####  7.1.3. Explicit Examples of Lagrangian Fillings

For the case of free $2$-graphs on a disk $\mathbb{D}^{2}$, this immediately
yields that the max-tb Legendrian $(2,n)$-torus positive link $\Lambda(2,n)$
has at least a Catalan $C_{n}$ number worth of exact Lagrangian fillings
[EHK16, Pan17b, STZ17, TZ18]. This is because $C_{n}$ counts binary trees,
which are equivalent to free 2-graphs. These exact Lagrangian fillings are
distinguished, up to Hamiltonian isotopy, through the use of cluster
coordinates – see Subsection 7.2.1. Now, the ability to increase
$N\in\mathbb{N}$ greatly expands232323This is particularly relevant for the
study of exact Lagrangian fillings, as it is expected that any
$\Lambda(\beta)$ with $\beta\in\mbox{Br}^{+}_{2}$ has only finitely many exact
Lagrangian fillings, and we will show in Theorem 7.14 that this is not the
case already for $N=3$. the class of Legendrian links for which their
Lagrangian fillings can be studied with $N$-graph calculus, including all
Legendrian positive braids $\Lambda(\beta)$, $\beta\in\mbox{Br}^{+}_{N}$ for
any $N\in\mathbb{N}$.

Example 1: Recently, the first examples of Legendrian links with infinitely
many exact Lagrangian fillings were described in the article [CG20]. We
exhibit them here in terms of 3-graphs. For any $(p,q)\in\mathbb{N}^{2}$, the
max-tb Legendrian $(p,q)$-torus positive link
$\Lambda(p,q)\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$ is the satellite of
the braid $\Delta(\sigma_{1}\sigma_{2}\cdot\ldots\cdot\sigma_{p-1})^{q}\Delta$
along the standard Legendrian unknot, where $\Delta=\Delta_{p}\in S_{p}$ is
the $p$-stranded half-twist. Let us now illustrate how to diagrammatically
visualize these infinitely many Lagrangian for the Legendrian link
$\Lambda(3,6)$.

######  Remark 7.5.

Similar $p$-graphs can be drawn for $\Lambda(p,q)$ for all
$(p,q)\in\mathbb{N}^{2}$ and they produce infinitely many Lagrangian fillings
if $p\geq 3,q\geq 6$ or $(p,q)=(4,4),(4,5)$. Alternatively, infinitely many
exact Lagrangian fillings for
$\Lambda(p,q)\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$, $p\geq 4,q\geq 7$ can
also be readily constructed from those of $\Lambda(3,6)$ [CG20, Corollary
1.5]. $\Box$

Consider the braid word
$\beta=(\sigma_{1}\sigma_{2})^{9}=\Delta(\sigma_{1}\sigma_{2})^{6}\Delta$ in
the 1-jet space $(J^{1}\mathbb{S}^{1},\xi_{\text{st}})$, where
$\Delta=\Delta_{3}\in S_{3}$ is the 3-stranded half-twist
$\Delta=\sigma_{1}\sigma_{2}\sigma_{1}$. This braid $\beta$ can be depicted as
a set of points in the circle $\mathbb{S}^{1}$ labeled with two colors,
corresponding to $\sigma_{1},\sigma_{2}$. Figure 90 shows this braid $\beta$
in two circles, the inner circle $\mathbb{S}^{1}\times\\{1\\}$ and outer
circle $\mathbb{S}^{1}\times\\{2\\}$ in the annulus
$\mathbb{S}^{1}\times[1,2]$. These two marked circles are labeled by
$\Lambda(3,6)$, as the Legendrian link associated to the marking $\beta$ is
the $(3,6)$-Legendrian link $\Lambda(3,6)$.

Figure 90. Legendrian weave whose Lagrangian projection defines an infinite
order element in the fundamental group of the space of Legendrian links
isotopic to $\Lambda(3,6)$. In particular, this Lagrangian concordance has
infinite order in the Lagrangian concordance monoid. Infinitely many
Lagrangian fillings for $\Lambda(3,6)$, and all torus links $\Lambda(n,m)$,
$n\geq 3,m\geq 6$, are obtained by concatenating this 3-graph.

The 3-graph $G\subseteq\mathbb{S}^{1}\times[1,2]$ depicted in Figure 90
describes a Legendrian surface
$\Lambda(G)\subseteq(J^{1}\mathbb{S}^{1}\times[1,2],\xi_{\text{st}})$ with
boundary $\Lambda(3,6)\sqcup\Lambda(3,6)$. By increasing the slope in the
radial direction, the Legendrian surface $\Lambda(G)$ can be assumed to have
no Reeb chords, and thus $\pi(\Lambda(G))$ is an exact Lagrangian cobordism
from $\Lambda(3,6)$ to itself. Since the graph $G$ has no trivalent vertices,
$\Lambda(G)$ has the topology of $\Lambda(3,6)\times[1,2]$ and it is in fact
an exact Lagrangian concordance. The remarkable property of the 3-graph $G$,
and its Lagrangian projection $\pi(\Lambda(G))$, is stated in the following:

######  Theorem 7.6 ([CG20]).

The $3$-graph exact Lagrangian concordance in Figure 90 has infinite order. In
particular, for any fixed exact Lagrangian filling of $\Lambda(3,6)$, iterated
concatenation of this $3$-graph yields infinitely many Lagrangian fillings of
the Legendrian link $\Lambda(3,6)\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$.
$\Box$

In fact, it is possible to describe the entire faithful modular
PSL($2,\mathbb{Z}$)-representation in [CG20] with the diagrammatics of
3-graphs. Similarly, the diagrammatics of 4-graphs give explicit spatial
wavefronts for the $M_{0,4}$-worth of the (Legendrian lift of the) Lagrangian
fillings for the Legendrian link
$\Lambda(4,4)\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$. The non-triviality,
and infinite order, of this Lagrangian concordance is detected by studying its
action on the cluster structure of the coordinate ring of the moduli space of
isomorphism classes of simple objects in
$\operatorname{Sh}_{\Lambda(3,6)}(\mathbb{R}^{2})$.

Example 2: Let us address the following question. Given a positive braid
$\beta$, and the Legendrian link $\Lambda=\Lambda(\Delta\beta\Delta)$, how do
we diagrammatically produce an $N$-graph which represents an embedded exact
Lagrangian filling for $\Lambda\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$ ?

Let us begin with a simple example, with
$\beta=\Delta^{2}=(\sigma_{1}\sigma_{2})^{3}$ the full-twist, which is
smoothly the $(3,3)$-torus link. The game is to draw $\tau_{i}$-edges along
the boundary $\partial\mathbb{D}^{2}$ of a (planar) 2-disk $\mathbb{D}^{2}$
according to the braid word $\beta$ and complete these edges to an $N$-graph
$G$ inside $\mathbb{D}^{2}$. The only rule is that the Legendrian weave
$\Lambda(G)$ should not have Reeb chords, or else it would yield an immersed
Lagrangian filling, and thus we require $G$ to be free.

Consider the free 3-graph $G_{1}$ in Figure 91 (upper Left). This represents
an embedded exact Lagrangian filling $L_{1}$ of the max-tb Legendrian
$(3,3)$-torus link
$\Lambda(3,3)=\Lambda(\Delta\beta\Delta)=\Lambda(\Delta^{4})$. We can now
apply the Legendrian mutation moves in Theorem 4.21 in order to produce
another Lagrangian filling $L_{2}$ which is not Hamiltonian isotopic to the
exact Lagrangian filling $L_{2}$. (Note that $L_{1}$ and $L_{2}$ are smoothly
isotopic relative to their boundaries, and $L_{2}$ will be also embedded
thanks to Lemma 7.4.) In Figure 91 we perform a Lagrangian disk surgery on
$L_{1}$ along a Lagrangian 2-disk which bounds the 1-cycle in
$H_{1}(L_{1},\mathbb{Z})$ graphically given by the $\sf Y$-cycle in surrounded
by the dashed green curve.

Figure 91. Four 3-graphs representing exact embedded Lagrangian fillings for
the maximal-tb $(3,3)$-torus link $\Lambda(3,3)$.

At this stage we can manipulate $L_{2}$ with Theorem 4.2, in this case Figure
91 (upper right) to 91 (bottom left) shows how to apply Move II to push-
through a hexagonal vertex through a trivalent vertex (as indicated by the
green arrow). This is an interesting move because it makes a new $1$-cycle for
$L_{2}$ readily visible, as represented by the blue monochromatic edge in 91
(bottom left) surrounded by a dashed green curve. We can perform Lagrangian
surgery at this monochromatic edge, as in Theorem 4.21, to obtain another
exact Lagrangian filling $L_{3}$, also embedded by Lemma 7.4. It is immediate
that $L_{1}$ and $L_{3}$ are not not Hamiltonian isotopic to $L_{2}$, as the
cluster coordinates associated to these 3-graphs, as explained in Subsection
7.2.1, show that $L_{1}$ and $L_{3}$ are not Hamiltonian isotopic. In
conclusion, the 3-graphs in Figure 91 represent three distinct embedded exact
Lagrangian fillings for $\Lambda(3,3)$.

Example 3: Let us illustrate what a generic $3$-graph diagram like for a
positive braid $\beta\in\mbox{Br}^{+}_{3}$. The pictures in the case of
$\beta\in\mbox{Br}^{+}_{N}$, $N\geq 3$ are alike, with as many as
$(N-1)$-colors instead. Let us consider a random braid

$\beta=(\sigma_{1}\sigma_{2}\sigma_{1})\sigma_{2}^{2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}^{3}\sigma_{2}^{3}(\sigma_{1}\sigma_{2}\sigma_{1}),$

which has no particular significance to us. To obtain exact Lagrangian
fillings, we draw blue and red edges around a circle
$\mathbb{S}^{1}\subseteq\mathbb{R}^{2}$, according to $\sigma_{1}$ or
$\sigma_{2}$, and construct 3-graphs with no Reeb chords and these boundary
constraints. Figure 92 shows four free $3$-graphs $G_{i}$, $i\in[1,4]$, such
that the Lagrangian projections
$\pi(\iota(\Lambda(G_{i})))\subseteq(\mathbb{D}^{4},\omega_{\text{st}})$ are
embedded exact Lagrangian fillings which are distinct up to Hamiltonian
isotopy for $i\neq j$, $i,j\in[1,4]$.

######  Remark 7.7.

From our experience drawing 3-graphs, the pictures in Figure 92 accurately
represent the generic appearance of exact Lagrangian fillings described by
free 3-graphs. We presently do not know any example of a Lagrangian filling
for a positive braid which does not arise as an $N$-graph, for some
$N\in\mathbb{N}$. $\Box$

Figure 92. Four exact embedded Lagrangian fillings for the braid $\beta$ in
Example 3. Their satellites in $(\mathbb{R}^{4},\omega_{\text{st}})$ are
smoothly isotopic relative to their boundaries, but not Hamiltonian isotopic.)

######  Remark 7.8.

There exists a technique for producing many such free $N$-graphs $G$, filling
$\beta$-boundary conditions at a circle and thus representing embedded exact
Lagrangian fillings. This is ongoing work by the first author, which in
particular proves that any Legendrian link $\Lambda(\beta)$ arising from a
positive braid $\beta\in\mbox{Br}^{+}_{N}$ admits an embedded Lagrangian
filling whose Legendrian lift is a Legendrian weave. In precise terms, it can
be proven that for each triangulation of a $|\beta|$-gon, one can assign a
free $N$-graph which represented an embedded Lagrangian filling of $\beta$,
where $|\beta|$ is the length of the positive braid $\beta$. $\Box$

###  7.2. Microlocal monodromies and cluster structures

In this section, we demonstrate how notions of cluster theory are borne out
with $N$-graphs. This is an important ingredient in showing that microlocal
monodromies can be used to distinguish exact Lagrangian fillings, as we do in
Section 7.3 and as has been mentioned previously.

