horizontal strip (counting from the bottom). This produces a horizontal
$s_{1}$-edge, depicted in blue in Figure 43.(6). That move itself moves a
piece of the boundary of the conjugate surface (which used to be at the bottom
of the fourth strip) towards a piece of the boundary of the conjugate surface
near the hexavalent vertex. This allows for the introduction of two
Reidemeister II1 moves which insert two horizontal $s_{2}$-edges. The precise
location is drawn in Figure 44.(7), and it is readily seen that a triangular
region with only one sheet is created in the middle. Therefore, we can and do
perform a Reidemeister III2 move creating a hexavalent vertex in that
triangular region which emanates $s_{2}$ and $s_{3}$-edges. This is depicted
in Figure 44.(8).

Figure 44. (7) Two Reidemeister RII1 being applied to the top two sheets,
creating the two additional red edges. (8) The hybrid Lagrangian obtained
after applying a Reidemeister RIII2 move to the second and third sheets,
creating the hexavalent vertex. (9)-(10)-(11) Consists of a sequence of
Reidemeister RII2 moves, one creating the blue edge in (8), another red edge
in (9) and the two green edges in (11). (12) The hybrid Lagrangian surface
after applying the final Reidemeister RIII2 move.

This process of inserting Reidemeister II1 moves and then a Reidemeister III2
move can now be iterated as follows, always inserting the moves above. At the
$j$th step, we perform $j$ Reidemeister II1 moves, creating horizontal weave
$s_{i}$-edges that spell $s_{1}s_{2}\ldots s_{j}$ top to bottom. Figure 44.(9)
and (10) show this for $n=5$ at the step $j=2$; the first iterative step $j=1$
was Figures 43.(6) and 44.(7) and (8).151515In a sense, Figures 43.(4) and (5)
are the 0th iterative step. Then, two Reidemeister II1 moves are created at
the right and left (and on top) of the unique vertical weave $s_{j+1}$-edge,
giving rise to two horizontal $s_{j+1}$-edges which are incident to vertices
of a (newly created) triangular region. This is drawn in Figure 44.(11). To
finalize the $j$th iterative step, a Reidemeister III2 is performed at that
triangular region, creating a hexavalent vertex emanating $s_{j+1}$\- and
$s_{j+2}$-edges. The iteration ends when $j=n-3$, creating a weave with
$s_{1}$\- up to $s_{n-1}$-edges. Figure 44.(12) depicts the result of these
iterations. Finally, the unique vertical long $\sf I$-cycle on the weave, with
a vertical $s_{n-1}$-edge on towards the top, can be vertically lengthened
beyond all weave edges above it. Indeed, all the weave edges above that
vertical $s_{n-1}$-edge are $s_{j}$-edges with $j\leq n-3$ and thus the
vertical $s_{n-1}$-edge can cross them with tetravalent vertices. This
concludes the construction of the weave for $i=1$ and any $n\in\mathbb{N}$.

The case of arbitrary $i\in[1,n-1]$ readily follows from the case $i=1$ by
observing that the construction for $i=1$ only involves the horizontal edges
of the plabic fence right (at and) above the vertical edge. In consequence,
for any $i\in[1,n-1]$, we proceed as in the $i=1$ case for the portion of the
plabic fence consisting of the horizontal edges $i$th through $n$th, counting
from the bottom, and apply the Type 1 column construction for the horizontal
edges $1$st through $(i-1)$st.

Figure 45. On the left, the weave we obtain for Type II columns. On the right,
the weave $\mathfrak{c}_{i}^{\uparrow}(\mathtt{w}_{0,n})$ for Type II columns
(see [CasalsWeng22, Sec. 3.3.2]). These two weaves differ by a sequence of
push-through moves, the move II in Figure 18.

Finally, note that at the right and left the ends of these hybrid surfaces,
hybrids between a conjugate surface and a weave, coincide, regardless of
whether they are obtained from a Type 1 or a Type 2 column. Thus, these hybrid
surface can be horizontally concatenated according to the plabic fence, since
all the pieces have the same lateral boundary conditions. In order to remove
the white (empty) areas and obtain a weave in the end, one then applies (the
inverse of) Reidemeister II2 moves. This concludes the construction of the
Hamiltonian isotopy from a conjugate surface to (the Lagrangian projection of)
a Legendrian weave. This concludes the case of a Type 2 column.

Third, the case of a Type 3 column follows an argument analogous to the above
and it is left for the reader. In fact, the resulting weaves are equivalent to
the weaves $\mathfrak{l}_{i}^{b}$, or $\mathfrak{l}_{i}^{w}$ introduced in
[CasalsWeng22, Sec. 3.3.3]. An explicit instance of the necessary sequence of
moves (from Figure 27) is depicted in Figure 46. The general case is a direct
generalization of this sequence of moves.

At this stage we have found weaves for the Type 1, Type 2 and Type 3 columns
and their lateral ends (both at the right and left) match. Therefore, we can
horizontally concatenate these weaves to obtain a hybrid surface. In order to
turn the hybrid surface entirely into a weave, it now suffices to apply
${n-1\choose 2}$ inverse Reidemeister II2 moves, which will remove the white
(empty) areas and obtain a weave in the end. In addition, a sequence of push-
through moves, pushing the unique trivalent vertex upwards, shows that this
weave is equivalent to the weave
$\mathfrak{c}_{i}^{\uparrow}(\mathtt{w}_{0,n})$ in Subsection 2.2.2 Figure 17,
see also [CasalsWeng22, Sec. 3.3.2]; Figure 45 illustrates this equivalence in
an example. ∎

Figure 46. (1) A Type 3 column in a grid plabic graph. (2) The conjugate
surface associated to a Type 3 column. (3) Performing three RII1 moves. (4)
Inserting a hexavalent vertex with an RIII2 move. (5) Creating three more
weave edges (one blue and two red) with three RII1 moves. (6) Performing a
RIII2 move that creates a hexavalent vertex. (7)-(8) Introducing a sequence of
RII1 moves. (9) The result of applying the final RIII2 move.

Given a plabic fence with white vertices on top of its vertical edges, the
proof described above can also be applied with the modification that now the
process creates the weave downwards. Thus the base case is $i=n$, at the top
horizontal edge of the plabic fence. The resulting weave is equivalent to
$\mathfrak{c}_{i}^{\downarrow}(\mathtt{w}_{0,n})$ from [CasalsWeng22, Sec.
3.3.2].

Note that in the correspondence between conjugate Lagrangians and Legendrian
weaves, each black vertex on the $i$-th level of the bipartite graph in a Type
II column corresponds uniquely to an $s_{i}$-trivalent vertex in the
corresponding weave $\mathfrak{c}_{i}^{\uparrow}(\mathtt{w}_{0,n})$. In the
conjugate Lagrangian, the 1-cycles are given by the null components in the
plabic fence between two vertical edges on the same level, while in the
Legendrian weave, the 1-cycles are given by long $\sf I$-cycles, which connect
two trivalent vertices adjoining edges on the same level. Keeping track of the
1-cycles of the Lagrangian fillings under the Hamiltonian isotopy, we can show
that these 1-cycles correspond to each other, as explained in the following
corollary:

Figure 47. (1) The relative 1-cycles in the Type II column corresponding to
null regions with boundaries. (2) The relative 1-cycles after a Reidemeister
III1 move. (3) The relative 1-cycles after all Reidemeister III2 moves. (4)
The relative 1-cycles in the weave we obtain for Type II columns before
applying push throughs. (5) The relative $\sf I$-cycles in the weave
$\mathfrak{c}_{i}^{\uparrow}(\mathtt{w}_{0,n})$ after applying push throughs.

######  Corollary 4.1.

Consider the Hamiltonian isotopies of the hybrid Lagrangians in Theorem 1.1.
Then the following holds:

  1. (1)

Each 1-cycle in the conjugate Lagrangian from the rectangular null regions
between two vertical edges on $i$-th row and $j$ and $j^{\prime}$-th columns
corresponds to the long $\sf I$-cycle in the Lagrangian projection of the
Legendrian weave connecting the two trivalent vertices on the $j$ and
$j^{\prime}$-th columns adjoining $s_{i}$-edges.

  2. (2)

Each relative 1-cycle in the conjugate Lagrangian from the half-open
rectangular null regions from leftmost/rightmost vertical edges on $i$-th to
the left/right boundary of the plabic fence corresponds to the relative long
$\sf I$-cycle in the Lagrangian projection of the Legendrian weave from
leftmost/rightmost the trivalent vertex adjoining $s_{i}$-edges to the
boundary of the weave.

