$P_{k}(a)$, as a function of an input $a$, as:

$(P_{k}(a))_{ij}=\begin{cases}1&i=j\text{ and }i\neq k,k+1\\\
1&(i,j)=(k,k+1)\text{ or }(k+1,k)\\\ a&i=j=k+1\\\
0&\text{otherwise.}\end{cases}$

Namely, $P_{k}(a)$ is the identity matrix except for the $2\times 2$ submatrix
given by rows and columns $k$ and $k+1$, which is
$\left(\begin{smallmatrix}0&1\\\ 1&a\end{smallmatrix}\right)$. Consider the
full twist $\Delta^{2}$ in $n$-strands and set $\ell=\ell(\beta\Delta^{2})$
for the length of $\beta\Delta^{2}$. Given a braid word
$\beta\Delta^{2}=\sigma_{k_{1}}\cdots\sigma_{k_{\ell}}$, where
$\sigma_{i}\in\operatorname{Br}_{n}$ are Artin generators in $n$-strands, let
$c=|\pi_{0}(\Lambda_{\beta\Delta^{2}})|$ and choose
$i_{1},\ldots,i_{c}\in\mathbb{N}$ such that the $i_{j}$-th strand in
$\beta\Delta^{2}$ is in its $j$-th connected component, $j\in[c]$. Define
$D(\mathbf{t}_{\beta\Delta^{2}})$ to be the diagonal $n\times n$ matrix with
$(k,k)$-entry equal to $t_{j}$ if $k=i_{j}$ and $1$ otherwise, let
$\mathbf{1}$ be the $n\times n$ identity matrix and set

$P_{\beta\Delta^{2}}(z_{1},\ldots,z_{\ell};t_{1},\ldots,t_{c}):=P_{k_{1}}(z_{1})P_{k_{2}}(z_{2})\cdots
P_{k_{\ell}}(z_{\ell})D(\mathbf{t}_{\beta\Delta^{2}}).$

Then, by [CN22, Proposition 5.2], $X(\Lambda_{\beta},T)$ is isomorphic to the
affine variety:

$X(\Lambda_{\beta},T)\cong\\{(z_{1},\ldots,z_{\ell};t_{1},\ldots,t_{c}):\mathbf{1}+P_{\beta\Delta^{2}}(z_{1},\ldots,z_{\ell};t_{1},\ldots,t_{c})=0\\}\subset\mathbb{C}^{\ell}\times(\mathbb{C}^{\times})^{c}.$

The next result is proven in [GSW20a, Theorem 1.1]. It also follows from [CW,
Theorem 1.1], after noticing that $X(\Lambda_{\beta},T)$ is isomorphic to a
decorated moduli space of sheaves in $\mathbb{R}^{2}$ singularly supported in
a front for $\Lambda_{\beta}$.

######  Theorem 4.8 ([CW]).

Let $\beta$ be a positive braid word on $n$-strands,
$\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ its associated Legendrian
link and a set of marked points $T\subset\Lambda_{\beta}$, one per component.
Then the coordinate ring of regular functions
$\mathbb{C}[X(\Lambda_{\beta},T)]$ is a cluster algebra. In addition, it has
the following properties:

  * (a)

Let $(L_{\beta},\Gamma_{\beta})$ be the pair of an embedded exact Lagrangian
filling $L_{\beta}$ of $\Lambda_{\beta}$ and an $\mathbb{L}$-compressing
system $\Gamma_{\beta}=\Gamma(\mathbb{G}(\beta))$ associated to the plabic
fence $\mathbb{G}(\beta)$. Then there exists a canonical cluster seed
$\mathfrak{c}(L_{\beta},\Gamma_{\beta})$ associated to $L_{\beta}$ and it has
quiver $Q(\mathbb{G}(\beta))$.

  * (b)

Let $(L,\Gamma)$ be an exact Lagrangian filling with an
$\mathbb{L}$-compressing system obtained from the Lagrangian filling
$(L_{\beta},\Gamma_{\beta})$ by a sequence of Lagrangian disk surgeries along
an ordered collection of curves
$\gamma_{1},\ldots,\gamma_{k}\in\Gamma_{\beta}$.282828Here it is a hypothesis
that such sequence of Lagrangian disk surgeries is such that the initial
$\mathbb{L}$-compressing system $\Gamma_{\beta}$ carries through a sequence of
$\mathbb{L}$-compressing systems along the disk surgeries and the last system
is $\Gamma$. Then

    * (i)

There is a canonical cluster seed $\mathfrak{c}(L,\Gamma)$ associated to
$(L,\Gamma)$ and the cluster variables in the cluster seed
$\mathfrak{c}(L,\Gamma)$ are computed by microlocal merodromies along the
Poincaré dual relative cycles of the curves in the $\mathbb{L}$-compressing
system $\Gamma$. In particular, cluster variables are indexed by the curves in
$\Gamma$.