To orient the discussion, we recall that the cluster structures on the Fock-
Goncharov moduli spaces of framed local systems described in [FG06b] were
given a sheaf-theoretic description in [STW16, STWZ19]. In these works, the
spectral surface associated to a bipartite graph, as defined in [Gon17,
Section 2.2], is described symplectically as an exact Lagrangian filling of
the zigzag Legendrian curve. In the case of bipartite graphs associated to an
$N$-triangulation, as in [Gon17, Section 1], the zigzag curves isotope to
concentric circles around the vertices of the triangulation, and the singular
support of such a configuration translates to the data of a local system with
a monodromy-invariant flag at each vertex. Sheaf quantization [GKS12] then
implies that local systems on the exact filling embed as a cluster chart of
objects, the chart being provided by the bipartite graph (and its dual
quiver), and the cluster coordinates given by microlocal monodromies. The
intersection form in $H_{1}(L,\mathbb{Z})$, or its negative, corresponds to
the skew-symmetric bilinear form in cluster theory. For us, the crucial point
is that we can represent all these Lagrangian fillings by $N$-graphs, as in
the diagrammatics of Subsection 7.1, and the cluster coordinates can be read
directly from the $N$-graph, as we will now explain.

######  Remark 7.9.

In [TZ18], the case of Legendrian surfaces defined by trivalent 2-graphs was
studied, giving a sheaf-theoretic description of the constructions in [DGG16].
In this setting, the microlocal monodromy functor $\mu mon$ induces, at the
level of moduli of objects, a morphism from the sheaf moduli space to the
cluster chart defined by the triangulation dual to the 2-graph. The image is a
(holomorphic) Lagrangian in a (holomorphic) symplectic leaf, as in [DGG16], in
a manner compatible with quantization of algebra of functions.242424In work in
progress with Linhui Shen, the second author will develop the relation to
cluster theory more systematically, and prove Lagrangianicity of the moduli
space. Furthermore, in that work, the potential describing the local exact
structure of the Lagrangian was interpreted as a generator of BPS states or
disk invariants, following the analysis of Aganagic-Vafa [AV00, AV12]. Here we
generalize some of the constructions to $N$-graphs, $N\geq 2$. $\Box$

In this article, the Legendrian surfaces are described by $N$-graphs, a more
complex construction, but we will now explain how the basic features should
persist. That is, the microlocal monodromy functor allows us to read cluster
coordinates for the moduli spaces of isomorphism classes of simple objects in
$Sh_{\Lambda}(\mathbb{R}^{2})$, equivalently augmentation varieties, directly
from $N$-graphs with boundary $\Lambda$. Examples of these constructions are
provided below.

####  7.2.1. Microlocal monodromies as cluster coordinates

By definition, microlocal monodromy is a functor

$\mu mon:Sh_{\Lambda}\to Loc(\Lambda)$

from the category $Sh_{\Lambda}$ of sheaves microsupported on the Legendrian
surface $\Lambda$, as defined in Subsection 5.3, to the category of local
systems on $\Lambda$ [STZ17]. This functor carries microlocal rank-one sheaves
$F\in Sh^{1}_{\Lambda}$, i.e. simple sheaves, to rank one local systems on the
surface $\Lambda$. Since it is locally defined, the monodromy of the local
system $\mu mon(F)$ around a loop $\gamma\in H_{1}(\Lambda)$ can be evaluated
by restricting the constructible sheaf $F$ to an annular tubular neighborhood
of $\gamma$. Below, these annuli are depicted as thin purple loops. In short,
the calculation for Legendrian weaves can be done using the microlocal
monodromy functor $\mu mon$ as it is used for knots, as described in [STZ17].

The main point in these computations is that the stalk $\mu mon(F)|_{\lambda}$
at a point $\lambda\in\Lambda$ is the cone of the restriction map
corresponding to $\lambda$, and for flags this is the inclusion of subspaces,
whence cones become cokernels. The transversality of adjacent flags ensures
that these cokernels propogate as a local system. Let us now perform these
calculations for 1-cycle $\gamma\in H_{1}(\Lambda,\mathbb{Z})$, starting at
the $\sf I$-cycle represented by a monochromatic edge.

Let us consider a monochromatic edge with label $\tau_{i}$, as depicted in
Figure 93.

$a$$b$$c$$d$$e$ Figure 93. Neighborhood of a monochromatic edge $e$ with the
data determining a constructible sheaf $F$. As we show, the microlocal
monodromy $\mu mon(F)$ along the 1-cycle $\gamma(e)$ is given by the cross-
ratio $\langle a,b,c,d\rangle$.

Near such a monochromatic edge, a sheaf object in a simply connected face is
specified by the data of a quadruple of flags. Each of these flags has the
same subspaces $\mathscr{F}^{j}$ in each region for $j\neq i$, and for $j=i$
we additionally require the data in each region of a line $l$ in the two-
dimensional space $V:=\mathscr{F}^{i+1}/\mathscr{F}^{i-1}.$ This is the data
of four lines $a,b,c,d\subseteq V$, as specified in Figure 93. Restricted to
the purple oval shown, we have a cylindrical braid of type
$\beta=\sigma_{i}^{4}$, where $\sigma_{i}$ is the lift of the transposition
$\tau_{i}$ from the Coxeter group $S_{N}$ to the braid group $\mbox{Br}_{N}$.
Given the prescribed transversality imposed by the flag moduli of an
$N$-graph, we further know that the cyclic chain of inequalities $a\neq b\neq
c\neq d\neq a$ holds. We thus have the chain of isomorphisms of cokernels

$a\cong V/b\cong c\cong V/d\cong a,$

which computes the microlocal monodromy. In this case, the isomorphism that we
obtain is the cross ratio

$\langle a,b,c,d\rangle=\frac{a\wedge b}{b\wedge c}\cdot\frac{c\wedge
d}{d\wedge a}$

of the four lines $a,b,c,d,$ and it is equal to the cluster coordinate
associated to 1-cycle $\gamma$ as prescribed in [FG06b, Section 9].

Let us now consider the cluster coordinate associated to a $\sf Y$-cycle,
which is a new type of $1$-cycle, as it only appears for $N\geq 3$. Figure 94
depicts a $\sf Y$-cycle, drawn as a purple circle, along with the data
determining a constructible sheaf in a neighborhood of this 1-cycle.

$\bullet$$\bullet$$\bullet$$(a,A)$$(c,C)$$(b,B)$$(a,ab)$$(b,ab)$$(c,bc)$$(b,bc)$$(c,ac)$$(a,ac)$
Figure 94. Neighborhood of a $\sf Y$-cycle with the data determining a
constructible sheaf $F$. As we compute, the microlocal monodromy $\mu mon(F)$
along the associated 1-cycle $\gamma$ is given by the triple ratio of the
three transverse flags.

Following the notation in Section 6, we denote by $ab$ the unique plane
containing the two lines $a$ and $b$, while $AB$ denotes the intersection of
the planes $A$ and $B$. The braid associated to the $\sf Y$-cycle $\gamma$, as
drawn by the purple circle in Figure 94, is given by
$\beta=(\sigma_{i}\sigma_{i+1}\sigma_{i})^{3}$, where $\sigma_{i}$ corresponds
to the crossing coming from a $\tau_{i}$-edge. By considering the three-
dimensional vector space $V:=\mathscr{F}^{i+2}/\mathscr{F}^{i-1}$, a given
flag is specified by a line and a plane in $V$. Since the word
$\sigma_{i}\sigma_{i+1}\sigma_{i}$ represents the half-twist $\Delta$ for
flags on $V$, and $\tau_{i}\tau_{i+1}\tau_{i}$ is the Coxeter element in
$S_{3}$, the complete data specifying a constructible sheaf near the $\sf
Y$-cycle is given by three transverse flags $(a,A),(b,B),(c,C)$ in $V$. In
this notation, the line is written in lower case and the covector defining the
the plane in upper case, thus $(a,A)$ determines a flag. Now, the microlocal
monodromy functor $\mu mon$ along $\gamma$ is computed as the composition of
the isomorphisms

$a\cong V/B\cong c\cong V/A\cong b\cong V/C\cong a.$

Let $v_{a}\in a,v_{b}\in b,v_{c}\in c,v_{d}\in d$ be non-zero vectors defining
the corresponding one-dimensional lines. Then the parallel transport from $a$
to $c$ in this basis is given by the quotient $B(a)/B(c)$, where $B(a)$ is the
pairing between the vector $v_{a}$ and the covector $B$. Iterating these
isomorphisms, we conclude that the microlocal monodromy along the ${\sf
Y}$-cycle is given by

$\langle(a,A),(b,B),(c,C)\rangle:=\frac{B(a)C(b)A(c)}{B(c)C(a)A(b)}.$

This expression is precisely the triple product of transverse flags as defined
in [FG06b], and thus we have shown that the microlocal monodromy along a $\sf
Y$-cycle determines a cluster coordinate.

####  7.2.2. Legendrian Mutations are cluster transformations

The coordinate transformations upon Legendrian mutations can also be computed,
as we will demonstrate in an example. The conclusion is that Legendrian
mutations induce cluster transformations. The case of a monochromatic edge
follows from the analysis in [TZ18, STWZ19], and we now study the mutation at
a $\sf Y$-cycle. To do so, consider the local geometry shown in Figure 95. We
want to compute how the cluster coordinate associated to the unique
monochromatic (blue) edge – as in Subsection 7.2.1 – changes as we perform a
Legendrian mutation along the $\sf Y$-cycle specified by the unique hexagonal
vertex.

$\bullet$$\bullet$$\bullet$$\bullet$$(a,A)$$(c,C)$$(b,B)$$(a,ab)$$(b,ab)$$(c,bc)$$(b,bc)$$(c,ac)$$(a,ac)$$(b,B^{\prime})$
Figure 95. The geometric setup before performing a Legendrian mutation at the
$\sf Y$-cycle, where the cluster coordinate associated to the monochromatic
edge is given by the cross-ratio $\langle B,bc,ab,B^{\prime}\rangle$.

The monochromatic blue edge has monodromy equal to the cross ratio $z:=\langle
B,bc,ab,B^{\prime}\rangle$ of the four planes in the projective line of planes
containing $b$. (This can be computed directly or by intersecting the four
lines with any transverse line – see Subsection 7.2.1.) Now, after Legendrian
mutation at the ${\sf Y}$-cycle, the resulting 3-graph is shown Figure 96.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$(a,ab)$$(b,ab)$(AB,A)(AB,B)$(c,bc)$$(b,bc)$(BC,B)(BC,C)$(c,ac)$$(a,ac)$(AC,C)(AC,A)$(b,B^{\prime})$$(a,A)$$(c,C)$$(b,B)$
Figure 96. The result of applying a Legendrian mutation to Figure 95 along the
Y-cycle, along with the data of a constructible sheaf.

The 1-cycle determined by the blue monochromatic edge in Figure 95 becomes a
(bichromatic edge) 1-cycle contained in the 3-graph shown in Figure 97, which
is itself a piece of Figure 96, in its upper-right corner252525The trivalent
blue vertex in Figure 97 is the unique trivalent blue vertex in Figure 96. The
trivalent red vertex in Figure 97 is the rightmost trivalent red vertex in
Figure 96. :

$\bullet$$\bullet$$(b,B^{\prime})$$(b,B)$(AB,B)${(AB,A)}$$(AC,A)$${(a,A)}$(a,ab)$(b,ab)$
Figure 97. Local geometry near the 1-cycle after mutation.