###### Proof.

This follows from the algorithm in the proof of Theorem 1.1 together with the
local characterization of 1-cycles in Corollary 3.4. In fact, since both the
null regions in the plabic fence and the long $\sf I$-cycles in the Legendrian
weave is the union of null regions and $\sf I$-cycles with boundaries in each
Type II column, it suffices for us to keep track of the relative 1-cycles in
each Type II column.

In each Type II column, there are three different types of relative 1-cycles
coming from null regions with boundaries, as shown in Figure 47.(1).
$\gamma_{\text{I}}$ are the two null regions on the $i$-th level of the plabic
fence which are bounded from one side by the vertical edge in the plabic
fence, $\gamma_{\text{II}}$ are the null regions on the $(i-1)$-th level of
the plabic fence right above the unique vertical edge in the plabic fence, and
$\gamma_{\text{III},i^{\prime}}$ are the other null regions on the
$i^{\prime}$-th level of the plabic fence with boundaries.

Then we keep track of the three classes of relative 1-cycles in the algorithm
of Theorem 1.1 using Corollary 3.4. Figures 47.(2) shows that after the
Reidemeister III1 move, both $\gamma_{\text{I}}$ and $\gamma_{\text{II}}$
winds around the blue trivalent vertex by $2\pi/3$, and Figures 47.(3) depicts
the resulting 1-cycles after all Reidemeister III2 moves. In Figure 47.(4),
$\gamma_{\text{I}}$ becomes the relative $\sf I$-cycle ending at the blue
trivalent vertex, $\gamma_{\text{II}}$ becomes the relative $\sf Y$-cycle
starting from the $(i-1)$-th blue edges, bifurcating at the blue-red
hexavalent vertex and ending at the blue trivalent vertex, and
$\gamma_{\text{III},i^{\prime}}$ becomes the relative $\sf I$-cycles starting
and ending at the $i^{\prime}$-th blue edges.

Finally we apply the push through moves in Subsection 2.2.2 Figure 17. After
applying push through once, $\gamma_{\text{I}}$ becomes an $\sf I$-cycle
ending at the red trivalent vertex adjoining $s_{2}$-edges and
$\gamma_{\text{II}}$ becomes an $\sf I$-cycle starting and ending at the
$(i-1)$-th blue edges. Therefore, by iteratively applying push throughs, we
can conclude that all these 1-cycles become the $\sf I$-cycles as in Figure
47.(5). ∎

Similar to Corollary 47, we can prove the correspondence between null regions
and long $\sf I$-cycles for a grid plabic graph, which is set up
combinatorially in [CasalsWeng22].

###  4.3. From conjugate fillings of $n$-triangulations to weaves

In this section we prove Theorem 1.1 for $n$-triangulations, showing that the
conjugate Lagrangian fillings associated to $A_{n}^{*}$-bipartite graphs of
$n$-triangulations are Legendrian weave fillings.

###### Proof of Theorem 1.1: $n$-triangulations.

The proof is locally in each triangle inside the ideal triangulation. Figure
48 explains the proof for $n=2$, Figure 49 explains the proof for $n=3$, and
Figure 50 explains the inductive part of the proof. Let us now provide the
necessary details for the general argument. We construct Hamiltonian isotopies
fixing the boundary of the triangle and proceed by induction. First, consider
the case $n=2$ in Figure 48.(1). we apply a Reidemeister III1 move for the
black vertex at the center, and thus obtain a trivalent vertex at the center;
the resulting hybrid Lagrangian is drawn in Figure 48.(2). Then, for any pair
of adjacent triangles we perform two (inverse) Reidemeister II moves for
conjugate surfaces161616This is a local move between conjugate surfaces, with
no hybrid Lagrangian involved. as in Proposition 2.8, so that the hybrid
Lagrangian surface in each triangle becomes that depicted in Figure 48.(3).
Finally, we consider a Hamiltonian isotopy transforming Figure 48.(3) into
Figure 48.(4), which is the required final weave in the $n=2$ case.

Figure 48. The correspondence between conjugate Lagrangian fillings of an
$2$-triangulation and the Legendrian weave of a $2$-triangulation.
Figure 49. The correspondence between conjugate Lagrangian fillings of an
$3$-triangulation and the Legendrian weave of a $3$-triangulation.The first
row is obtained by Hamiltonian isotopies fixing the boundary. The second row
is how we do Hamiltonian isotopies near the boundary.

Let us proceed with the induction step and assume the correspondence between
conjugate Lagrangians and Legendrian weaves for $n$-triangulations. For the
$(n+1)$-triangulation, we start with the alternating Legendrian (and the
corresponding conjugate Lagrangian) and we first consider the top $n$ rows of
the black and white vertices which form the bipartite graph of an
$n$-triangulation, as depicted in Figure 50.(1). By the induction hypothesis,
we can apply Hamiltonian isotopies, relative to the boundary, that introduce
the Legendrian weave of an $n$-triangulation, where all the strands in the
$n$-graph end at the crossings in the front projection of the Legendrian link;
we have depicted this in Figure 50.(2).

In order to construct the weave for the $(n+1)$-triangulation, we first apply
Reidemeister III1 moves at all the $n$ triangles corresponding to the interior
black vertices in the last row (the bottom row) of the bipartite graph, which
remained fixed under the previous Hamiltonian isotopy. This introduces $n$
trivalent vertices in the last row, as depicted in Figure 50.(3). Second,
consider the $n-1$ triangles above the $n$ trivalent vertices on the bottom
row, whose three vertices are the two $s_{1}$-edges in blue from the adjacent
trivalent vertices on the bottom and another $s_{1}$-edge in blue from the
weave of the $n$-triangulation on the top. We then apply Reidemeister III2
moves at each of these $n-1$ triangles. This introduces $n-1$ hexagonal
vertices emanating $s_{1}$ and $s_{2}$-edges, as depicted in Figure 50.(4).

We keep working iteratively. Consider the $n-2$ null triangles above the $n-1$
hexagonal vertices, whose three vertices are the two $s_{2}$-edges in red from
the adjacent hexagonal vertices on the bottom and another $s_{2}$-edge in red
from the weave of the $n$-triangulation on the top. We then apply Reidemeister
III2 moves at all these $n-2$ triangles: this introduces $n-1$ hexagonal
singularities emanating $s_{2}$ and $s_{3}$-edges, as depicted in Figure
50.(5). Iteratively, we can perform Reidemeister III2 moves at $n-j$ triangles
whose three vertices are two $s_{j}$-edges from the bottom and another
$s_{j}$-edge from the weave of the $n$-triangulation on the top. These move
introduces $n-j$ new hexagonal vertices emanating $s_{j}$ and $s_{j+1}$-edges.
This process yields the Legendrian weave of an $(n+1)$-triangulation, except
that now all the strands in the $(n+1)$-graph end at some crossings in the
front projection of the Legendrian link; see Figure 50.(5). At this stage,
this is the Legendrian weave that that we obtain by performing Hamiltonian
isotopies fixing the boundary of each triangle. In order to produce a weave,
we now further perform Hamiltonian isotopies which are not compactly supported
in each triangle.

Indeed, we perform Reidemeister II moves, which are Hamiltonian isotopies near
the boundary of each triangle. This yields a Legendrian weave of an
$(n+1)$-triangulation that fills the Legendrian $(n+1)$-satellite of the
outward unit conormal of the circles at the marked points. Continuing, we
apply Reidemerister II moves (of conjugate surfaces) near all the remaining
black vertices on the boundary of the triangle where we still have locally
conjugate surface fillings, and by Proposition 2.8 we can resolve these
crossings, as shown in Figure 50.(6).

Now we apply Reidemerister II2 moves near all the crossings where the
$s_{1}$-edges in the weaves (colored in blue) end, and resolve the crossings
and extend the blue strands to the boundary. This is illustrated in Figure
50.(7). Then inductively we apply Reidemeister II2 moves at the crossings
where all the $s_{j}$-edges end, to resolve these crossings and extend the
edges to the boundary of the triangle, as in Figure 50.(8)–(9). This finishes
the proof of the general case.

Figure 50. Given the correspondence between conjugate Lagrangian fillings of
an $n$-triangulation and the Legendrian weave of a $n$-triangulation, how we
build the correspondence for $(n+1)$-triangulations. The first two rows are
obtained by Hamiltonian isotopies fixing the boundary. The last row is how we
do Hamiltonian isotopies near the boundary.

∎

Note that in the correspondence between conjugate Lagrangians and Legendrian
weaves, each black vertex in the interior of the bipartite graph corresponds
uniquely to a blue trivalent vertex. In the conjugate Lagrangian, the 1-cycles
in the interior are given by hexagonal null regions, consisting of three
adjacent black vertices, while in the Legendrian weave, the 1-cycles in the
interior are given by $\sf Y$-cycles arising from three adjacent blue
trivalent vertices. We can also keep track of the 1-cycles of the Lagrangian
filling under the Hamiltonian isotopy, as concluded in the following
corollary:

######  Corollary 4.2.

Consider the Hamiltonian isotopies of the hybrid Lagrangians in Theorem 1.1.
Then the following holds:

  1. (1)

Each relative 1-cycle in the conjugate Lagrangian from a rectangular null
region (yellow cycles in Figure 51 left) corresponds to the relative $\sf
I$-cycle in the Lagrangian projection of the Legendrian weave (yellow cycles
in Figure 51 right) arising from the corresponding blue trivalent vertex.