    * (ii)

The cluster seed $\mathfrak{c}(L,\Gamma)$ is
$\mu_{v_{k}}\cdots\mu_{v_{1}}(\mathfrak{c}(L_{\beta},\Gamma_{\beta}))$, the
cluster seed obtained by the sequence of cluster mutations along the vertices
$v_{i}\in Q(\mathbb{G}(\beta))_{0}$ associated to
$\gamma_{i}\in\Gamma(\mathbb{G}(\beta))$, $i\in[k]$.

Technically, [CW] shows that $\mathbb{C}[X(\Lambda_{\beta},T)]$ is an upper
cluster algebra with the above properties. That said, it is locally acyclic,
see e.g. [CGG+22, Theorem 7.13], and thus the cluster algebra coincides with
the upper cluster algebra, as proven in [Mul14].

For Theorem 1.1, it suffices to focus on the mutable part of the cluster
algebra $\mathbb{C}[X(\Lambda_{\beta},T)]$: it is well-established that there
are $b_{1}(L)$ mutable vertices in each seed of
$\mathbb{C}[X(\Lambda_{\beta},T)]$, where $b_{1}(L)$ coincides for all
embedded exact orientable Lagragian fillings $L$, as the frozen variables have
only to do with the set of marked points $T$, cf. [CW, GSW20a].

######  Remark 4.9.

Note that the class of Legendrian links $\Lambda_{\beta}$ is such that
$\mathbb{C}[X(\Lambda_{\beta},T)]$ has been proven to be a cluster algebra.
This is unknown for a general Legendrian link
$\Lambda\subset(\mathbb{R}^{3},\xi_{st})$. Among other reasons, this is due to
the lack of definition of a cluster structure on a general derived stack,
which is a matter of homotopical algebraic geometry. That said, the technique
developed to prove Theorem 1.1 and Corollary 1.3 should likely apply to
general Legendrian links $\Lambda\subset(\mathbb{R}^{3},\xi_{st})$ once it is
understood how to make sense of cluster structures on $X(\Lambda,T)$ and they
are proven to exist. $\Box$

###  4.8. Proof of Theorem 1.1

First, we choose the Lagrangian filling $L$ and the $\mathbb{L}$-compressing
system $\Gamma$ in the statement to be $L:=L_{\beta}$ and
$\Gamma:=\Gamma(\beta)$. This filling and $\mathbb{L}$-compressing system were
both introduced in Subsection 4.3. As stated there, the curve configurations
$\mathcal{C}(\Gamma(\beta))=\mathcal{C}(\mathbb{G}(\beta))$ coincide. Here
$\mathbb{G}(\beta)$ is the plabic fence associated to $\beta$, as described in
Section 3, and $\mathcal{C}(\mathbb{G}(\beta))$ is the associated
configuration, as constructed in Section 3.1. We shorten notation to
$\mathcal{C}(\beta):=\mathcal{C}(\mathbb{G}(\beta))$.

Consider the cluster seed $\mathfrak{c}(L_{\beta},\Gamma_{\beta})$ in
$\mathbb{C}[X(\Lambda_{\beta},T)]$ associated to $(L_{\beta},\Gamma_{\beta})$.
By construction, the vertices of the quiver in
$\mathfrak{c}(L_{\beta},\Gamma_{\beta})$ are given by curves in
$\mathcal{C}(\beta)$ and the arrows record signed intersections, cf. [GSW20a]
or [CW, Section 4]. Thus the quiver in the seed
$\mathfrak{c}(L_{\beta},\Gamma_{\beta})$ coincides with the quiver
$Q(\mathcal{C}(\beta))$ associated to the curve configuration
$\mathcal{C}(\beta)$ in $L_{\beta}=\Sigma(\mathbb{G}(\beta))$. Note that the
curves in $\mathcal{C}(\beta)$ are smooth, oriented, embedded and form a basis
of $H_{1}(L_{\beta},\mathbb{Z})$. Therefore, $\mathcal{C}(\beta)$ is indeed a
curve configuration.

It follows from Properties (P1) and (P2) in Subsection 4.6, or direct
inspection, that $\mathcal{C}(\beta)$ is reduced and $Q(\mathcal{C}(\beta))$
contains no 2-cycles. Equally important, Proposition 3.12 implies that the QP
$(Q(\mathcal{C}(\beta)),W(\mathcal{C}(\beta)))$ is non-degenerate.