By applying Move II, we can push the red trivalent vertex in Figure 97 through
the hexavalent vertex. This allows us to represent the 1-cycle as a
monochromatic edge again, as shown in Figure 98:

$\bullet$$\bullet$$(b,B^{\prime})$$(b,B)$$(b,bAC)$$(b,ab)$ Figure 98. The
constructible sheaf near the 1-cycle after Legendrian mutation and Move II.
The new coordinate is thus the cross-ratio $\langle
B,bAC,ab,B^{\prime}\rangle$.

The required conclusion, stating that the new cross-ratio $z^{\prime}=\langle
B,bAC,ab,B^{\prime}\rangle$ is obtained by a cluster transformation, follows
from this:

######  Lemma 7.10.

Let $x=\langle(a,A),(b,B),(c,C)\rangle$ be the triple ratio of flags and
$z=\langle B,bc,ab,B^{\prime}\rangle$ the cross-ratio of lines. Denote by
$z^{\prime}=\langle B,bAC,ab,B^{\prime}\rangle$ the new microlocal monodromy.
Then

$z^{\prime}=z(1+x).$

###### Proof.

By $\operatorname{PGL}_{3}$ invariance, we may assume that

$a=\begin{pmatrix}1\\\ 0\\\ 0\end{pmatrix},A=(0,0,1),b=\begin{pmatrix}0\\\
0\\\ 1\end{pmatrix},B=(1,0,0),c=\begin{pmatrix}1\\\ -1\\\
1\end{pmatrix},C=(1,1+x,x).$

Since the cross-ratio $z$ is prescribed, we find that $B^{\prime}=(z,1,0)$,
and similarly

$AC=\begin{pmatrix}1+x\\\ -1\\\ 0\end{pmatrix}.$

This implies that $bAC=(1,1+x,0)$, and thus $z^{\prime}=z(1+x)$. ∎

Note that $z^{\prime}=z(1+x)$ in Lemma 7.10 is the transformation expected for
a cluster-X transformation.262626The rule for a cluster-X transformation upon
mutating at loop $k$ is that the monodromy $z_{i}$ transforms to $1/z_{k}$ if
$i=k$ and otherwise $z_{i}^{\prime}=z_{i}(1+z_{k}^{-{\rm
sgn}\epsilon_{ik}})^{-\epsilon_{ik}},$ where $\epsilon_{i,k}$ is the skew-
symmetric cluster form. We get agreement on the nose if we make this form the
_negative_ of the intersection pairing. This concludes that a Legendrian
mutation at the ${\sf Y}$-cycle induces a cluster transformation for the
microlocal monodromy coordinate at the monochromatic blue edge in Figure 95.
The computation is analogous if we choose a different blue monochromatic edge
to be added near the ${\sf Y}$-cycle. In particular, if we had chosen instead
the blue edge attaching at the lower-right of the Y-cycle and pointing upward,
and again called its monodromy $z$, then we would have $A^{\prime}=(0,z,1)$
and would obtain

$z^{\prime}=\langle
ab,aBC,A,A^{\prime}\rangle=z\left(1+\frac{1}{x}\right)^{-1},$

in agreement with the cluster transformation.272727We remark that the case
$x=-1$ is not a generic configuration of flags, since in this case $c\in ab,$
and thus not in the domain of the birational cluster map.

######  Example 7.11.

Flip of a $N$-triangulation. Let $(C,\tau)$ be a punctured surface $C$, $\tau$
an ideal triangulation and $\tau^{\prime}$ an ideal triangulation obtained
from $\tau$ by a flip. Denote by $t_{N}$, resp. $t^{\prime}_{N}$, the
$N$-triangulation refinement of $\tau$, resp. $\tau^{\prime}$. It is an
exercise [Gon17, Prosition 1.1] to show that the Legendrian weave
$\Lambda(G(t^{\prime}_{N}))$ differs from $\Lambda(G(t_{N}))$ by a sequence of
${N+1\choose 3}$ 2-graph mutations, i.e. $\Lambda(G(t^{\prime}_{N}))$ can be
obtained from $\Lambda(G(t_{N}))$ by performing ${N+1\choose 3}$ Legendrian
mutations along 1-cycles represented by monochromatic edges.

For instance, [Gon17, Figure 9] translates into four monochromatic edge
mutations for a flip in a $N=3$ triangulation, as we have depicted in Figure
99. We can see how to perform the corresponding moves for $3$-triangulations
with $3$-graphs. Indeed, referring to the notation in Figure 105, perform a
monochromatic edge mutation at $z$ and $w$, then perform Move III, a flop of
the two trivalent and two hexagonal vertices in the center, and proceed with a
mutation at the remaining two monochromatic edges. In conclusion, the
constructions of this paper can therefore be used to give a geometric
understanding of the intermediate quivers arising when flipping
$N$-triangulations.

Figure 99. Flip in a 3-triangulation realized as four monochromatic edge
mutations. In general, the Legendrian weaves associated to two
$N$-triangulations which differ by a flip of the underlying (1-)triangulation
differ by a sequence of ${N+1\choose 3}$ such 2-graph mutations. Note the Move
III flop isotopy in-between the two mutation pairs.

###  7.3. N-graph Realization of Quiver Mutations

In this subsection we explain how to use $N$-graphs in order to construct
infinitely many Lagrangian fillings for certain Legendrian links in the
standard contact 3-sphere $(\mathbb{S}^{3},\xi_{\text{st}})$. These Lagrangian
fillings are distinguished by the microlocal monodromies/cluster coordinates
in Subsection 7.2.

Let $\beta\in\mbox{Br}^{+}_{N}$ be a positive braid, with a fixed braid word
$w(\beta)$. Consider an $N$-graph $G\subseteq\mathbb{D}^{2}$ on the 2-disk
such that the labels of the edges of $G$ near $\partial\mathbb{D}^{2}$, read
cyclically, form the word $w(\beta)$. Following Subsection 7.1, the Lagrangian
projection
$L(G)=\pi(\iota(\Lambda(G)))\subseteq(\mathbb{R}^{4},\omega_{\text{st}})$ of
$\iota(\Lambda(G))\subseteq(\mathbb{R}^{5},\xi_{\text{st}})$ is an exact
Lagrangian filling of the Legendrian link
$\Lambda(\beta)\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$ associated to the
positive braid $\beta$. All the $N$-graphs $G\subseteq\mathbb{D}^{2}$ which
feature in this subsection will be free, and thus the Lagrangian projections
are embedded, equivalently $\Lambda(G)$ has no Reeb chords.

Now consider a free $N$-graph $G\subseteq\mathbb{D}^{2}$,
$b_{1}(G):=\mbox{rk}(H_{1}(\Lambda(G),\mathbb{Z}))$ and a basis

$B=\\{[\gamma_{1}],\ldots,[\gamma_{b_{1}(G)}]\\}$

for $H_{1}(\Lambda(G),\mathbb{Z})\cong\mathbb{Z}^{b_{1}(G)}$, equivalently a
basis for the first homology group of its Lagrangian projection $L(G)$. For a
choice of basis $B$, we denote by $Q(B)$ the intersection quiver of the
1-cycles $\gamma_{i}$, $i\in[1,b_{1}(G)]$. The vertices $v_{i}$ of the quiver
$Q(B)$ are in bijection with elements of the homology basis $B$, and the
number of arrows between two distinct vertices $v_{i},v_{j}$ is given by the
geometric intersection number $|\gamma_{i}\cap\gamma_{j}|$. The direction of
each arrow is given by the sign of each geometric intersection, and there are
no loops, i.e. no edges from a vertex $v_{i}$ to itself. The quiver obtained
by mutation of a quiver $Q$ at the vertex $v_{i}$ will be denoted
$\mu_{i}(Q)$.

We will study the realization of quiver mutations, algebraic in nature, as
Legendrian mutations of free $N$-graphs, which are geometric. Suppose that
there exists a subset $B_{\mu}$ of classes $[\gamma_{i}]$, $i\in[1,k]$, for
some $k\leq b_{1}(G)$, such that $\gamma_{i}\in B_{\mu}$ is represented by a
3-graph cycle with no multiplicity. That is, each 1-cycle $\gamma_{i}$ is
represented by either a $\sf Y$-cycle, a tree, a monochromatic edge $\sf
I$-cycle or a long edge. Let $\\{x_{1},\ldots,x_{k}\\}$ be the cluster
coordinates associated to $\\{\gamma_{1},\ldots,\gamma_{k}\\}$ via microlocal
monodromies, as in Subsection 7.2.1.

######  Remark 7.12.

In general, this set of cluster coordinates $\\{x_{1},\ldots,x_{k}\\}$ is only
a partial subset of the entire cluster seed $\\{x_{1},\ldots,x_{b_{1}(G)}\\}$
for $H_{1}(L(G),\mathbb{Z})$. The ability to work with a subset is an
advantage that allows for our methods to be applied in more generality. From
the viewpoint of cluster algebras, the vertices of $Q(B)$ which are not in
$Q(B_{\mu})$ are to be considered as frozen vertices, and the variables
$\\{x_{k+1},\ldots,x_{b_{1}(G)}\\}$ as frozen coordinates. $\Box$

By Subsection 4.8, and Lemma 7.4, we can perform a Legendrian mutation at
$\gamma_{i}\in B_{\mu}$ and obtain a free $N$-graph $\mu_{i}(G)$. The
intersection quiver $Q(\mu_{i}(B))$ associated to the mutated basis
$\mu_{i}(B)$ is the mutated quiver $\mu_{i}(Q(B))$. The 1-cycle in the mutated
graph $\mu_{i}(G)$ corresponding to $\gamma_{j}\in B$, under mutation at
$\gamma_{i}$, is denoted by $\mu_{i}(\gamma_{j})$. By Subsection 7.2.2, the
cluster coordinate associated to $\mu_{i}(\gamma_{j})$ is given by the $j$-th
coordinate in the cluster transformation of $\\{x_{1},\ldots,x_{b_{1}(G)}\\}$
at $x_{i}$. Therefore, the exact Lagrangian filling represented by the free
$N$-graph $\mu_{i}(G)$ has intersection quiver $\mu_{i}(Q(B))$ and cluster
coordinates obtained by mutation of the cluster seed
$\\{x_{1},\ldots,x_{b_{1}(G)}\\}$ for $L(G)$ at $x_{i}$. In conclusion, if the
1-cycles are represented by trees, performing one quiver (or cluster seed)
mutation as a Legendrian mutation is possible, and the microlocal monodromies
after the Legendrian mutation accurately reflect cluster mutation.

######  Remark 7.13.

The challenging aspect of the geometric side is that iterating this procedure
is not necessarily possible, or at least readily accessible. This aspect is
not reflected in the algebra of quiver mutations (or cluster coordinate
mutations) since, by definition, two opposite edges between vertices are
canceled282828In previous attempts to geometrically iterate Lagrangian
mutations, such as [STW16, Section 2], this obstruction manifests itself as
embedded curves becoming immersed upon performing Dehn twists, a problem which
presently has no known solution.. $\Box$

The technology of 3-graphs and their mutations, as developed in Subsection
4.8, allows us to iterate Legendrian mutations in an abundance of cases,
including arbitrarily high genus. We will illustrate explicit cases in which
an infinite sequence of quiver mutations can be realized as an infinite
sequence of $N$-graph mutations. These cases can be inserted in (infinitely
many) other examples, and the first consequence is the production of new
families of Legendrian links with infinitely many exact Lagrangian fillings:

######  Theorem 7.14.

Let
$\Lambda_{s,t}=\Lambda(\beta_{s,t})\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$
be the Legendrian link given by the standard satellite of the positive braid

$\beta_{s,t}=(\sigma_{1}^{3}\sigma_{2})(\sigma_{1}^{3}\sigma_{2}^{2})^{s}\sigma_{1}^{3}\sigma_{2}(\sigma_{2}^{2}\sigma_{1}^{3})^{t}(\sigma_{2}\sigma_{1}^{3})(\sigma_{2}^{t+1}\sigma_{1}^{2}\sigma_{2}^{s+2}),\qquad
s,t\in\mathbb{N},s,t\geq 1.$

Then $\Lambda_{s,t}\subseteq(\mathbb{S}^{3},\xi_{\text{st}})$ admits
infinitely many embedded exact Lagrangian fillings in
$(\mathbb{D}^{4},\lambda_{\text{st}})$ realized as $3$-graphs
$G_{s,t}\subseteq\mathbb{D}^{2}$ and their Legendrian mutations.