  2. (2)

Each 1-cycle in the conjugate Lagrangian from hexagonal null regions (orange
cycles in Figure 51 left) corresponds to the $\sf Y$-cycle in the Lagrangian
projection of the Legendrian weave (orange cycles in Figure 51 right) arising
from the corresponding three adjacent blue trivalent vertices.

Furthermore, the null regions around the marked points on $\Sigma$ corresponds
to the 1-cycles with lowest levels in the Legendrian weave around the
punctures.

###### Proof.

This follows from the algorithm in the proof of Theorem 1.1 together with the
local characterization of 1-cycles in Corollary 3.4. An example is shown in
Figure 51. ∎

Figure 51. The correspondence between 1-cycles in the conjugate Lagrangian and
in the Legendrian weave for a 3-triangulation. The yellow cycles arising from
the rectangular null regions on the boundary correspond to the relative $\sf
I$-cycles, while the orange cycles arising from the hexagonal null regions in
the interior correspond to the $\sf Y$-cycles.

Note that we can also consider the conjugate Lagrangian filling
$L(\mathbb{G}_{n})$ associated to the $A_{n}$-bipartite graph $\mathbb{G}_{n}$
of the $n$-triangulation. Since $L(\mathbb{G}_{n})\not\subset
T^{*}\Sigma_{\text{op}}$, this conjugate Lagrangian does not directly come
from a free Legendrian weave in $J^{1}\Sigma_{\text{op}}$. Nevertheless, we
are still able to describe the corresponding Legendrian surface filling of the
link $\bigcirc_{n-1}$ in $J^{1}\Sigma$ as a Legendrian weave away from
neighbourhoods of the marked points. The statement reads:

######  Theorem 4.3.

For an ideal triangulation on a surface $\Sigma$ with marked points, the
conjugate Lagrangian associated to the $A_{n}$-bipartite graph
$\mathbb{G}_{n}$ associated to the $n$-triangulation is Hamiltonian isotopic
to the Lagrangian projection of the Legendrian
$\widetilde{L}(\mathfrak{w}_{n})$ associated to the $n$-triangulation as in
Figure 52.

In particular, should we consider the compactification of the conjugate
Lagrangian surface, by adding the necessary capping disks at each of its
boundary components, then Theorem 4.3 implies that the resulting Hamiltonian
isotopy class of exact Lagrangian in $T^{*}\Sigma$ admits a representative
whose lift is a (non-free) Legendrian weave.

######  Corollary 4.4.

Consider the Hamiltonian isotopies of the Lagrangians in Theorem 4.3. Then the
following holds:

  1. (1)

Each relative 1-cycle in the conjugate Lagrangian from a rectangular null
region corresponds to the relative $\sf I$-cycle Legendrian surface.

  2. (2)

Each 1-cycle in the conjugate Lagrangian from hexagonal null regions
corresponds to the $\sf Y$-cycle in the Legendrian surface.

Figure 52. The conjugate Lagrangian filling $L(\mathbb{G}_{n})\subset
T^{*}\Sigma$ and the corresponding hybrid Legendrian surface filling of
$\bigcirc_{n-1}$, being a Legendrian weave away from neighbourhoods of the
marked points.

The proofs are exactly the same as Theorem 1.1 and Corollary 4.2.

###  4.4. From pinching sequences to weaves

Let us now prove that embedded exact Lagrangian fillings for Legendrian
positive braid closures obtained through pinching sequences are Hamiltonian
isotopic to (Lagrangian projections of) Legendrian weaves. The result reads as
follows:

######  Theorem 4.5.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid and $\Delta$ be the half
twist, $\Lambda_{\beta\Delta^{2}}\subset T^{*,\infty}\mathbb{D}^{2}$ be the
Legendrian cylindrical braid closure of $\beta\Delta^{2}$, and
$\Lambda_{\beta}^{\prec}\subset\mathbb{D}^{3}\subset
T^{*,\infty}\mathbb{D}^{2}$ the Legendrian rainbow braid closure of $\beta$.
Consider a pinching sequence $\sigma\in S_{\ell(\beta)}$.

Then the embedded exact Lagrangian filling $L_{\sigma}$ associated to the
pinching sequence $\sigma$ is Hamiltonian isotopic to the Lagrangian
projection of a Legendrian weave. In fact, the weave associated to the
Hamiltonian isotopy class given by $\sigma=\mbox{id}$, i.e. left to right
pinching, coincides with the weave associated to the plabic fence
$\mathbb{G}_{\beta}$ given by $\beta$.

###### Proof.

Consider $\sigma=\mbox{id}$. Following Example 2.15, each crossing in $\beta$
in the Lagrangian projection corresponds to the Reeb chord in the unique
region on the left of the corresponding crossing in $\beta$ in the front
projection. We prove that each Legendrian weave
$\mathfrak{c}_{i}^{\uparrow}(\mathtt{w}_{0,n})$ defines the elementary
cobordism pinching the leftmost Reeb chord.

In the front projection, in order to pinch the Reeb chord corresponding to the
crossing $s_{i}$ in the leftmost closed bigon region, we first need to perform
a sequence of Reidemeister III moves to the strands crossing the closed region
such that they are moved away from the closed region. The closed bigon region
is bounded by the $i$-th $s_{1}$-crossing in the half twist and the
$s_{i}$-crossing. The strands crossing this region are the 1-st to the $i$-th
strand. See Figure 53 (1). These Reidemeister III moves introduce a number of
hexagonal vertices in the weave following Figure 36, bringing the $i$-th
$s_{1}$-strand to the top, as depicted in Figure 53 (2)–(3). The resulting
weave is $\mathfrak{n}_{i}^{\uparrow}(\mathtt{w}_{0,n})$. Then we pinch the
Reeb chord in the closed region, as explained in Subsection 3.5 (see Figure
37), which introduces an $s_{i}$-trivalent vertex in the weave, as in Figure
53 (4). The resulting weave for the elementary cobordism is exactly
$\mathfrak{c}_{i}^{\uparrow}$. Finally, we need Reidemeister III moves to
bring the braid word back to the half twist, whose resulting weave is
$\mathfrak{n}_{i}^{\uparrow}(\mathtt{w}_{0,n})^{\text{op}}$, as in Figure 53
(5). The concatenation of these weaves is exactly
$\mathfrak{c}_{i}^{\uparrow}(\mathtt{w}_{0,n})$.

Figure 53. The Reeb chord pinching in the front projection. (1) The Legendrian
front of the braid word $\Delta s_{i}$. (2) The front after a Reidemeister III
move emanating $s_{1}$ and $s_{2}$-crossings (introducing a hexagonal vertex
in the weave). (3) The front after a Reidemeister III move emanating $s_{2}$
and $s_{3}$-crossings (introducing a hexagonal vertex). (4) The front after
Reeb chord pinching (introducing a trivalent vertex). (5) The front after
applying Reidemeister III moves backwards.

In the Lagrangian projection, following [Hughes21A, Section 4], the above
elementary cobordism can be realized by a single Reeb chord pinching with no
Reidemeister moves in the Lagrangian projection. The key observation is that
in the front projection, none of the bigons (in which there exists a Reeb
chord) are contained in the bigon on the left of the first crossing in $\beta$
(in which there exists a Reeb chord corresponding to the first crossing of
$\beta$ in the Lagrangian projection). Therefore, under small perturbation of
the front in Figure 53.(1), we may assume that the Reeb chords in other bigons
are moved away from the bigon on the left of the first crossing in $\beta$.
That means all Reeb chords corresponding to the crossings in the half twist
$\Delta$ are beyond the left of the figure while all chords corresponding to
the crossings after $s_{i}$ in $\beta$ are beyond the right of the figure.
Hence we may assume that the Lagrangian projection is on the left of Figure
54. Therefore, all the Reidemeister III moves in the front do not change the
Lagrangian projection. At this stage, the Reeb chord pinching as presented in
in Subsection 3.5 does not introduce any additional Reidemeister moves in the
Lagrangian projection. This is shown explicitly on the right of Figure 54.
Therefore, no Reidemeister moves are introduced in the Lagrangian projection.

Figure 54. The front projection (on the top) and the corresponding Lagrangian
projection (on the bottom) in Reeb chord pinching.