Let $(v_{1},\ldots,v_{\ell})$ be the sequence of mutable vertices in
$Q(\mathcal{C}(\beta))$ given to us in item $(i)$ of the statement. Let us
construct an embedded exact Lagrangian filling $L_{k}$ for the seed
$\mu_{v_{k}}\ldots\mu_{v_{1}}(\mathfrak{c}(L,\Gamma))$ along with an
$\mathbb{L}$-compressing system for $L_{k}$. Let us denote by
$\mathscr{D}_{0}=\mathscr{D}_{\beta}$ the collection of
$\mathbb{L}$-compressing disks associated to $\Gamma(\beta)$. Now, the vertex
$v_{1}$ corresponds to an $\mathbb{L}$-compressible curve
$\gamma_{1}\in\mathcal{C}(\Gamma(\beta))$. In order to obtain a Lagrangian
filling in the seed
$\mu_{1}:=\mu_{v_{1}}(\mathfrak{c}(L_{\beta},\Gamma_{\beta}))$ we apply
Proposition 4.7 with $L=L_{\beta}$ and $D=D_{v_{1}}\in\mathscr{D}_{0}$ the
$\mathbb{L}$-compressing disk associated to $\gamma_{1}$. The hypothesis of
reducedness and non-degeneracy of the initial QP are indeed satisfied by the
previous paragraph.

Proposition 4.7 now produces an $\mathbb{L}$-compressing system
$\mathscr{D}_{1}$ for the Lagrangian filling
$L_{1}:=\mu_{D_{v_{1}}}(L_{\beta})$ whose associated QP is reduced and non-
degenerate. In addition, the Lagrangian disks in $\mathscr{D}_{1}$ are in a
specified bijection with the disks in $\mathscr{D}_{0}$. This bijection is
established as follows. The $\mathbb{L}$-compressing disks are indexed by the
curves in the respective configurations. Lemma 4.5 implies that the curves
undergo a $\gamma$-exchange under Lagrangian disk surgery. By construction,
curves in a configuration before and after a $\gamma$-exchange are in a
specified bijection, as the vertices of the corresponding quiver are
identified (via the identity), cf. Section 2.3.1. A reduction of a
configuration also determines a unique bijection, as Lemmas 2.5 and 2.7 show
that the vertices of the quiver remain (identically) the same under triple
point moves and local bigon moves. Therefore, the Lagrangian disks in
$\mathscr{D}_{1}$ are indeed in a specified bijection with the disks in
$\mathscr{D}_{0}$.

Consider the next vertex $v_{2}$ of $Q(\mathcal{C}(\beta))$ at which we must
mutate. By the bijection above, this specifies a unique
$\mathbb{L}$-compressing disk $D_{v_{2}}$ in $\mathscr{D}_{1}$ in the
$\mathbb{L}$-compressing system of $\mu_{D_{v_{1}}}(L_{\beta})$. Since the QP
associated to $\mathscr{D}_{1}$ is reduced and non-degenerate, we can apply
Proposition 4.7 to $\mu_{D_{v_{1}}}(L_{\beta})$ and $\mathscr{D}_{1}$. This
produces a Lagrangian filling
$L_{2}=\mu_{D_{v_{2}}}\mu_{D_{v_{1}}}(L_{\beta})$ in the cluster seed
$\mu_{v_{2}}\mu_{v_{1}}(\mathfrak{c}(L,\Gamma))$ with an
$\mathbb{L}$-compressing system $\mathscr{D}_{2}$. Since the QP associated to
$\mathscr{D}_{2}$ is again reduced and non-degenerate, we can iteratively
apply Proposition 4.7 without any constraints. By Theorem 4.8, this procedure
indeed constructs the required Lagrangian fillings
$L_{k}=\mu_{D_{v_{k}}}\ldots\mu_{D_{v_{2}}}\mu_{D_{v_{1}}}(L_{\beta})$ with
$\mathbb{L}$-compressing systems
$\Gamma_{k}=\mathscr{D}_{k}=\mu_{D_{v_{k}}}\ldots\mu_{D_{v_{2}}}\mu_{D_{v_{1}}}(\mathscr{D}_{\beta})$
in the required cluster seeds. This proves item $(i)$ of the statement. The
construction we used, applying Proposition 4.7, implies item $(ii)$. $\Box$