###### Proof.

The argument is uniform for all $s,t\in\mathbb{N}$ and all the difficulties,
and their solutions, are already present for the simplest case.292929We thank
Dylan Thurston for useful discussions on quivers and their mutations. In
particular, for providing the infinite sequence of mutations that we use in
this proof. Let us thus assume $s=t=1$ for now. First, we need to construct a
free 3-graph $G=G_{1,1}$ which represents a Lagrangian filling for the
Legendrian link $\Lambda(\beta)$ associated to $\beta=\beta_{1,1}$. This
3-graph is shown in Figure 100:

Figure 100. The 3-graph $G$ and the initial Quiver $Q$.

The exact Lagrangian $L(G)$ associated to $G$ is a genus-4 surface with two
boundary components, and thus $b_{1}(G)=9$. Let us consider the subset
$B_{\mu}=\\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}\\}$ given
by the following 1-cycles: $\gamma_{1}$ is represented by the yellow 1-cycle
in Figure 100, which is a tree of $\sf Y$-pieces, and
$\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$ are represented by monochromatic
edges, in purple in Figure 100. In addition, we consider the 1-cycle
$\gamma_{6}$ represented by the monochromatic edge, in green. The intersection
quiver $Q=Q(B_{\mu}\cup\\{\gamma_{6}\\})$ is given by the quiver drawn in
Figure 100. The quiver $Q$ is of infinite mutation type, as it is
associated303030Precisely, the quiver $Q$ corresponds to the rank 6
paracompact hyperbolic Coxeter group $\overline{L}_{5}=[3^{[1,1,1,1,1]}]$. to
hyperbolic Coxeter diagram [Law17, Table 1]. In fact, we claim that the
sequence of quiver mutations $\mu_{s_{n}}$, where

$s_{n}=\begin{cases}1&\mbox{if }n\equiv 1\\\ 2&\mbox{if }n\equiv 2\\\
3&\mbox{if }n\equiv 3\\\ 4&\mbox{if }n\equiv 4\\\ 5&\mbox{if }n\equiv
5\end{cases}\pmod{5},$

is an infinite sequence of quiver mutations. Indeed, each time we apply the
sequence of mutations $\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}$, the number of
arrows from the vertex $v_{1}$ to $v_{6}$ increases by two, and the number of
arrows from $v_{i}$ to $v_{6}$, for $i\in[2,5]$ increases by one. In
particular, at the $k$th iteration there are $2k+1$ arrows from $v_{1}$ to
$v_{6}$ and $k$ arrows from $v_{i}$ to $v_{6}$, $i\in[2,5]$. Now we reach the
core of the issue, which is realizing this infinite sequence of quiver
mutations as Legendrian mutations of 3-graphs. For that, we observe the
following two properties:

Let us perform a Legendrian mutation along the $\sf Y$-tree which represents
the 1-cycle $\gamma_{1}$. The resulting 3-graph, which is free by Lemma 7.4,
is depicted in Figure 101:

Figure 101. Mutated 3-graph $\mu_{1}(G)$ at the 1-cycle $\gamma_{1}$ (yellow)
corresponding to vertex 1 in $Q$, and its associated intersection quiver
$\mu_{1}(Q)$.

Now, upon this Legendrian mutation at $\gamma_{1}$ the 1-cycles
$\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$ are still represented by
monochromatic edges. These new 1-cycles
$\mu_{1}(\gamma_{2}),\mu_{1}(\gamma_{3}),\mu_{1}(\gamma_{4}),\mu_{1}(\gamma_{5})$
are circled in purple in Figure 101. The figure also displays the mutated
quiver $\mu_{i}(Q(B_{\mu}\cup\\{\gamma_{6}\\}))$ and the cycle $\gamma_{6}$ in
green. Similarly, upon this Legendrian mutation, the 1-cycle
$\mu_{1}(\gamma_{1})$ is still represented by an embedded $\sf Y$-tree, as
depicted in yellow in Figure 101.

These properties hold true as we now perform Legendrian mutations at the
monochromatic edges $\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$. The free
3-graph resulting from these four mutations is drawn in Figure 102:

Figure 102. Mutated 3-graph $\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}(G)$ and its
associated intersection quiver $\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}(Q)$.

The claim is that we can iterate the sequence of mutations
$\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}$ geometrically as Legendrian mutation of
the free 3-graph, and these two properties hold. That is, at any stage in the
sequence of mutations $\mu_{s_{n}}$ we have that

  * (i)

the 1-cycles $\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$ are represented by
monochromatic edges,

  * (ii)

the 1-cycle $\gamma_{1}$ is represented by an embedded $\sf Y$-tree, with no
multiplicities.

In fact, the $\sf Y$-tree representing $\gamma_{1}$ always has exactly four
$\sf Y$-pieces. These four pieces have been surrounded by a dashed pink circle
in Figures 100 through 104. The two items above can be readily verified, as
follows. The behavior of the mutated $3$-graph near each of the the
monochromatic edges is as depicted in Figure 103:

Figure 103. The effect of the sequence of mutations
$\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}$ near the monochromatic edges
$\gamma_{i}$, $i\in[2,5]$. The only mutations from these five that affect
$\gamma_{i}$ are $\mu_{1}$ and $\mu_{i}$.

It thus follows that $\gamma_{i}$, $i\in[2,5]$, remains a monochromatic edge
upon any iteration of the $3$-graph mutation
$\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}$. Similarly, according to the Legendrian
mutation rules of Subsection 4.8, each $\sf Y$-piece of the $\sf Y$-tree
representing $\gamma_{1}$ itself mutates to a $\sf Y$-piece, and mutating at
$\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}$ preserves this property. Thus
the pattern persists upon any iteration. The two properties $(i)$ and $(ii)$
now allow us to perform the sequence of mutations $\mu_{s_{n}}$ up to any
point in the sequence. For instance, the sequence of mutations
$\mu_{1}\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}$ applied to $G$ lead to the
3-graph in Figure 104:

Figure 104. Mutated 3-graph $\mu_{1}\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}(G)$
and its associated intersection quiver
$\mu_{1}\mu_{5}\mu_{4}\mu_{3}\mu_{2}\mu_{1}(Q)$.

In order to pairwise distinguish the exact Lagrangian fillings associated to
the sequence of $3$-graphs $(\mu_{s_{n}}\mu_{s_{n-1}}\cdots\mu_{1})(G)$, up to
Hamiltonian isotopy, we use the microlocal monodromies
$\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\\}$ along the 1-cycles $\gamma_{i}$,
$i\in[1,6]$, and their mutations. By Subsection 7.2.2, the cluster seed
$\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\\}$ associated to the quiver $Q$
mutates to the cluster seed associated to
$(\mu_{s_{n}}\mu_{s_{n-1}}\cdots\mu_{1})(Q)$ upon performing the Legendrian
3-graph mutations $(\mu_{s_{n}}\mu_{s_{n-1}}\cdots\mu_{1})(G)$. Since the
quivers $(\mu_{s_{n}}\mu_{s_{n-1}}\cdots\mu_{1})(Q)$ are distinct, and so are
the associated cluster seeds, it follows that the associated Lagrangian
fillings are distinct. This concludes the proof for $s=t=1$.

The general case $s,t\in\mathbb{N}$ is proven with the same argument. Indeed,
the free 3-graph $G_{1,1}$ in Figure 100 generalizes to a 3-graph whose
boundary is $\beta_{s,t}$, just by adding $s$ copies of the leftmost pattern
in $G_{1,1}$, to the left, and $t$ copies of the rightmost pattern in
$G_{1,1}$, to the right. In this general case, it is still true that
$\gamma_{1}$ is represented313131In this case the $\sf Y$-tree has ($s+t+2$)
$\sf Y$-pieces, $s+1$ to the left and $t+1$ to the right of the base root. by
a $\sf Y$-tree and the remaining
$\\{\gamma_{2},\gamma_{3},\ldots,\gamma_{s},\gamma_{s+1},\ldots,\gamma_{s+t+3}\\}$
cycles are represented by monochromatic edges. The argument is then identical,
with the infinite sequence of mutations given by

$s_{n}=i,\qquad\mbox{if }n\equiv i\pmod{s+t+3},\quad 1\leq i\leq s+t+3.$

The reader can directly verify that this is an infinite sequence of mutations,
as the multiplicity of the arrows to the cycle $\gamma_{s+t+4}$ – generalizing
the green cycle $\gamma_{6}$ in Figure 100 – increases as we apply the
mutations $\mu_{s+t+3}\mu_{s+t+2}\cdots\mu_{2}\mu_{1}$. ∎

######  Remark 7.15.

For $s=t=1$, note that the sequence $\mu_{s_{n}}$ never mutates at the 1-cycle
$\gamma_{6}$, i.e. at the sixth vertex $v_{6}$ in $Q$. It is nevertheless
crucial to include $\gamma_{6}$ in the quiver as well as the cluster variable
$x_{6}$, with its subsequent mutations. Note that the 1-cycle $\gamma_{6}$ is
initially represented by an embedded curve in the 3-graph, but this curve
develops immersed points as we iterate the sequence of mutations $\mu_{s_{n}}$
according to Subsection 4.8. This still allows us to define the cluster
coordinate associated to it but we would not be able to mutate along such a
1-cycle just with the rules developed in Subsection 4.8. (This is just a side
remark, since the argument for Theorem 7.14 does not require mutating at
$v_{6}$.) $\Box$

The Legendrian links in Theorem 7.14 are relatively simple. For instance, the
Legendrian knot associated to $\beta_{1,1}$ is genus-4 two-component link. One
of the components is an unknot and the other is the $(2,7)$-torus knot
$7_{1}$. Note that $\Lambda(\beta_{1,1})$ is (smoothly) distinct from the
$(3,6)$-torus link that the first author studied in [CG20], which also has
genus-4. Thus, not only does Theorem 7.14 bring a new method to construct
infinitely many Lagrangian fillings, but it in fact provides new Legendrian
links with infinitely many Lagrangian fillings.

######  Remark 7.16.

Note that the $\overline{L}_{5}$ quiver that we used in Theorem 7.14 appears
as a subquiver of the intersection quiver for several other positive braids.
Following L. Lewark’s positive braid table323232Lukas Lewark’s Positive Knots
Table: Braids and Trees at “http://lewark.de/lukas/braids.html ”. each of the
following positive genus-6 braids, $14n_{5644}$, $15n_{118169}$,
$16n_{144958}$, $16n_{149517}$, $16n_{173894}$, $16n_{175324}$ and
$16n_{339638}$, to name a few, contain $\overline{L}_{5}$ in their
intersection quiver. We believe that an argument similar to Theorem 7.14
should prove that the maximal-tb representative of each of these links has
infinitely many exact Lagrangian fillings. $\Box$

Finally, the contrast between Theorem 7.14 and [CG20, Corollary 1.5] is
interesting. The former constructs an infinite family of Lagrangian fillings
for a Legendrian link by directly using Legendrian mutations, which are
themselves distinguished by their effect – as cluster mutations – on the
microlocal monodromies. The latter result [CG20] is entirely about
constructing infinite order Lagrangian concordances, coming from Legendrian
loops of positive braids, and the infinite family of Lagrangian fillings is a
byproduct of such construction. In particular, $N$-graph calculus should apply
to much more general Legendrian links, and does not require knowing about the
existence of an infinite order element in their Lagrangian concordance monoid.