Finally, we claim that the weave $\mathfrak{n}(\mathtt{w}_{0,n})$ defines the
minimal cobordism of the Legendrian unlink. Indeed, since in
$T^{*,\infty}\mathbb{D}^{2}$, the Legendrian knot on the boundary is isotopic
to the Legendrian unlink, and it follows from [EliPol96] that
$\mathfrak{n}(\mathtt{w}_{0,n})$ has to define the minimal embedded Lagrangian
cobordism.

For a general permutation $\sigma\in S_{\ell(\beta)}$, to show that
$L_{\sigma}$ is Hamiltonian isotopic to a Lagrangian projection of a
Legendrian weave, it suffices to notice that a pinching sequence defines a
decomposable Lagrangian filling, which in the front projection is a sequence
of pinching saddle cobordisms, Lagrangian 0-handes, Legendrian Reidemeister
III moves. Since each of these elementary cobordisms can be realized by
Legendrian weaves (Section 3.5), we can conclude that the Lagrangian filling
is Hamiltonian isotopic to the projection of a weave. ∎

Reeb pinching sequences can be combined with Legendrian isotopies to produce
new embedded exact Lagrangian fillings [CasalsNg21]. The hexavalent vertices
in a weave capture Legendrian Reidemeister III moves. Thus, in general, the
above argument shows that a decomposable Lagrangian filling obtained via
pinching saddle cobordisms, Lagrangian 0-handes, Legendrian Reidemeister III
moves and cyclic rotation (of the braid word) is Hamiltonian isotopic to the
Lagrangian projection of a Legendrian weave.171717In general, Reidemeister
moves might be required in the Lagrangian projection.

###  4.5. Numerical comparisons on Lagrangian fillings with each method

For any Legendrian $(2,k)$-torus link, the number of graphical conjugate
Lagrangian fillings is the same as the number of free Legendrian weaves. In
general, the number of conjugate Lagrangian fillings can be strictly less than
the number of free Legendrian weaves.

For a Legendrian positive $(n,k)$-torus link, the number of graphical
conjugate Lagrangian fillings corresponds to the number of certain reduced
plabic graphs of an $(k+n)$-gon (Example 2.9), which is also the number of
cluster charts in $\mathrm{Gr}(n,k+n)$ defined by Plücker coordinates
[OPS15WeakSep] (see Section 6.2). Indeed, each Lagrangian filling defines a
distinguished cluster chart in the moduli space of sheaves (which we discuss
in Section 5), and fillings that define different charts can be shown to be
different; see [GaoShenWeng20, CasalsWeng22].181818A precise relation between
Lagrangian fillings and cluster varieties is conjectured in [Casals20Sk,
Conjecture 5.1]; we will discuss it partially in Section 6.. The number of
Legendrian weaves is also well studied, here are some explicit examples:

######  Example 4.6.

For Legendrian $(n,k)$-torus links, where either $n\geq 3,k\geq 6$ or $n,k\geq
4$, the number of graphical conjugate Lagrangian fillings (which is the number
of reduced plabic graphs of a $(k+n)$-gon) is always finite. In contrast, we
showed in [CasalsGao20] that there are infinitely many free Legendrian weaves
that fill the torus link; see also [CasalsZas20]*Theorem 7.6 and
[CasalsWeng22]. Hence the number of graphical conjugate Lagrangians is
strictly smaller than the number of free Legendrian weaves. $\Box$

######  Example 4.7.

For the Legendrian $(3,3)$-torus link, there are 34 graphical conjugate
Lagrangian fillings, corresponding to the number of Plücker-type cluster
coordinates in $\mathrm{Gr}(3,6)$ [KK21Gr37]*Theorem 4.2. In constrast,
[Hughes21D] shows that the number of different free Legendrian weaves is (at
least) 50, which is the number of all cluster charts in $\mathrm{Gr}(3,6)$.
Conjecturally, all conjugate Lagrangian fillings should be Hamiltonian
isotopic to the Lagrangian projection of a Legendrian weave.

We can also consider Lagrangian fillings coming from pinching sequences, i.e.
concatenations of standard local models of 1-handle attachments and 0-handle
attachments (with no Reidemeister moves in the Lagrangian projection). For
different Lagrangian projections of the Legendrian link, i.e. braid
representatives of the $(3,3)$-torus link, the number of Lagrangian fillings
coming from pinching sequences will be different. For example, for the braid
$(s_{1}s_{2})^{3}$, the number of different fillings given by pinching
sequences is 46, while for the braid $(s_{1}s_{2}s_{2})^{2}$ or
$(s_{2}s_{1}s_{1})^{2}$, the number is 42. Again, all these fillings coming
from pinching sequences are Hamiltonian isotopic to the Lagrangian projection
of a Legendrian weave, but there are strictly more Legendrian weave fillings.
$\Box$

######  Example 4.8.

For the Legendrian $(3,4)$-torus link, there are 259 graphical conjugate
Lagrangian fillings, in bijection with the number of Plücker-type cluster
coordinates in $\mathrm{Gr}(3,7)$; see [KK21Gr37]*Theorem 5.2. In contrast,
[AnBaeLee21ADE] shows that the number of different free Legendrian weaves is
(at least) 833, which is the number of all cluster charts in
$\mathrm{Gr}(3,7)$). Finally, the number of Lagrangian fillings coming from
pinching sequences for the braid $(s_{1}s_{2})^{4}$ is 633, which we have
verified by computer.191919We thank D. Weng for sharing with us his code
computing the different clusters that can be obtained by pinching Reeb chords
in positive braids. In consequence, free Legendrian weaves still provide the
maximal number of Lagrangian fillings.

##  5\. Sheaf Quantization of Conjugate Fillings and Legendrian weaves

Sections 2, 3 and 4 prove results about Hamiltonian isotopy classes of
Lagrangian fillings. In order to connect the comparison results between
Lagrangian fillings to the comparison result of cluster coordinates in Section
6, we need to first enhance the geometric objects, i.e. Lagrangian fillings,
with some algebraic structures, i.e. constructible sheaves or microlocal
sheaves. We use the notation from Appendix A.

Let $L\subseteq(T^{*}\Sigma,\lambda_{st})$ be an exact Lagrangian orientable
surface with non-empty Legendrian boundary
$\partial_{\infty}L\subseteq(T^{*,\infty}\Sigma,\ker(\lambda_{st}))$, and
$\mathcal{L}\in\operatorname{Loc}^{pp}(L)$ a local system with perfect stalks.
Consider the category of sheaves singularly supported on $\widetilde{L}$ with
perfect stalks
$\operatorname{Sh}_{\widetilde{L}}^{pp}(\Sigma\times\mathbb{R})$ and the
microlocal functor
$m_{\widetilde{L}}:\operatorname{Sh}_{\widetilde{L}}^{pp}(\Sigma\times\mathbb{R})\longrightarrow\operatorname{Loc}^{pp}(L)$,
where $\widetilde{L}\subseteq(J^{1}\Sigma,\ker(dz-\lambda_{st}))$ is the
Legendrian lift of $L$.202020Here we have trivially identified
$\operatorname{Loc}(\widetilde{L})=\operatorname{Loc}(L)$, as local systems
only depend on $L$ itself and not its Hamiltonian isotopy class. The first
precise definition we need is the following:

######  Definition 5.1.

Let $L\subseteq(T^{*}\Sigma,\lambda_{st})$ be an exact Lagrangian orientable
surface with non-empty boundary, with Legendrian lift
$\widetilde{L}\in(J^{1}\Sigma,\xi_{st})$, and
$\mathcal{L}\in\operatorname{Loc}^{pp}(L)$ a local system. A sheaf
$\mathcal{F}\in\operatorname{Sh}_{\widetilde{L}}^{pp}(\Sigma\times\mathbb{R})$
is said to be a sheaf quantization of $(L,\mathcal{L})$ if
$m_{\widetilde{L}}(\mathcal{F})=\mathcal{L}$. By definition, a sheaf
quantization functor of $L$ is a functor of dg-categories

$\psi_{\widetilde{L}}:\operatorname{Loc}^{pp}(L)\longrightarrow\operatorname{Sh}_{\widetilde{L}}^{pp}(\Sigma\times\mathbb{R})$

such that $\psi_{\widetilde{L}}\circ
m_{\widetilde{L}}=\mathrm{id}_{\operatorname{Loc}(\Lambda)}$.