###  4.9. A related statement

Let $\Lambda\subset(\mathbb{R}^{3},\xi_{st})$ be a Legendrian link and
$L\subset(\mathbb{R}^{4},\lambda_{st})$ an embedded exact Lagrangian filling.
Suppose that $\Gamma$ is a collection of $\ell$ oriented simple
$\mathbb{L}$-compressible curves in $L$ which are linearly independent in
$H_{1}(L;\mathbb{Z})$, but not necessarily spanning. Let us refer to such a
collection as a partial $\mathbb{L}$-compressing system of rank $\ell$. A
curve QP $(Q(\Gamma),W(\Gamma))$ is still defined.292929In fact,
$(Q(\Gamma),W(\Gamma))$ is defined for any collection of oriented simple
curves, even if they are not linearly independent in $H_{1}(L;\mathbb{Z})$.
Linear independence is only used in proving Proposition 3.15. Thus one may
proceed by just assuming that the assumptions in this proposition, non-
degeneracy and the bigon condition, hold instead. If we want to emphasize the
dependence on $L$, we write $Q(L,\Gamma)=Q(\Gamma)$. If such QP
$(Q(\Gamma),W(\Gamma))$ is non-degenerate, one may proceed as above and
conclude a statement in line with Theorem 1.1 for more general Legendrian
links $\Lambda\subset(\mathbb{R}^{3},\xi_{st})$. Following the same steps in
the proof of Theorem 1.1, we obtain the following:

######  Theorem 4.10.

Let $\Lambda\subset(\mathbb{R}^{3},\xi_{st})$ be a Legendrian link,
$T\subset\Lambda$ a set of marked points with one marked point per component
and $X(\Lambda,T)$ the affine scheme given by the spectrum of $H^{0}$ of the
Legendrian contact dg-algebra of $\Lambda\subset(\mathbb{R}^{3},\xi_{st})$.
Suppose that there exists an orientable embedded exact Lagrangian filling
$L\subset(\mathbb{R}^{4},\lambda_{st})$ of $\Lambda$ and a partial
$\mathbb{L}$-compressing system $\Gamma$ for $L$ of rank $\ell$ such that
$(Q(\Gamma),W(\Gamma))$ is non-degenerate. Then:

  * (i)

If $\mu_{v_{\ell}}\ldots\mu_{v_{1}}$ is any sequence of mutations, where
$v_{1},\ldots,v_{\ell}$ are mutable vertices of the quiver $Q(L,\Gamma)$, then
there exists a sequence of embedded exact Lagrangian fillings $L_{k}$ of
$\Lambda$ with associated quivers
$Q(L_{k},\Gamma_{k})=\mu_{v_{k}}\ldots\mu_{v_{1}}(Q(L,\Gamma))$, for all
$k\in[\ell]$.

  * (ii)

Each embedded exact Lagrangian filling $L_{k}$ is equipped with a partial
$\mathbb{L}$-compressing system $\Gamma_{k}$ of rank $\ell$ such that
Lagrangian disk surgery on $L_{k}$ along any Lagrangian disk in
$\mathscr{D}(\Gamma_{k})$ yields an $\mathbb{L}$-compressing system.
Furthermore, $\Gamma_{k+1}$ is equivalent to such a partial
$\mathbb{L}$-compressing system via a sequence of triple point moves and local
bigon moves.

In addition, if there exists a sub-algebra $A\subset\mathbb{C}[X(\Lambda,T)]$
which is a cluster algebra and $(L,\Gamma)$ defines a cluster seed
$\mathfrak{c}(L,\Gamma)$ for $A$, then
$\mathfrak{c}(L_{k},\Gamma_{k})=\mu_{v_{k}}\ldots\mu_{v_{1}}(\mathfrak{c}(L,\Gamma))$
in $A\subset\mathbb{C}[X(\Lambda,T)]$, for all $k\in[\ell]$. In particular,
the map $\mathfrak{C}:\mbox{Lag}^{c}(\Lambda)\longrightarrow\mbox{Seed}(A)$ is
surjective, i.e. there exists an embedded exact Lagrangian filling endowed
with an $\mathbb{L}$-compressing system realizing every cluster seed in $A$.
$\Box$

The advantage of Theorem 4.10 is that it applies to general
$\Lambda\subset(\mathbb{R}^{3},\xi_{st})$. The disadvantage is that, in a
given instance being studied, one must construct a partial
$\mathbb{L}$-compressing system $\Gamma$ and verify the hypothesis that
$(Q(\Gamma),W(\Gamma))$ is non-degenerate. The existence of a sub-algebra $A$
that is cluster can be sometimes established using the Starfish lemma [FWZ,
Prop. 6.4.1]. In the case of Theorem 1.1, we have built above a specific
$\mathbb{L}$-compressing system $\Gamma=\Gamma_{\beta}$ for any
$\Lambda_{\beta}\subset(\mathbb{R}^{3},\xi_{st})$ and proven, with Proposition
3.12, the non-degeneracy of its associated QP. For such links, it follows from
[CW] that such a sub-algebra $A$ exists. In fact, we can take
$A=\mathbb{C}[X(\Lambda_{\beta},T)]$, cf. [CW, CGG+22].

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