##  8\. Moduli Space for $N$-triangles and Non-Abelianization

In this final section, we focus on $N$-graphs associated to
$N$-triangulations, as introduced in Section 3. This class of $N$-graphs $G$
yields Legendrian weaves $\Lambda(G)$ whose Lagrangian projections are related
to the Goncharov-Kenyon conjugate Lagrangian surfaces [GK13, STWZ19]. These
Lagrangian surfaces have also appeared in the context of Gaiotto-Moore-
Neitzke’s spectral networks [GMN13, Nei14]. In particular, we prove Theorem
1.9, which computes the flag moduli space $\mathcal{M}(G)$ for $G$ any
$N$-triangle $t_{N}$, matching the algebraic results in [GMN14, Section 8] and
[FG06b, Section 9].

###  8.1. Flag moduli space of the $N$-triangle

Let us compute the flag moduli space associated to the $N$-graph $G(t_{N})$ of
an $N$-triangle $t_{N}$, as we defined in Section 3.2 (see Figure 15). The
result reads as follows:

######  Theorem 8.1.

Let $G(t_{N})$ be the $N$-graph associated to an $N$-triangle $t_{N}$. The
flag moduli space of $G(t_{N})$ is a ${N-1\choose 2}$-dimensional complex
torus, i.e.

$\mathcal{M}(t_{N},G(t_{N});k)\cong(k^{*})^{{N-1\choose 2}}.$

This rest of this subsection is devoted to the proof of Theorem 8.1. The
statement of Theorem 8.1 is an instance of how incidence geometry problems
connect to the contact topology of Legendrian surfaces. Indeed, although our
proof is entirely within projective geometry, the conclusion from Theorem 8.1
ought to be read as the fact that the moduli space
$\mathcal{M}(t_{N},G(t_{N});k)$ is parametrized by the toric coordinates
provided by the holonomies
$\mbox{Hom}(H_{1}(\Lambda(G(t_{N}),\mathbb{Z})),k^{*})$. For $k=\mathbb{C}$,
this complex torus should be related to the complex torus appearing in Fock-
Goncharov [FG06b, FG06a] in their study of cluster varieties, see [Kuw20,
Theorem 8.3].

Theorem 8.1 can also be interpreted as follows. The triangle $t_{N}$ is
topologically a disk $\mathbb{D}^{2}$ with boundary a circle
$\partial\mathbb{D}^{2}=\mathbb{S}^{1}$. The Legendrian weave

$\Lambda(G(t_{N}))\subset(J^{1}(\mathbb{D}^{2}),\xi_{\text{st}})$

has a Lagrangian projection $L:=\pi(\Lambda(G(t_{N})))$, which is an exact
Lagrangian submanifold, where $\pi:J^{1}(\mathbb{D}^{2})\longrightarrow
T^{*}\mathbb{D}^{2}$ is the projection along the standard (vertical) Reeb
flow. The Lagrangian $L$ has boundary in $T^{*}\mathbb{D}^{2}|_{S^{1}}\cong
J^{1}(S^{1}),$ and it is checked that $\partial L$ is the cylindrical
Legendrian braid $\Delta^{3},$ where $\Delta$ is the half-twist positive braid
corresponding to a longest word in the Weyl group, i.e. the Garside element.
Since $G(t_{N})$ is free, $L$ is an embedded exact Lagrangian filling of
$\partial L$. Now, by looking at the boundary circle $\mathbb{S}^{1}$ and
considering the moduli space à la [STZ17], we conclude that the moduli space
of Lagrangian fillings should carry a cluster structure: the flag moduli space
$\mathcal{M}(t_{N},G(t_{N});k)$ is one such chart.

In fact, by an argument akin to Lemma 5.6, the flags at two vertices of the
triangle $t_{N}$ determine the flags along the edge they bound, and therefore
the flags along the boundary circle $\delta\mathbb{D}^{2}=\mathbb{S}^{1}$ must
be determined by the flags at the vertices, themselves three mutually
completely transverse flags in the flag variety $\mathcal{B}$. This space of
triples of mutually transverse flags is one of the Richardson
varieties333333We thank Ian Le for many discussions on the Richardson variety.
$\mathcal{R}$. Now, by the $\operatorname{PGL}_{N}$ action, two totally
transverse flags can be put in standard position $B$, $B^{-}$, with residual
symmetry the Cartan $H$ of diagonal matrices up to scale. Then the moduli
space $\mathcal{R}/H$ is a cluster variety and the exact Lagrangian filling
$L$ provides a cluster chart via its moduli of local systems

$Loc(L)\cong\mbox{Hom}(H_{1}(\Lambda(G(t_{N}),\mathbb{Z})),k^{*})\cong(k^{*})^{{N-1\choose
2}}.$

This torus can be checked to agree with that of 8.1. Note also that, following
Section 7 many other cluster charts and exact Lagrangian fillings can be found
by performing $N$-graph mutations. Let us now prove our result:

###### Proof of Theorem 8.1.

Let us argue by induction on $N$, where the base case $N=2$ follows from the
fact that $\operatorname{PGL}_{2}$ acts transitively on triples of distinct
points. Let us assume that

$\mathcal{M}(t_{N},G(t_{N});k)\cong(k^{*})^{{N-1\choose 2}}$

for the $N$-graph of an $N$-triangle. Consider an $(N+1)$-triangle with one
side being an arbitrary fixed preferred base, and thus the row associated to
this base contains $2N-3$ triangles. It is combinatorially apparent that the
complement of this row in $t_{N+1}$ is in fact an $N$-triangle $t_{N}$, and
thus we can construct $t_{N+1}$ by adding such base row to $t_{N}$. This
combinatorial splitting is translated into a containment of an $N$-graph
$G(t_{N})$ within $G(t_{N+1})$. Let us describe such splitting in the
$(N+1)$-graph by providing its construction starting from the $N$-graph
$G(t_{N})$.

Start with the $N$-graph $G(t_{N})$ – see Section 3.2 – and consider the
${N\choose 2}$ edges intersecting the base side of $t_{N}$. The edges are
depicted vertically and the base side horizontally – see Figure 15. These are
$\tau_{i}$-edges, $i=1,\ldots,N-1$, with exactly $(N-1-k)$ $\tau_{k}$-edges.
The $(N+1)$-graph $G(t_{N+1})$ can be described in the following $N$ stages:

  * 1.

First, insert an $(N-1,N)$-hexagonal point in the unique $\tau_{N-1}$ edge in
the base side of $G(t_{N})$. The $\tau_{N}$-edge aligned with the previously
existing $\tau_{N-1}$-edge is continued down vertically. The remaining two
$\tau_{N}$-edges are extended horizontally, respectively to the left and to
the right, and the remaining two $\tau_{N-1}$ edges are continued down
diagonally, in south-east and south-west direction respectively.

  * 2.

Second, continue down the $\tau_{i}$-edges, $i=1,\ldots,N$, until the two
$\tau_{N-1}$-edges intersect with the two originally existing
$\tau{N-2}$-edges. In the moment of collision, insert two
$(N-2,N-1)$-hexagonal vertices at the intersection point matching the two
incoming $\tau_{N-1}$ and $\tau{N-2}$ trajectories. We extend the two
$\tau_{N-1}$-edges adjacent to the incoming $\tau_{N-2}$-edges horizontally to
the left and to the right.

The remaining two pairs of three edges, each with two $\tau_{N-2}$-edges and a
$\tau_{N-2}$-edge, are continued down, with the $\tau_{N-1}$-edges continued
vertically and the $\tau_{N-2}$-edges continued diagonally in the south-east
or south-west directions, accordingly.

  * 3.

Iteratively, we proceed as follows in the $l$th stage, $2\leq l\leq N-1$. We
continue down the $\tau_{i}$-edges, $i=1,\ldots,N$, and at this stage the only
edges being continued diagonally down are $\tau_{N-l+1}$-edges. There are
$2l-2$ of such edges, which can be gathered in two groups, internal and
external.

By definition, there are two external edges, which are the leftmost and
rightmost $\tau_{N-l+1}$-edges, respectively continuing south-west and south-
east. For these two external edges, we insert two $(N-l,N-l+1)$-hexagonal and
describe the $N$-graph as described in Stage 2. The $2l-4$ internal
$\tau_{N-l+1}$-edges, which continue down diagonally, ought to intersect with
$\tau_{N-l}$-edges, which continue down vertically.

For the $2(l-2)$ internal edges, there are $(l-2)$ such intersections, since
an intersection occurs for each pair. For each such an intersection, insert a
$(N-l,N-l+1)$-hexagonal vertex, and continue the outgoing three edges down as
described by the local model for the hexagonal vertex. Hence, for each of
these hexagonal vertices, the outgoing $\tau_{N-l+1}$-edge continues
vertically down whereas the two $\tau_{N-l}$-edges continue down diagonally.
Thus, at the $l$th stage we have inserted exactly $l$ $(N-l,N-l+1)$-hexagonal
points.

  * 4.

In the $N$th stage, all $\tau_{i}$-edges, $i\leq 2\leq N$ continue down
vertically and we are left with $2(N-1)$ $\tau_{1}$-edges continuing
diagonally. In line with the previous stages, there are two external
$\tau_{1}$-edges and $2(N-2)$ internal edges. Insert two $\tau_{1}$-trivalent
vertices at the end of the two external $\tau_{1}$-edges. The internal edges
will meet in consecutive pairs at $N-2$ intersection points. In this final
stage we insert a $\tau_{1}$-vertex in each of these intersection points, and
continue the remaining $\tau_{1}$-edge vertically down.

Let us now compute the flag moduli space $\mathcal{M}(t_{N},G(t_{N+1});k)$
using this inductive construction of $G(t_{N+1})$. A crucial fact to be used
is Lemma 5.6, i.e. at a hexagonal vertex, four consecutive flags uniquely
determine the remaining two flags. Let us assume that we have chosen a point
in $\mathcal{M}(t_{N},G(t_{N});k)$ and we thus have the data of a flag
$\mathscr{F}$ in $\mathbb{P}^{N}$ for each open region in
$\mathbb{D}^{2}\setminus G(t_{N})$. This data needs to be considered in the
moduli space of flags, given that $G(t_{N+1})$ is an $(N+1)$-graph, and thus
we fix an embedding $i_{0}:\mathbb{P}^{N}\longrightarrow\mathbb{P}^{N+1}$ and
the corresponding inclusion
$\operatorname{PGL}_{N}\subseteq\operatorname{PGL}_{N+1}$. Let us then start
the construction $G(t_{N+1})$ from $G(t_{N})$ by stages, as described above,
and prove the statement in Theorem 8.1.

In the first stage, the flag data at the inserted $(N-1,N)$-hexagonal point in
the $\tau_{N-1}$-edge is uniquely determined by a choice of a codimension-2
projective subspace $H^{2}$ in $\mathbb{P}^{N}$, transverse to
$i_{0}(\mathbb{P}^{N})$. Note that the intersection of $H^{2}$ and
$i_{0}(\mathbb{P}^{N})$ is uniquely determined by the flag data coming from
$\mathcal{M}(t_{N},G(t_{N});k)$. We claim that this choice in the first stage
can be absorbed by the symmetry group $\operatorname{PGL}_{N+1}$.