By Theorem 5.5.(1) below, there exists a sheaf quantization functor for
Legendrian lifts of Lagrangian fillings. In addition, such functor is fully
faithful. This sheaf quantization functor admits a left-adjoint which is
between the corresponding categories of compact objects (in the unbounded dg-
categories)

$\psi_{\widetilde{L}}^{L}:\operatorname{Sh}^{cpt}_{\widetilde{L}}(\Sigma\times\mathbb{R})\longrightarrow\operatorname{Loc}^{cpt}(L).$

Thus, a sheaf quantization of $L$ leads to an open embedding

$\mathbb{R}\mathcal{M}\big{(}\psi_{\widetilde{L}}^{L}\big{)}:\mathbb{R}\mathcal{L}oc(L)\hookrightarrow\mathbb{R}\mathcal{M}\big{(}\Sigma\times\mathbb{R},\widetilde{L}\big{)}$

between the associated derived stacks of pseudo-perfect objects in
$\operatorname{Sh}^{cpt}_{\widetilde{L}}(\Sigma\times\mathbb{R})$ and
$\operatorname{Loc}^{cpt}(L)$ [ToenVaquie07, Section 3]. Note that these
derived stacks parametrize objects in the subcategories
$\operatorname{Loc}^{pp}(L)$ and
$\operatorname{Sh}^{pp}_{\widetilde{L}}(\Sigma\times\mathbb{R})$.

By Theorem 5.5.(2), the proper push-forward $\pi_{\Sigma,!}$ of the projection
$\Sigma\times\mathbb{R}\longrightarrow\mathbb{R}$ yields a fully faithful
functor

$\pi_{\Sigma,!}:\operatorname{Sh}_{\widetilde{L}}^{pp}(\Sigma\times\mathbb{R})\longrightarrow\operatorname{Sh}_{\Lambda}^{pp}(\Sigma),$

where
$\Lambda:=\partial_{\infty}L\subseteq(T^{*,\infty}\Sigma,\ker(\lambda_{st}))$
is the ideal Legendrian boundary of $L$. The functor $\pi_{\Sigma,!}$ admits a
left-adjoint which is between the corresponding categories of compact objects
(in the unbounded dg-categories)

$\pi_{\Sigma,!}^{L}:\operatorname{Sh}^{cpt}_{\Lambda}(\Sigma)\longrightarrow\operatorname{Sh}^{cpt}_{\widetilde{L}}(\Sigma\times\mathbb{R})$

and thus we obtain an open embedding of the corresponding derived stacks:

$\mathbb{R}\mathcal{M}\big{(}\pi_{\Sigma,!}^{L}\big{)}:\mathbb{R}\mathcal{M}\big{(}\Sigma\times\mathbb{R},\widetilde{L}\big{)}\hookrightarrow\mathbb{R}\mathcal{M}(\Sigma,\Lambda).$

Again, these moduli parametrize objects in the subcategories
$\operatorname{Sh}^{pp}_{\widetilde{L}}(\Sigma\times\mathbb{R})$ and
$\operatorname{Sh}^{pp}_{\Lambda}(\Sigma)$. By restricting to Abelian local
systems of (differential graded) perfect $\Bbbk$-modules, the composition
$\mathcal{M}\big{(}\psi_{\widetilde{L}}^{L}\circ\pi_{\Sigma,!}^{L}\big{)}$
yields two open embeddings of algebraic stacks

$H^{1}(L;\Bbbk^{\times})\hookrightarrow\mathcal{M}_{1}(\Sigma,\Lambda)_{0},\;\mbox{and}\;H^{1}(L\setminus
T,\Lambda\setminus
T;\Bbbk^{\times})\hookrightarrow\mathcal{M}_{1}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0},$

where
$\mathcal{M}_{1}(\Sigma,\Lambda)_{0}\subseteq\mathbb{R}\mathcal{M}(\Sigma,\Lambda)$
is the moduli space of microlocal rank 1 sheaves singular support on the
Legendrian and vanishing stalk near them marked points of $\Sigma$, and
$\mathcal{M}_{1}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}$ its framed version
with base points $T\subseteq\Lambda$, see Appendix B.4. In Section 6 we
explain that these embeddings define cluster charts.

######  Remark 5.2.

In general, the definition of sheaf quantization (and a sheaf quantization
functor) is more general than Definition 5.1, in that
$\operatorname{Loc}^{pp}(L)$ is substituted by the global sections of the
Kashiwara-Schapira stack on the Lagrangian cone over the Legendrian lift of
$L$; see Appendix B. Indeed, it is not generally the case that this category
of global sections is equivalent to $\operatorname{Loc}^{pp}(L)$: see the
obstructions discussed in [Gui19, Section 10]. That said, for an orientable
exact Lagrangian surface with non-empty Legendrian boundary these obstruction
vanish, and we can equivalently work with $\operatorname{Loc}^{pp}(L)$. (In
the main body of the section, in order to better keep track of signs, we will
work over dg-derived category of twisted local systems
$\operatorname{Loc}^{pp}_{\epsilon}(\mathcal{I}_{\widetilde{L}})$ on the frame
bundle $\mathcal{I}_{\widetilde{L}}$. See Section B.2.2.) $\Box$

In brief, the goal in this section is to describe sheaf quantizations for
Lagrangian fillings through explicit combinatorial models in the two cases of
conjugate Lagrangian surfaces and Legendrian weaves. The necessary subtleties
and results about sheaf categories and constructible sheaves are provided in
Appendices A and B.

###  5.1. Generalities on constructible sheaves and sheaf quantization

Let $\operatorname{Sh}(M)$ be the dg-derived category of sheaves on a smooth
manifold $M$. In [KSbook, Chapter V], M. Kashiwara and P. Schapira introduced
the notion of singular support of a sheaf in $\operatorname{Sh}(M)$
(Definition B.1) and show in [KSbook, Theorem 6.5.4] that the singular support
of is a coisotropic subset in $T^{*,\infty}M$. By [KSbook, Theorem 8.4.2], if
the singular support of a sheaf is a (possibly singular) Legendrian in the
ideal contact boundary $T^{*,\infty}M$, then that sheaf itself must be a
constructible sheaf.

Let $\Lambda\subset T^{*,\infty}M$ be a Legendrian submanifold. The front
projection $\pi(\Lambda)\subset M$ gives a stratification of $M$ and a
(constructible) sheaf in $\operatorname{Sh}_{\Lambda}^{pp}(M)$ is a stratified
locally constant sheaf with respect to the stratification $\pi(\Lambda)$. The
co-directions of the Legendrian $\Lambda$ determine directions of transport
maps between (derived) sections of such sheaves. Discrete, rather
combinatorial, descriptions of such sheaves can be obtained by applying work
of R. MacPherson, see [STZ17, Section 3.3], [CasalsWeng22, Section 4.1] and
references therein. For instance, [STZ17]*Section 3 discusses the following
examples, which cover the case of Legendrian links:

######  Example 5.3.

In the case that $\pi(\Lambda)$ is locally a hyperplane in $\mathbb{R}^{n+1}$
as in Figure 55 (left), with covectors pointing upward, then the sheaf is
locally determined by the following diagram

$F_{d}\longrightarrow F_{u}$

of sheaves of chain complexes of $\Bbbk$-modules. In the case that
$\pi(\Lambda)$ is locally two transversely intersecting hyperplanes in
$\mathbb{R}^{n+1}$ as in Figure 55 (middle), covectors both pointing upward,
then the sheaf is locally determined by a diagram

$\textstyle{F_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{lu}}$$\textstyle{F_{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{dl}}$$\scriptstyle{\phi_{dr}}$$\textstyle{F_{u}}$$\textstyle{F_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{ru}}$

where the total complex
$\mathrm{Tot}\big{(}F_{d}\xrightarrow{(\phi_{dl},\phi_{dr})}F_{l}\oplus
F_{r}\xrightarrow{(-\phi_{lu},\phi_{ru})}F_{u}\big{)}\simeq 0$ is required to
be acyclic; see [STZ17]*Theorem 3.12. Finally, in the case that $\pi(\Lambda)$
is locally a smooth family of cusps in $\mathbb{R}^{n+1}$, as in Figure 55
(right), with covectors pointing upward, the sheaf is locally determined by a
diagram

$F_{o}\xrightarrow{\phi_{d}}F_{i}\xrightarrow{\phi_{u}}F_{o}$

where $\phi_{u}\circ\phi_{d}$ is a quasi-isomorphism. In the examples we
study, namely Legendrian rainbow braid closures, we can also assume that
$\phi_{u}\circ\phi_{d}=\mathrm{id}$ [STZ17]*Proposition 3.22. $\Box$

Figure 55. The local combinatoric models for sheaves in
$Sh^{b}_{\Lambda}(\mathbb{R}^{n+1})$ where on the left $\pi(\Lambda)$ is a
hyperplane, in the middle $\pi(\Lambda)$ is two transverse hyperplanes, and on
the right $\pi(\Lambda)$ is a smooth family of cusps.