In order to understand the symmetry group, it is convenient to represent an
element in $\operatorname{PGL}_{N+1}$ via the projective matrix

$\left[\begin{array}[]{cccccc}a_{1,1}&a_{1,2}&\ldots&a_{1,N}&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&a_{1,N+1}\\\
a_{2,1}&a_{2,2}&\ldots&a_{2,N}&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&a_{2,N+1}\\\
\vdots&\vdots&\ddots&\vdots&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&\vdots\\\
a_{N,1}&a_{N,2}&\ldots&a_{N,N}&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&a_{N,N+1}\\\
\cline{1-6}\cr
a_{N+1,1}&a_{N+1,2}&\ldots&a_{N+1,N}&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&a_{N+1,N+1}\\\
\end{array}\right],$

where the subgroup $\operatorname{PGL}_{N}\subseteq\operatorname{PGL}_{N+1}$
is defined by

$\operatorname{PGL}_{N}=\\{A\in\operatorname{PGL}_{N+1}:a_{N+1,N+1}=1,a_{i,N+1}=a_{N+1,i}=0,1\leq
i\leq N\\}.$

In these coordinates, we can assume that the subgroup
$K\subseteq\operatorname{PGL}_{N+1}$ fixing our fixed hyperplane
$i_{0}(\mathbb{P}^{n})\subseteq\mathbb{P}^{N-1}$ is cut out by the equations

$K:=\\{A\in\operatorname{PGL}_{N+1}:a_{N+1,i}=0,1\leq i\leq N\\}.$

As a result, the remaining $\operatorname{PGL}_{N+1}$-symmetries (once the
flag moduli space $\mathcal{M}(t_{N},G(t_{N});k)$ is fixed, and thus the
symmetries of $\operatorname{PGL}_{N}$ have been used) consist of projective
transformations of the form

$\left[\begin{array}[]{cccccc}a_{1,1}&a_{1,2}&\ldots&a_{1,N}&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&c_{1}\\\
a_{2,1}&a_{2,2}&\ldots&a_{2,N}&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&c_{2}\\\
\vdots&\vdots&\ddots&\vdots&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&\vdots\\\
a_{N,1}&a_{N,2}&\ldots&a_{N,N}&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&c_{N}\\\
\cline{1-6}\cr
0&0&\ldots&0&\lx@intercol\hfil\lx@intercol\vrule\lx@intercol&c_{N+1}\\\
\end{array}\right],$

where $a_{i,j}$ are fixed, $1\leq i,j\leq N$, $c_{i}\in k$, $1\leq i\leq N$,
and $c_{N+1}\in k^{*}$ are free. Then, in this coordinate system, we can
assume that the choice of the codimension-2 projective subspace $H^{2}$ uses
the gauge provided by $c_{1},c_{2}\in k$.

In the second stage, two $(N-2,N-1)$-hexagonal vertices are inserted. For each
of them, the flag data is fixed by induction in three out of the six regions
near the hexagonal vertex. Hence, there is a choice of a codimension-3
projective subspace $H^{3}$ in each of these two vertices. Let us fix one of
these choices by using the free coordinate $c_{3}\in k$ and notice that the
other choice has an a priori moduli of $k$. Nevertheless, the
$\tau_{N-3}$-edge that interacts with the $\tau_{N-2}$ edges in the third
stage forces that moduli to be $k^{*}$, since the two newly chosen flags must
be $\tau_{N-3}$-transverse. Thus in the second stage we have used the symmetry
provided by $c_{3}\in k$ and we are left with a $k^{*}$ contribution to the
flag moduli.

In the $l$th stage, $3\leq l\leq N-1$, we proceed inductively as follows. We
partition the $(N-l,N-l+1)$-hexagonal vertices inserted in this stage into two
groups: external, containing two of them, and internal, containing $(l-2)$ of
them. By definition, the two external $(N-l,N-l+1)$-hexagonal vertices are the
leftmost and rightmost vertices. Each of these two external vertices have
flags fixed in three out of the six regions, by the process in the $(l-1)$st
stage. Thus, as in the second stage, there is exactly one choice of flag at
each of these $(N-l,N-l+1)$-hexagonal vertices which determines each of their
respective neighborhoods. This corresponds to a choice of codimension-$l$
projective subspace $H^{l+1}$ in accordance with the incidence conditions
imposed by the given flags. Proceeding as in the second stage, we fix one of
these choices with the free variable $c_{l+1}$ and the remaining choice
contributes $k^{*}$ to the flag moduli.

The $(l-2)$ internal $(N-l,N-l+1)$-hexagonal vertices have flags fixed in four
out of the six regions, given the process in the $(l-1)$st stage. By Lemma
5.6, these hexagonal vertices are uniquely determined in their neighborhoods.
Thus, although the $k^{*}$ contribution of one of the external hexagonal
vertices interacts with an internal vertex, no contributions to the flag
moduli space come directly from the internal vertices.

The argument then develops iteratively in the above manner until the $(N-1)$st
stage is completed. The last $N$th stage consists of the insertion of $N$
$\tau_{1}$-trivalent vertices. Following the same pattern as before, only the
two external trivalent vertices contribute to the flag moduli, since each of
the internal trivalent vertices have their three surrounding flags determined
at the $(N-1)$st stage. In this last stage, the variables $c_{i}$, $1\leq
i\leq N$ have been fixed and the only remaining degree of free symmetry is
$c_{N+1}\in k^{*}$. Let us use such symmetry to fix the choice in one of the
two external trivalent vertices, and thus the contributions of this last stage
to the flag moduli space is the $k^{*}$ choice of the remaining point coming
from the remaining external trivalent vertex.

The conclusion in the statement Theorem 8.1 now follows by gathering the
contributions of the flag moduli space at each stage. Indeed, the first stage
has no contribution, whereas each of the $(N-1)$ stages, from the second to
the last $N$th stage, has a $k^{*}$ flag moduli space contribution. By the
inductive hypothesis, the desired flag moduli space is

$\mathcal{M}(t_{N+1},G(t_{N+1});k)\cong\mathcal{M}(t_{N},G(t_{N});k)\times(k^{*})^{N-1}\cong(k^{*})^{{N-1\choose
2}}\times(k^{*})^{N-1}\cong(k^{*})^{{N\choose 2}},$

which corresponds to the statement, as required. ∎

######  Remark 8.2.

Note that the inductive combinatorial description of $G(t_{N+1})$ in terms of
$G(t_{N})$ used in the proof of Theorem 8.1 can be used to provide a third
alternative definition of the local $N$-graph $G(t_{N})$, in addition to the
descriptions introduced in Subsection 3.2. $\Box$

####  8.1.1. Tetrahedral Triangulations at $N=3$ and $N=4$

Let us study the Legendrian weaves $\Lambda(G(\tau))$ and flag moduli space
$\mathcal{M}(\mathbb{S}^{2},G(\tau))$ associated to 3- and 4-graphs $G(\tau)$
for the tetrahedral 3- and 4- triangulations $\tau$ of the 2-sphere
$\mathbb{S}^{2}$. The case $N=2$ has been discussed in Subsection 6.1 above,
where $\Lambda(G)\cong\mathbb{T}^{2}_{c}$ is the Legendrian Clifford Torus and
$\mathcal{M}(G)\cong\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$. Let us denote
the pair of pants $\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ by $\mathscr{H}$.

Let us consider the 3-graph $G^{(3)}=G(\tau^{(3)})\subseteq\mathbb{S}^{2}$
associated to the tetrahedral 3-triangulation $\tau^{(3)}$ of the 2-sphere
$\mathbb{S}^{2}$, according to the construction in Section 3. We want to
compute its flag moduli space $\mathcal{M}(G)$. This will be done directly by
using the $N$-graph calculus computations in Section 4. Indeed, it is proven
in Subsection 4.6 that in this case the (satellite of the) Legendrian weave
$\Lambda(G)$ is Legendrian isotopic to the four-fold connected sum of the
Clifford torus $\mathbb{T}_{c}^{2}$. Hence, we obtain that
$\mathcal{M}(\mathbb{S}^{2},G^{(3)})\cong\mathscr{H}^{4}$. From the
description in Theorem 8.1, we are also giving a contact geometric proof of
the following

######  Corollary 8.3 ([FG06b]).

The moduli of four generic flags in $\mathbb{C}^{3}$ is isomorphic to
$\mathscr{H}^{4}$. $\Box$

The same argument, using $N$-graph calculus also allows us to study the flag
moduli space $\mathcal{M}(\mathbb{S}^{2},G^{(4)})$, where
$G^{(4)}=G(\tau^{(4)})\subseteq\mathbb{S}^{2}$ is the 4-graph associated to
the tetrahedral 4-triangulation $\tau^{(4)}$ of the 2-sphere $\mathbb{S}^{2}$.
It is left as an exercise for the reader to use Theorem 8.1 and conclude that
$\mathcal{M}(\mathbb{S}^{2},G^{(4)})$ is isomorphic, as an algebraic variety,
to

$\mathcal{M}(\mathbb{S}^{2},G^{(4)})=\\{(z_{1},w_{1},\ldots,z_{5},w_{5})\in(\mathbb{C}^{*})^{9}:(1-\kappa)w_{i}z_{i}-z_{i}+1=0,1\leq
i\leq 5\\}\times(\mathscr{H})^{4},$

where $\kappa=1-z_{1}z_{2}z_{3}z_{4}z_{5}\in\mathbb{C}^{*}$. The exercise is
solved in [DGG16, Section 6.3.2] in the language of the $3d$ $N=2$
superconformal field theory $T_{4}[\Delta,\Pi]$.

###  8.2. A Computation of the Non-Abelianization Map

We conclude the main body of the manuscript by exploring the relationship
between Legendrian weaves and the works [FG06b, AV00, AV12, Pal15] in some
explicit examples. In particular, we present a case in which the non-
Abelianization map featured in [GMN13, GMN14] can be realized by the
microlocal monodromies associated to constructible sheaves microlocally
supported along Legendrian weaves.

The context is described as follows. Let $(C,\tau_{N})$ be a polygon endowed
with an ideal $N$-triangulation $\tau_{N}$, and choose a wavefront for
$\Lambda(G(\tau_{N}))$ with no Reeb chords, such that the Lagrangian
projection is a smooth exact Lagrangian $L$ embedded in the cotangent bundle
$(T^{*}C,\lambda_{\text{st}})$. This Lagrangian projection $L$ has a sheaf
quantization [NZ09] to a rank-$N$ sheaf on $C$ with no singular support, i.e.
a local system in $C$. Now, of course, all local systems on polygons are
trivial, but the crucial point is that the Lagrangian covering gives a
preferred343434In the Floer-theoretic languange of the Fukaya category, the
basis elements are the intersections of the exact Lagrangian with the
cotangent fibers. basis for the fibers of the local system, which can undergo
changes à la handle-slides in the Morse context – see [GKS12]. Here, the
Lagrangian covering is given by the restriction $\pi|_{L}:L\longrightarrow C$
of the projection $\pi:T^{*}C\longrightarrow C$ onto the zero section. Now,
the $N$-graphs and the microlocal monodromies, as discussed in Section 7.2,
precisely encode these changes. In our context, the non-Abelianization map is
the construction that recovers the constructible sheaf from its microlocal
monodromy.

We illustrate this in the following example. Figure 105 shows the 3-graph $G$
associated to two adjacent 3-triangles. Suppose that we are given a local
system on $\Lambda(G)$. Denote by $x,y$ the two monodromies of the
corresponding Legendrian weave around the two Y-cycles, and $z,w$ the two
microlocal monodromies along the two $\sf I$-cycles represented by the two
(red) monochromatic edges.

Figure 105. The 3-graph associated to an adjacent pair of $3$-triangles. The
monodromies along the four 1-cycles are labeled $x,y,z,w$.

The Legendrian weave $\Lambda(G)$ is a thrice-punctured genus-one surface and
these four 1-cycles are a basis for $H_{1}(\Lambda(G),\mathbb{Z})$. We would
like to reconstruct the flag data, specifying a constructible sheaf, from the
monodromies $x,y,z,w$ of the local system. Indeed, this will realize the Non-
Abelianization map [GMN13] from rank-one local systems on the (spectral, or
conjugate) Lagrangian – parametrized by monodromies – to decorated rank-two
local systems on the base surface. Since the base surface here is
contractible, the only degrees of freedom are the choices of flags at
vertices. The map is computed as follows.

Let $(a,A),(b,B),(c,C)$ be the flags at the vertices of the left triangle, and
let $(d,D)$ be the remaining flag. We would like a birational map from the
monodromies $(x,y,z,w)$ to the choice of flags. By using the
$\operatorname{PGL}_{3}$-action, we may assume $(a,A),(b,B),(c,C)$ are as in
Subsection 7.2.2 above, with triple product $x$. Then the flag $(c,D)$ is
determined by the cross ratios $z$ and $w$, and the triple product $y$. For
instance, $z$ is the cross ratio $\langle b,BC,AB,BD\rangle$ while we find
$w=\langle a,AD,AB,AC\rangle.$ These determine $D$, whence the triple product
$y$ fixes $d\in D$. Direct computation shows

$d=\begin{pmatrix}-\frac{x}{q}(1+x)\\\ x(1+y)\\\ -py(1+x)\end{pmatrix},\quad
D=(pq,p(1+x),x).$

This thus recovers [FG06b, Pal15] from the perspective of $N$-graphs.

### Example: Tetrahedron with 3-Triangulation.

Let us conclude this subsection by analyzing the genus-4 Legendrian weave
$\Lambda$ in Example 4.6 from the microlocal perspective. We also compute,
following [TZ18], the primitive which characterizes (a discrete cover of)
$\mathcal{M}(G)$ as an exact Lagrangian subvariety. Following [AV00, AV12],
this primitive – the superpotential of an effective 4d theory – is interpreted
as a generating function of BPS numbers, and should have integrality
properties. We check this for this example.

Consider the tetrahedron with its unique 3-triangulation, as in Example 4.6,
which gives rise to a 3-graph $G$. An object in the category of simple
constructible sheaves
$Sh^{1}_{\Lambda(G)}(\mathbb{S}^{2}\times\mathbb{R},\Lambda)_{0}$ microlocally
supported along $\Lambda(G)$ is defined by a four-tuple of transverse flags in
$V\cong\mathbb{C}^{3}$, placed at the vertices of the tetrahedron, as in
Figure 94.

Note that there are $4\cdot 3=12$ total nodes, and the Legendrian surface
indeed has genus $g=4$. We therefore have $2g=8$ cluster variables, specified
by the monodromies around each of the eight loops, which themselves are a
basis for $H_{1}(\Lambda(G),\mathbb{Z})\cong\mathbb{Z}^{8}$. Four of the
monodromies are the triple ratios along the faces. Let us label the faces by
the three unordered vertices it contains, e.g. we write $x_{123}$ for the
monodromy of the loop deteremined by the minimal triangle at the center of the
face (123): it is the triple ratio of the three flags at vertices 1, 2 and 3.
There are $4\times 3=12$ more edge monodromies, but we will find $4\cdot 2=8$
relations among all these 16 total, giving 8 independent monodromies as
expected for a genus-$4$ surface. Let us compute the edge monodromies.

First, following [TZ18], for each edge $e$ we define a corresponding
coordinate $x_{e}$ to be the _negative_ of the cross ratio.353535We believe
the sign appears due to the fact that we should be considering _twisted_ local
systems, i.e. lifts to the circle bundle of the surface that have monodromy
$-1$ over the circle fibers, as in [FG06b] and [GMN13, Section 10]. Now there
are two relations for each vertex: first, the product of the edge coordinates
around the encircling triangular face is unity; second, the product of edge
and ${\sf Y}$-monodromies encircling the vertex at a greater distance is
unity. There are thus $8$ independent coordinates, and we can take two from
each of the triangles surrounding the four vertices. Let us then write

$x_{12}=-\frac{w_{1}\wedge v_{4}}{v_{4}\wedge v_{2}}\frac{v_{2}\wedge
v_{3}}{v_{3}\wedge w_{1}}$

for the coordinate associated to the edge of the triangle encircling vertex
$1$ and traversing the one-simplex of the triangulation between vertices $1$
and $2$, where $v_{i}$, $i\in[1,4]$, are generators of lines and $w_{i}$ are
generators for planes (thought of as anti-symmetric two-vectors) — and
likewise for the other edges. Then the relation for the triangle encircling
vertex $1$ is $x_{12}x_{13}x_{14}=1$, and likewise for the other vertices.
Recall that we have similarly denoted by $x_{123}$ the inverse of the
coordinate corresponding to the Y-cycle in the face containing vertices $1$,
$2$ and $3$ — and likewise for the coordinates $x_{ijk}$, $i,j,k\in[1,4]$. For
the first vertex we have the unital relation

$x_{123}x_{21}x_{142}x_{41}x_{134}x_{31}=1,$

and likewise for the other three vertices. This expresses the flag moduli in
terms of generators, given by $x_{ij},x_{ijk}$, and relations, as above.

Let us verify that these coordinates define a (holomorphic) Lagrangian
embedding of flag moduli space $\mathcal{M}(G)$ associated to the genus-4
Legendrian $\Lambda(G)$ into the moduli space of framed local systems for
$C=\mathbb{S}^{2}$. The symplectic 2-form is computed from the intersection
form to be

$\omega=-d\log x_{12}\wedge d\log x_{13}+d\log x_{23}\wedge d\log x_{24}-d\log
x_{34}\wedge d\log x_{31}+d\log x_{41}\wedge d\log x_{42}.$

We can directly compute the following four relations

$x_{12}=\frac{-1}{1+x_{13}},\qquad x_{24}=\frac{-1}{1+x_{23}},\qquad
x_{34}=\frac{-1}{1+x_{31}},\qquad x_{42}=\frac{-1}{1+x_{41}},$

which readily imply that the embedding of the flag moduli space
$\mathcal{M}(G)$ in each of the cluster charts for the moduli space of framed
local systems is Lagrangian. This holomorphic Lagrangian $\mathcal{M}(G)$ is
in fact exact and we can compute a primitive function $W$ for the restriction
of the Liouville 1-form $\lambda_{\text{st}}$. This would allow us to write
$\mathcal{M}(G)$ as the graph $\Gamma_{dW}$ of the 1-form $dW$ in this chart.
This primitive encodes the BPS states associated to some Lagrangian filling,
given by the Lagrangian projection of $\Lambda(G)$, determined by a phase and
a framing (implicit here) as in [TZ18, Section 4.8] – see also [AV00, AV12].
For that, we define the variables

$U_{1}=-x_{13},\quad V_{1}=-x_{12},\quad U_{2}=-x_{23},\quad V_{2}=-x_{24},$
$U_{3}=-x_{31},\quad V_{3}=-x_{34},\quad U_{4}=-x_{41},\quad V_{4}=-x_{42}.$

Also, recall that if we have $U+V^{-1}=1$ with $U=e^{u}$ and $V=e^{v}$, then
we can write

$v=-\log(1-U)=\partial_{u}{\rm Li}_{2}(U).$

Hence, since we have $U_{i}+V_{i}^{-1}=1$ for all $i$, with symplectic 2-form
$\omega=\sum_{i}du_{i}\wedge dv_{i}$, we conclude that
$\mathcal{M}(G)=\Gamma_{dW}$ where

$W(U_{1},U_{2},U_{3},U_{4})=\sum_{i=1}^{4}{\rm Li}_{2}(U_{i}).$

This computation for the BPS potential is in line with the results in [TZ18,
Section 5].

Finally, let us review how geometric methods, as developed in Section 4, would
lead to this result. Instead of the algebraic computation above, we could have
directly used the diagrammatic calculus, as in Example 4.6, and deduced that
our Legendrian weave $\Lambda(G)\cong\\#_{i=1}^{4}\mathbb{T}^{2}_{c}$ is the
Legendrian connected sum of four Clifford 2-tori $\mathbb{T}^{2}_{c}$. Since
the generating function of BPS numbers for $\mathbb{T}^{2}_{c}$ is given by
one dilogarithm ${\rm Li}_{2}(U)$, by direct computation, and the potential
$W$ is additive under connected sum, we could have directly deduced that
$W(U_{1},U_{2},U_{3},U_{4})=\sum_{i=1}^{4}{\rm Li}_{2}(U_{i})$. This concludes
that our algebraic computation above is consistent with the contact topology
of the underlying Legendrian weave.

##  Appendix A Soergel Calculus and Legendrian Weaves

In this appendix, we provide a construction and a concise speculation
regarding the symplectic geometrization of Soergel Calculus via Legendrian
weaves. The following discussion owes a good deal to B. Elias and E. Gorsky,
as explained in the introduction, to whom we are very grateful. Soergel
calculus, as developed by B. Elias, M. Khovanov and G. Williamson [EK10,
EW16], provides a diagrammatic presentation of the category of Soergel
bimodules, which itself categorifies the Hecke algebra. The similarities
between Elias’ diagrammatic calculus and our Legendrian weaves are apparent.
Legendrian weaves can be understood as a geometric approach to the study of
the algebra of certain complexes of Soergel bimodules. We explain this below.

######  Remark A.1.

Soergel bimodules are essential to categorifications of knot invariants
[Rou06, Soe07, Kho07, EK10]. The link between these and moduli spaces of
sheaves for Legendrian braid closures was described in [STZ17, Section 6].
From this perspective, it is not unnatural to seek a connection between planar
Soergel structures and planar structure defined by Legendrian weaves, the two-
dimensional version of braids. $\Box$

The category of Soergel bimodules is the Karoubi completion of the subcategory
of Bott-Samelson bimodules, arising as the equivariant cohomology of a closed
Bott-Samelson variety, and thus it suffices to understand the relation to this
latter class of bimodules. The key connection between the present work and
Soergel bimodules is that a subclass of Legendrian weaves yields exact
Lagrangian cobordisms between Legendrian links, which are themselves
represented as positive braids. The moduli space of microlocal constructible
sheaves supported on a singular compactification of a positive braid is a
closed Bott-Samelson variety, and our Legendrian weaves, understood as
Lagrangian cobordisms – and singularly compactified – induce morphisms between
these closed Bott-Samelson varieties.

Thus, we are able to geometrize the diagrammatics of Soergel calculus by
considering the $D_{4}^{-}$-singularity for the trivalent vertices in [EK10,
EW16], the $A_{3}$-swallowtail singularity for the univalent vertex and the
$A_{1}^{3}$-singularity for their hexagonal vertices. (The Soergel calculus we
geometrize corresponds to the $m=2$ Coxeter exponent.)

######  Remark A.2.

Exact Lagrangian cobordisms are directed, due to the convexity directionality
in symplectic topology. The dissonance arises from the fact that, as of today,
Soergel calculus only considers closed Bott-Samelson varieties, whereas the
moduli space of microlocal sheaves supported on a positive braid is an open
Bott-Samelson variety. Thus, the Soergel calculus is geometrized by singular
compactifications of our Legendrian weaves, and our Legendrian weave calculus,
without compactification, should naturally induce a Soergel calculus for open
Bott-Samelson varieties. $\Box$

For instance, the $A_{3}$-Zamolodchikov relation from Soergel calculus
corresponds to the $A_{1}^{4}$-Reidemeister move in Legendrian weave calculus,
as depicted in Figure 106.