####  5.1.1. Combinatorial microlocal functor

In the combinatorial description of Example 5.3, the functor
$m_{\Lambda}:\operatorname{Sh}_{\Lambda}^{pp}(M)\longrightarrow\operatorname{Loc}^{pp}(\Lambda)$
can be also computed explicitly, following [STZ17]*Section 5, as follows. (For
the general framework, see Appendix B.2). Consider a Legendrian
$\Lambda\subset T^{*,\infty}M$, and a generic point $p\in\Lambda$ such that
$\pi(\Lambda)$ is a smooth hypersurface near $\pi(p)$, as in Figure 55 (left).
Then the microstalk of $\mathcal{F}\in\operatorname{Sh}_{\Lambda}^{pp}(M)$ at
$p$ can be computed as

$m_{\Lambda,p}(\mathcal{F})=\mathrm{Cone}(F_{d}\rightarrow F_{u}).$

Given a path $\gamma\subseteq\Lambda$ connecting two points $p,q\in\Lambda$,
the cone construction above yields a local system
$m_{\Lambda,\gamma}(\mathcal{F})$, often referred to as the microlocalization
of $\mathcal{F}$ along $\gamma$, and thus the microstalk of $\mathcal{F}$ is
independent of the point we choose in $\Lambda$. By definition, a sheaf
$\mathcal{F}$ is said to be of microlocal rank $r$ sheaf if the microstalk is
concentrated in a single degree and has rank $r$.

######  Example 5.4.

Consider a path $\gamma$ in $\Lambda$ such that $\pi(\gamma)$ passes through a
smooth family of transverse double points, as in Figure 55 (right). Then the
parallel transport map from bottom right to top left is computed by

$m_{\Lambda,dr}(\mathcal{F})=\mathrm{Cone}\big{(}F_{d}\xrightarrow{\phi_{dr}}F_{r}\big{)}\xrightarrow{\sim}\mathrm{Cone}\big{(}F_{l}\xrightarrow{\phi_{lu}}F_{u}\big{)}=m_{\Lambda,ul}(\mathcal{F}).$

Similarly, the parallel transport map from top right to bottom left is
computed by

$m_{\Lambda,ur}(\mathcal{F})=\mathrm{Cone}\big{(}F_{r}\xrightarrow{\phi_{ru}}F_{u}\big{)}\xrightarrow{\sim}\mathrm{Cone}\big{(}F_{d}\xrightarrow{\phi_{dl}}F_{l}\big{)}=m_{\Lambda,dl}(\mathcal{F}).$

In general, we equip the front diagram $\pi(\Lambda)$ with a Maslov potential
and the microstalk carries a degree shift by the Maslov potential.

In the sheaf quantization of Lagrangian fillings, we need to study both the
case that $\Lambda$ is a Legendrian link and the case that $\Lambda$ is a
Legendrian surface (the Legendrian lift of a Lagrangian filling of a
Legendrian link). In higher dimension, including Legendrian surfaces, the
signs of microlocal monodromies need to be fixed and that requires fixing a
spin structure on the 2-skeleton of the Legendrian. In particular, the sign
issue appears when we compute microlocal merodromy along relative 1-cycles
(see [CasalsWeng22, Section 4.5]) so as to obtain cluster variables over
$\mathbb{Z}$. This is discussed in Appendix B.2.

####  5.1.2. Existence of sheaf quantization for Lagrangian fillings

Let
$\operatorname{Sh}_{\Lambda}^{pp}(M)\subseteq\operatorname{Sh}_{\Lambda}(M)$
be the full subcategory of constructible sheaves with singular support in
$\Lambda$ and $\Bbbk$-perfect stalks, and consider the dg-category
$\operatorname{Loc}^{pp}_{\epsilon}(\mathcal{I}_{\widetilde{L}})$ of pseudo-
perfect twisted local systems on the (relative) frame bundle
$\mathcal{I}_{\widetilde{L}}$ defined in Appendix B.2. The superscript pp
means pseudo-perfect and see Appendix A for descriptions of these categories.
By [GKS12, Theorem 3.7], the dg-category $\operatorname{Sh}^{pp}_{\Lambda}(M)$
is an invariant of a Legendrian $\Lambda$ under Legendrian isotopies; see
Theorem B.17 and Appendix A.(b).

Results of X. Jin and D. Treumann [JinTreu17], following S. Guillermou’s
[Gui12], can be adapted to show that a sheaf quantization result exists for
Lagrangian fillings, explaining how Lagrangian fillings of $L$ equipped with
local systems gives rise to sheaves in $\operatorname{Sh}^{pp}_{\Lambda}(M)$.

######  Theorem 5.5 (Theorem B.20).

Let $\Lambda\subset T^{*,\infty}M$ be a Legendrian and $L\subset T^{*}M$ a
relatively spin exact Lagrangian filling of $\Lambda$ with zero Maslov class.
Then there exists Legendrian lift $\widetilde{L}\subset J^{1}(M)$ of $L\subset
T^{*}M$ whose primitive is bounded from below and

  1. (1)

There exists a fully faithful functor
$\Psi_{\widetilde{L}}:\operatorname{Loc}^{pp}_{\epsilon}(\mathcal{I}_{\widetilde{L}})\rightarrow\operatorname{Sh}^{pp}_{\widetilde{L}}(M\times\mathbb{R})$
such that the stalk at $M\times\\{-\infty\\}$ is acyclic and
$m_{\widetilde{L}}\circ\Psi_{\widetilde{L}}\simeq\mathrm{id}$.

  2. (2)

The proper push forward functor
$\pi_{M,!}:\operatorname{Sh}^{pp}_{\widetilde{L}}(M\times\mathbb{R})\rightarrow
Sh^{pp}_{\Lambda}(M)$ via the projection $\pi_{M}:M\times\mathbb{R}\rightarrow
M$ is fully faithful.

Let $\Pi_{M}:(J^{1}M,\xi_{st})\longrightarrow(T^{*}M,\lambda_{st})$ be the
projection forgetting the $j_{0}$-value of a section.

######  Proposition 5.6 (Theorem B.22).

Let $L\subset(T^{*}M,\lambda_{st})$ be a relatively spin exact Lagrangian
filling of a Legendrian submanifold $\Lambda\subset T^{*,\infty}M$,
$\widetilde{L}\subset(J^{1}M,\xi_{st})$ its Legendrian lift and
$\mathcal{L}\in\operatorname{Loc}^{pp}_{\epsilon}(\mathcal{I}_{\widetilde{L}})$
a twisted local system. Consider a Hamiltonian
$H:T^{*}M\longrightarrow\mathbb{R}$ that is homogeneous at infinity,
$\varphi_{t}\in\mbox{Ham}(T^{*}M,d\lambda_{st})$ its time-$t$ flow,
$t\in[0,1]$, which extends to an homonymous contactomorphism
$\varphi_{t}\in\mbox{Cont}(T^{*,\infty}M,\ker\lambda_{st})$ of the ideal
contact boundary, $t\in[0,1]$,
$\widetilde{H}=\Pi^{*}_{M}(H):J^{1}M\longrightarrow\mathbb{R}$ the pull-back
and $\widetilde{\varphi}_{t}\in\mbox{Cont}(J^{1}M,\xi_{st})$ its time-$t$
flow.

Suppose that
$\widetilde{\mathcal{F}}_{t}\in\operatorname{Sh}^{pp}_{\widetilde{L}_{t}}(M\times\mathbb{R})$
is a sheaf quantization of $(\varphi_{t}(L),\varphi_{t*}\mathcal{L})$ and
consider the proper push-forward
$\mathcal{F}_{t}:=\pi_{M,!}(\widetilde{\mathcal{F}}_{t})\in\operatorname{Sh}^{pp}_{\varphi_{t}(\Lambda)}(M)$,
$t\in[0,1]$. Then:

  * (i)

The sheaf kernel convolution
$\Phi_{\widetilde{H}}:\operatorname{Sh}^{pp}_{\widetilde{L}}(M\times\mathbb{R})\longrightarrow\operatorname{Sh}^{pp}_{\varphi_{1}(\widetilde{L})}(M\times\mathbb{R})$
associated to $\widetilde{H}$ satisfies
$\Phi_{\widetilde{H}}(\widetilde{\mathcal{F}}_{0})\simeq\widetilde{\mathcal{F}}_{1}$.