Figure 106. Contact Isotopy among Legendrian weaves, relative to the
boundaries. The lack of Reeb chords allows us to interpret these as exact
Lagrangian cobordisms between the positive braids
$\sigma_{i+1}\sigma_{i}\sigma_{i-1}\sigma_{i+1}\sigma_{i}\sigma_{i+1}$ and
$\sigma_{i-1}\sigma_{i}\sigma_{i-1}\sigma_{i+1}\sigma_{i}\sigma_{i-1}$. The
fact that these Lagrangian cobordisms are Hamiltonian isotopic implies that
the morphism induced between the associated Bott-Samelson bimodules must
coincide. $\Box$

Now, let us consider two positive braids
$\beta_{1},\beta_{2}\in\mbox{Br}^{+}_{n}$, $n\in\mathbb{N}$, and their
associated Legendrian (long) links
$\Lambda(\beta_{1}),\Lambda(\beta_{2})\subseteq(J^{1}[0,1],\xi_{\text{st}})$
[CG20, Section 2]. A Legendrian weave
$\Lambda\subseteq(J^{1}([0,1]\times[1,2]),\xi_{\text{st}})$ with no Reeb
chords and boundaries $\Lambda(\beta_{1})$ at $[0,1]\times\\{1\\}$, and
$\Lambda(\beta_{2})$ at $[0,1]\times\\{2\\}$, yields an embedded and exact
Lagrangian cobordism $L(\Sigma)$ from $\Lambda(\beta_{1})$ to
$\Lambda(\beta_{2})$ in the symplectization of $(J^{1}[0,1],\xi_{\text{st}})$,
as in Section 7. In particular, each trivalent vertex $\Sigma(G_{tri})$ and
hexagonal vertex $\Sigma(G_{hex})$ yield the following exact Lagrangian
cobordism:

  * (i)

The Lagrangian projection $L(G_{tri})$ of the Legendrian weave
$\Lambda(G_{tri})$ is a Lagrangian cobordism from the Legendrian tangle
$\Lambda(\beta_{1})$ given by one crossing in two strands
$\beta_{1}=\sigma_{i}$, to the Legendrian tangle $\Lambda(\beta_{2})$ given by
two crossing in two strands $\beta_{1}=\sigma^{2}_{i}$, where $i\in\mathbb{N}$
is labeling the transposition $\tau_{i}$ of the edges of $G_{tri}$. Smoothly,
this is a saddle cobordism obtained by an index-1 handle attachment to the
Lagrangian cone $\Lambda(\beta_{1})\times[0,\varepsilon]$ in the
symplectization, for $\varepsilon\in\mathbb{R}^{+}$ small.

  * (ii)

The Lagrangian projection $L(G_{hex})$ of the Legendrian weave
$\Lambda(G_{hex})$ is a Lagrangian concordance from the Legendrian tangle
$\Lambda(\beta_{1})$ given by three crossings in three strands
$\beta_{1}=\sigma_{i}\sigma_{i+1}\sigma_{i}$, to the Legendrian tangle
$\Lambda(\beta_{2})$ given by $\beta_{2}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$,
where $i\in\mathbb{N}$ is labeling the transpositions $\tau_{i},\tau_{i+1}$ in
the edges of $G_{hex}$. Smoothly, this is a Lagrangian surface obtained by
graphing a Reidemeister three move.

For simplicity, let us suppose that the relative homology
$H_{1}(L,\partial_{-}L;\mathbb{Z})$, which we denote by $H_{1}(L)$, is a free
$\mathbb{Z}$-module and the surface $L$ is spin, as is verified for the two
local cobordisms above. An exact Lagrangian cobordism
$L\subseteq(J^{1}[0,1],\xi_{\text{st}})\times[1,2]$ from $\Lambda(\beta_{1})$
to $\Lambda(\beta_{2})$ yields an algebraic map

$\Phi_{L}:\widehat{\mathcal{M}}(\Lambda(\beta_{1}))\longrightarrow\mathcal{M}(\Lambda(\beta_{2})),$

where $\widehat{\mathcal{M}(\Lambda(\beta_{1}))}$ is an algebraic
$(\mathbb{C}^{*})^{b_{1}(L)}$-bundle over $\mathcal{M}(\Lambda(\beta_{1}))$,
and $\mathcal{M}(\Lambda(\beta))$ denotes the moduli space of microlocal
rank-1 objects in the dg-category of of microlocal sheaves in
$\mathbb{S}^{1}\times$ microlocally supported on $\Lambda(\beta)$, as
described in [CG20, Section 3], [STWZ19, STZ17].

######  Remark A.3.

In the Floer-theoretic context, the map $\Phi_{L}$ is obtained by applying the
contravariant functor $Hom(\cdot,k)$ in the category of dg-algebras to the
morphism

$\Phi_{L}^{Fl}:\mathcal{A}(\Lambda(\beta_{2}))\longrightarrow\mathcal{A}(\Lambda(\beta_{1}))\otimes_{\mathbb{Z}}\mathbb{Z}[H_{1}(L)]$

of the Legendrian Contact dg-algebras $\mathcal{A}(\Lambda(\beta))$ associated
to Legendrian links $\Lambda(\beta)$. The Floer theoretic map $\Phi_{L}^{Fl}$
is described in [EHK16, Pan17b], and it is a count of holomorphic strips whose
boundary homology classes are encoded in $\mathbb{Z}[H_{1}(L)]$. To ease the
geometry, we have tensored by the flag moduli space map $\Phi_{L}$ above
$\mathbb{C}[H_{1}(L)]\cong\mathbb{Z}[H_{1}(L)]\otimes_{\mathbb{Z}}\mathbb{C}$
to base change the $\mbox{Spec}(\mathbb{Z}[H_{1}(L)])$-bundle to a complex
variety $\widehat{\mathcal{M}}(\Lambda(\beta_{1}))$. $\Box$

The relation to Soergel calculus now arises because the moduli space of simple
microlocal sheaves $\mathcal{M}(\Lambda(\beta_{1}))$ is (explicitly)
isomorphic to the open Bott-Samelson variety associated to $\beta$, also known
as the Broué-Michel variety of $\beta$ [STZ17, Tri19, CG20]. Let
$R=H^{*}(\mathcal{B})$ denote the cohomology of the complete flag variety for
$\operatorname{GL}(N,\mathbb{C})$, $N\in\mathbb{N}$, and $B_{s_{i}}$ the Bott-
Samelson Soergel ($R\otimes R$)-bimodule associated to a permutation $s_{i}\in
S_{N}$ in the Weyl group $S_{N}$ of $\operatorname{GL}(N,\mathbb{C})$. The
Rouquier complex $T_{i}:=[B_{s_{i}}\longrightarrow R]$ will be denoted by
$T_{i}$, for all $i\in\mathbb{N}$. Consider a braid

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

where $\sigma_{i_{1}}$ is the leftmost crossing in the front diagram of the
Legendrian braid, and the crossings are read from left to right. Then the
(singular) compactly supported cohomology of algebraic variety
$\mathcal{M}(\Lambda(\beta))$ is described by the tensor product

$T_{\beta}=T_{i_{1}}\otimes_{R}T_{i_{2}}\otimes_{R}\cdots\otimes_{R}T_{i_{l}},$

of Rouquier complexes.

######  Remark A.4.

Should the reader be interested in the closure of the Legendrian
$\Lambda(\beta)\subseteq(J^{1}S^{1},\xi_{\text{st}})$, instead of the long
link $(J^{1}[0,1],\xi_{\text{st}})$, the cohomology of the corresponding
moduli space $\mathcal{M}(\Lambda(\beta))$ is obtained by applying Hochschild
homology to the above complex $T_{\beta}$. In particular,
$H^{*}(\mathcal{M}(\Lambda(\beta)))$ coincides with the triply-graded homology
of the knot associated to $\beta$, equivalently, Khovanov-Rozansky link
homology – see [STZ17, Theorem 6.14]. $\Box$

In conclusion, the geometric map
$\Phi_{L}:\widehat{\mathcal{M}}(\Lambda(\beta_{1}))\longrightarrow\mathcal{M}(\Lambda(\beta_{2}))$
functorially induces

$H^{*}_{c}(\Phi_{L}):H^{*}_{c}(\widehat{\mathcal{M}}(\Lambda(\beta_{1})))\longrightarrow
H^{*}_{c}(\mathcal{M}(\Lambda(\beta_{2}))),$

which is a map of (products of) Rouquier complexes
$\widehat{T}_{\beta_{1}}\longrightarrow T_{\beta_{2}}$, where
$\widehat{T}_{\beta_{1}}$ is the compactly supported cohomology of
$\widehat{\mathcal{M}}(\Lambda(\beta_{1}))$, which contains the information of
the compactly supported cohomology $T_{\beta_{1}}$ of the open Bott-Samelson
variety for $\beta_{1}$.

Now, applying this to the two Lagrangian cobordisms associated to the
trivalent vertices $G_{tri}$ and the hexagonal vertices $G_{hex}$, we obtain
the following two maps:

  * (i)

The map $\Phi_{L(G_{tri})}:T_{s_{i}}\otimes
H^{*}(\mathbb{S}^{1})\longrightarrow T_{s_{i}}\otimes_{R}T_{s_{i}}$, where $i$
labels the $\tau_{i}$-edges of $G_{tri}$, and we have identified the fiber
bundle
$\widehat{\mathcal{M}}(\Lambda(\beta_{1}))\cong(\Lambda(\beta_{1}))\times\mathbb{C}^{*}$
with the Cartesian product, as in this case the bundle is topologically
trivial. The fact that there is one copy of
$\mathbb{C}^{*}=\mathbb{S}^{1}\times\mathbb{R}$ corresponds to the fact that
the Lagrangian cobordism $L(G_{tri})$ has a unique index 1 critical point and
its cocore carries the data $\mathbb{C}^{*}$.

  * (ii)

The map $\Phi_{L(G_{hex})}:T_{s_{i}}\otimes T_{s_{i+1}}\otimes
T_{s_{i}}\longrightarrow T_{s_{i+1}}\otimes T_{s_{i}}\otimes T_{s_{i+1}}$,
where in this case
$\widehat{\mathcal{M}}(\Lambda(\beta_{1}))\cong\mathcal{M}(\Lambda(\beta_{1}))$
as the Lagrangian $L(G_{hex})$ is a cylinder and $H_{1}(L)\cong\\{0\\}$ is
trivial.

In conclusion, the above discussion can summarized according to the following
tenet:

######  Principle A.5.

Let $T_{\beta_{1}},T_{\beta_{2}}$ be the Rouquier complexes associated to
positive braids $\beta_{1},\beta_{2}$ and $\Psi:T_{\beta_{1}}\longrightarrow
T_{\beta_{2}}$ the morphism given by a graph $G_{\Psi}$ with only (upwards)
trivalent and hexagonal morphisms in (open) Soergel calculus. Then the
Lagrangian projection of the Legendrian weave $\Lambda(G_{\Psi})$ yields an
embedded exact Lagrangian cobordism $L$ from $\Lambda(\beta_{1})$ to
$\Lambda(\beta_{2})$ and a geometric map

$\Phi_{L}:\widehat{\mathcal{M}}(\Lambda(\beta_{1}))\longrightarrow\Lambda(\beta_{2})$

such that $H^{*}_{c}(\Phi_{L})=\Psi$. $\Box$

The difference between the principle above and a theorem lies on the correct
definition of open Soergel calculus, of which we are not aware at this stage.
That said, since the trivalent and the hexagonal vertices are two of the main
building blocks for closed Soergel calculus, the above construction provides a
potential symplectic geometrization of open Soergel calculus, associated to
Rouquier complexes, instead of Soergel bimodules. In particular, in the
context of open Bott-Samelson varieties, the Lagrangian cobordisms above
indicate the need for additional data from $H_{1}(L)=\mathbb{Z}^{|V|}$ in
specifying a morphism, where $|V|$ is the number of trivalent vertices. The
development of open Soergel calculus, the computations establishing that our
geometric maps induce the expected algebraic maps, as well as the Lagrangian
description of the univalent vertex, will be the subject of upcoming and more
algebraic work.

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