  * (ii)

The sheaf kernel convolution
$\Phi_{H}:\operatorname{Sh}^{pp}_{\Lambda}(M)\longrightarrow\operatorname{Sh}^{pp}_{\varphi_{1}(\Lambda)}(M)$
associated $H$ satisfies $\Phi_{H}(\mathcal{F}_{0})\simeq\mathcal{F}_{1}.$

The convolutions of sheaf kernels $\Phi_{H}$ and $\Phi_{\widetilde{H}}$ from
[GKS12] are described in Appendix B.3. In brief, Theorem 5.5 states that a
sheaf quantization for a Lagrangian filling exists, and Proposition 5.6 shows
that such sheaf quantization is invariant under Hamiltonian isotopies. The two
statements above are slightly different from the results presented in
[JinTreu17]. In particular, we assume that the front of the Legendrian
$\tilde{L}$ is bounded from below, while Jin-Treumann assume boundedness from
above, and we use the proper push forward by the projection $\pi_{M!}$, while
Jin-Treumann use the push forward $\pi_{M*}$. (Our choices are given by the
geometry of Lagrangian surfaces in our setting.) Therefore, we technically
need to provide proofs of the above results, which can be obtained by
appropriately modifying the arguments in [JinTreu17]. These proofs are written
in Appendix B.3.

######  Remark 5.7.

The Hamiltonian invariance in Proposition 5.6, which allows for the Legendrian
boundary condition $\varphi_{t}(\Lambda)$ to vary, is needed in our context.
The fixed boundary condition in [JinTreu17, Section 3.20] is not enough
because in our setting (see Sections 3 and 4) the Legendrian boundaries
change, via a contact isotopy, when the Lagrangian fillings change, via a
Hamiltonian isotopy. In addition, we must also use the Hamiltonian invariance
statement for
$\mathcal{F}_{t}\in\operatorname{Sh}^{pp}_{\varphi_{t}(\Lambda)}(M)$, rather
than just the invariance of the sheaf quantization
$\widetilde{\mathcal{F}}_{t}\in\operatorname{Sh}^{pp}_{\widetilde{\varphi}_{t}(\widetilde{L})}(M)$.
Indeed, the toric charts for the cluster structures we use are defined by a
composition

$H^{1}(\varphi_{t}(L);\Bbbk^{\times})\hookrightarrow\mathcal{M}_{1}(M\times\mathbb{R},\widetilde{\varphi}_{t}(\widetilde{L}))_{0}\hookrightarrow\mathcal{M}_{1}(M,\varphi_{t}(\Lambda))_{0},$

into a derived stack of moduli of (pseudo-perfect) objects of
$\operatorname{Sh}^{cpt}_{\varphi_{t}(\Lambda)}(M)$, and not just in
$\operatorname{Sh}^{cpt}_{\widetilde{\varphi}_{t}(\widetilde{L})}(M\times\mathbb{R})$.
Thus, we need to show that the toric chart itself is a Hamiltonian invariant
and we do so by showing the invariance of the composition above.

####  5.1.3. The moduli space of microframed sheaves

Let $\Sigma$ be a smooth surface and $\Lambda\subset T^{*,\infty}\Sigma$ a
Legendrian link. By [ToenVaquie07], there exists a moduli derived stack
$\mathcal{M}(\Sigma,\Lambda)$ which parametrizes pseudo-perfect objects in the
smooth dg-category $\operatorname{Sh}^{cpt}_{\Lambda}(\Sigma)$. These can be
identified with objects in $\operatorname{Sh}^{pp}_{\Lambda}(\Sigma)$, which
are constructible sheaves with perfect stalks; see Appendix A and Definition
B.24. In particular, if we fix a Maslov potential, we can consider the locus
$\mathcal{M}_{r}(\Sigma,\Lambda)_{0}\subseteq\mathcal{M}(\Sigma,\Lambda)$ of
microlocal rank $r$ sheaves, concentrated in a given degree, with acyclic
stalks at the marked points of $\Sigma$, and respectively the locus
$\mathcal{M}_{r}(\Sigma\times\mathbb{R},\widetilde{L})_{0}\subseteq\mathcal{M}(\Sigma\times\mathbb{R},\widetilde{L})$
of microlocal rank $r$ sheaves, concentrated in a given degree, with acyclic
stalks at $M\times\\{-\infty\\}$. For the Legendrian links we study, these
microlocal rank $r$ sheaves have no negative self-extension groups, and thus
the moduli space $\mathcal{M}(\Sigma,\Lambda)_{0}$ is an Artin stack, though
typically not an algebraic variety. By Theorem 5.5, one can construct an open
embedding of moduli stacks

$\mathcal{L}oc_{r}(L)\hookrightarrow\mathcal{M}_{r}(\Sigma\times\mathbb{R},\widetilde{L})_{0}\hookrightarrow\mathcal{M}_{r}(\Sigma,\Lambda)_{0}$

which define open toric charts on the moduli
$\mathcal{M}_{r}(\Sigma,\Lambda)_{0}$; see Appendix B.4. As shown in
[CasalsWeng22], this Artin stack does only typically admit a cluster
$\mathcal{X}$-structure. In order to obtain a better suited moduli space, we
add the data of certain framings, as follows.

Let $\Lambda\subset T^{*,\infty}\Sigma$ be a Legendrian link equipped with a
Maslov potential and choose a finite set of base points
$T=\\{p_{1},\dots,p_{n}\\}\subset\Lambda$. By definition, the microframed
moduli of microlocal rank $r$ sheaves is defined as

$\mathcal{M}_{r}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}:=\left\\{(\mathcal{F},t_{1},\dots,t_{n})|\mathcal{F}\in\mathcal{M}_{r}(\Sigma,\Lambda)_{0},\,t_{i}:m_{\Lambda}(\mathcal{F})_{p_{i}}\xrightarrow{\sim}\Bbbk^{r}\right\\}.$

It parametrizes those sheaves in $\mathcal{M}_{r}(\Sigma,\Lambda)_{0}$ with
the additional data of a fixed trivialization of microstalks at the base
points. See Definition B.27 and Appendix B.4.212121The terminology
microframing is used to distinguish the notion of framing in [STWZ19]*Section
2.4. Their relation will be discussed in Appendix B.4. A first advantage is
that $\mathcal{M}_{r}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}$ is an algebraic
variety in the cases we study, whereas $\mathcal{M}_{r}(\Sigma,\Lambda)_{0}$
is not. From the perspective of cluster algebras,
$\mathcal{M}_{r}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}$ is better behaved than
the unframed $\mathcal{M}_{r}(\Sigma,\Lambda)_{0}$ in that
$\mathbb{C}[\mathcal{M}_{1}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}]$ is
actually a commutative cluster algebra.222222We expect
$\mathbb{C}[\mathcal{M}_{r}^{\mu,\textit{fr}}(\Sigma,\Lambda)]$ to be a non-
commutative cluster algebra for $r\geq 2$, in some sense of the definition of
a non-commutative cluster algebra, see [GonKon21].

Figure 56. The rectangular regions $D$ and the set of base points
$T=\\{p_{1},\dots,p_{n}\\}$ for a Legendrian (cylindrical or rainbow)
$n$-braid closure.

The symplectic geometric definition of
$\mathcal{M}_{r}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}$ can be described in a
Lie theoretic manner by using affine flags. The following result illustrates
this:

######  Proposition 5.8.

Let $\Lambda\subset T^{*,\infty}\Sigma$ be a Legendrian positive braid
closure, either cylindrical closure or rainbow closure, equipped with a Maslov
potential, $D\subset\Sigma$ an open disk such that $\Lambda\cap T^{*}D$
consist of $n$ parallel strands. Let $T=\\{p_{1},...,p_{n}\\}\subset\Lambda$
be a collection of base points in $D$, one on each strand. Then the
restriction $\mathcal{F}|_{D}$ of a microframed sheaf
$(\mathcal{F},t_{1},\dots,t_{n})\in\mathcal{M}^{\mu,\textit{fr}}_{r}(\Sigma,\Lambda)_{0}$
determines the complete flag

$0=F_{0}\subset F_{1}\subset F_{2}\subset\dots\subset F_{n}\cong\Bbbk^{n},$

and the microframing $(t_{1},\dots,t_{n})$ determines an assigned volume
$\omega_{i}$ on each $F_{i}\,(0\leq i\leq n)$.

###### Proof.

Consider the region $D=D_{0}\cup D_{1}\cup...\cup D_{n}$, microlocal rank 1
sheaves on $D$ is characterized by a complete flag

$B_{k}:0=V_{0}\subset V_{1}\subset...\subset V_{n}\cong\Bbbk^{n}.$

Given the microframing data, we have specified isomorphisms of microstalks

$t_{i}:\,m_{\Lambda^{\prec}_{\beta_{0}}}(\mathcal{F})_{p_{i}}=V_{i}/V_{i-1}\cong\Bbbk.$

Note that the preimage of the the unit $1\in\Bbbk$ determines a vector
$\overline{v}_{i}\in V_{i}/V_{i-1}$. In particular, when $i=1$ we have a
volume form $\omega_{1}=v_{1}$ on $V_{1}$. Inductively, a vector
$\overline{v}_{i}\in V_{i}/V_{i-1}$ and the given volume form $\omega_{i-1}$
on $V_{i-1}$ determines a unique volume form on $V_{i}$ by

$\omega_{i}=\omega_{i-1}\wedge v_{i},$

where $v_{i}$ is any representative in the equivalence class
$\overline{v}_{i}\in V_{i}/V_{i-1}$. Thus we get a collection of volume forms
by induction.

Conversely, consider a collection of volume forms
$\omega_{1},\dots,\omega_{n}$ on $V_{0},V_{1},\dots,V_{n}$. There exists a
vector $v_{i}\in V_{i}/V_{i-1}$ such that

$\omega_{i}=\omega_{i-1}\wedge v_{i}.$

In addition, the equivalence class $\overline{v}_{i}\in V_{i}/V_{i-1}$ is well
defined. The vector $\overline{v}_{i}$ on $V_{i}/V_{i-1}$ then determines a
trivialization $t_{i}:V_{i}/V_{i-1}\cong\Bbbk$ by $t_{i}(v_{i})=1$. ∎

In Appendix B.4, we discuss different notions of framings of sheaves in the
literature [STWZ19, CasalsWeng22] and explain how microlocal merodromy along
relative 1-cycles gives rise to functions the above moduli space.

###  5.2. Sheaf quantization of conjugate Lagrangians

For an alternating Legendrian $\Lambda\subset T^{*,\infty}\Sigma$ and its
conjugate Lagrangian filling $L\subseteq T^{*}\Sigma$, the sheaf quantization
of rank $r$ local systems $\operatorname{Loc}^{r}(L)$ in the conjugate surface
is studied in [STWZ19]*Section 4.3. This subsection briefly summarizes that
construction.

####  5.2.1. Alternating sheaves

Let $\Lambda\subset T^{*,\infty}\Sigma$ be an alternating Legendrian.
Following Theorem 5.5 (see also Theorem B.20 Appendix B.3), which adapts the
results of [JinTreu17], a conjugate Lagrangian filling $L$ gives rise to a
fully faithful sheaf quantization functor

$\operatorname{Loc}^{pp}(L)\hookrightarrow\operatorname{Sh}^{pp}_{\Lambda}(\Sigma).$

In [STWZ19]*Definition 4.13, the sheaves in
$\operatorname{Sh}^{pp}_{\Lambda}(\Sigma)$ which are the sheaf quantizations
of the corresponding conjugate Lagrangians are characterized as follows:

######  Definition 5.9.

Let $\Lambda\subset T^{*,\infty}\Sigma$ be an alternating Legendrian link and
$L$ its conjugate Lagrangian filling. An alternating sheaf is an object in
$\operatorname{Sh}_{\Lambda}^{pp}(\Sigma)$ whose support is contained in the
closure of the union of white and black regions.

In [STWZ19]*Theorem 4.17 & Proposition 4.18 it is shown that this definition
indeed captures the required notion:

######  Theorem 5.10 ([STWZ19]).

Let $\Lambda\subset T^{*,\infty}\Sigma$ be an alternating Legendrian link and
$L$ its conjugate Lagrangian filling. The subcategory of alternating sheaves
in $\operatorname{Sh}_{\Lambda}^{pp}(\Sigma)$ is equivalent to the category of
local systems on $L$. In fact,
$\mathcal{F}\in\operatorname{Sh}_{\Lambda}^{pp}(\Sigma)$ is the image of a
rank $r$ local system on $L$ under the Jin-Treumann sheaf quantization functor
(Theorem 5.5)

$\operatorname{Loc}^{pp}(L)\hookrightarrow\operatorname{Sh}^{pp}_{\widetilde{L}}(\Sigma\times\mathbb{R})\hookrightarrow\operatorname{Sh}^{pp}_{\Lambda}(\Sigma)$

if and only if it is an alternating sheaf with microlocal rank $r$.

Indeed, an alternating sheaf can be characterized in the following way,
following [STWZ19]*Proposition 4.15 & 16\. Locally near a crossing, suppose
that the white region $W$ is the first quadrant while the black region $B$ is
the third quadrant. Then the alternating sheaf $\mathcal{F}$ fits into an
exact triangle

$\mathcal{F}_{\overline{W}}[1]\to\mathcal{F}\to\mathcal{F}_{\overline{B}}\xrightarrow{+1}.$

Therefore, when $\mathcal{F}_{\overline{W}}$ and $\mathcal{F}_{\overline{B}}$
are given, the sheaf $\mathcal{F}$ is classified by
$Ext^{1}(\mathcal{F}_{\overline{B}},\mathcal{F}_{\overline{W}}[1])$. In
addition, the singular support condition
$SS^{\infty}(\mathcal{F})\subset\Lambda$ is satisfied only when the element in
$Ext^{1}(\mathcal{F}_{\overline{B}},\mathcal{F}_{\overline{W}}[1])$ is
invertible.

######  Example 5.11.

Let $\mathcal{F}_{\overline{W}}=\Bbbk_{W}^{r}[1]$,
$\mathcal{F}_{\overline{B}}=\Bbbk_{\overline{B}}^{r}$. Any nonzero element in
$Ext^{1}(\mathcal{F}_{\overline{B}},\mathcal{F}_{\overline{W}}[1])=M_{r\times
r}(\Bbbk)$ defines an alternating sheaf $\mathcal{F}$, which in this case is
the sheaf quantization of a rank $r$ local system on the conjugate Lagrangian
$L$.

####  5.2.2. Microlocal holonomies

By Theorem 5.10, an alternating sheaf comes from a local system of a conjugate
Lagrangian filling. The holonomies of the local system determine the
microlocal holonomies of the sheaf. Let us now explain how to compute the
unsigned microlocal monodromies in Section 5.1.1 and Example 5.4 (see also
Appendix B.2) along a 1-cycle $\gamma_{F}\in H_{1}(L;\mathbb{Z})$ for a null
region $F$, following [STWZ19]*Section 5.2.

Suppose that the front $\pi(\Lambda)$ is the union of the $x$-axis and
$y$-axis on the plane, and $\mathcal{F}$ is an alternating sheaf supported in
the first and third quadrant of the plane. Then the microlocal merodromy
$m_{L,\gamma}(\mathcal{F})$ along the path $\gamma$ from the positive $x$-axis
to the negative $y$-axis is the composition of

  1. (1)

the parallel transport of the microstalk along the horizontal strand of
$\pi(\Lambda)$;

  2. (2)

the isomorphism between the microstalk on the left of the horizontal strand of
$\pi(\Lambda)$ and the stalk in the third quadrant;

  3. (3)

the isomorphism between the stalk in the third quadrant and the microstalk at
the bottom of the vertical strand of $\pi(\Lambda)$.

Note that this is in fact the microlocal holonomy of the corresponding sheaf
$\widetilde{\mathcal{F}}\in\operatorname{Sh}^{pp}_{\widetilde{L}}(\Sigma\times\mathbb{R})$
from Theorem 5.5 Part (1). In an abuse of notation, so as to follow [STWZ19],
we still refer to it as the microlocal holonomy of the alternating sheaf. In
Section 6 we study cluster algebras over $\mathbb{Z}$. Therefore, we need to
fix the signs when computing these microlocal holonomies on Legendrian
surfaces. The details of how to achieve this are explained in Appendix B.2. In
consequence, we need to choose a coherent collection of sign curves (following
Appendix B.2 Definition B.14), which accounts for fixing a relative spin
structure on $L$. The definition of such a coherent collection reads as
follows:

######  Definition 5.12.

Let $L\subset T^{*}\Sigma$ be a conjugate Lagrangian filling of
$\Lambda\subset T^{*,\infty}\Sigma$. Consider line segments $\gamma_{v}\subset
L$ connecting two components of $\Lambda$ at each crossing $v\in\pi(\Lambda)$
such that $\pi(\gamma_{v})$ are embedded arcs in the white region. Let the
1-skeleton $(L,\Lambda)_{\leq 1}$ of $(L,\Lambda)$ be the union of the
boundary $\Lambda$ and the line segments $\gamma_{v}$. By definition, a
coherent collection of sign curves associated to the 1-skeleton is a graph
$P\subseteq\Sigma$ whose vertices are the black vertices and boundary points
in $\Lambda$, such that the number of curves starting from the black vertices
equals their degrees in the bipartite graph.

We leave it to the readers to verify that this definition is compatible with
Appendix B.2, accounting for relative spin structures on $L$, and also the
computation [GonKon21]*Section 3.2 where the $-1$’s in the monodromies appear.

Finally, in the context of conjugate Lagrangian surfaces, the first hint of
the appearance of cluster $\mathcal{X}$-structures, observed for partial
$\mathcal{X}$-structures in [STWZ19, Section 5.1] and proven in general in
[CasalsWeng22, Corollary 1.2], is illustrated by the precise rational
transformation that the microlocal monodromy along 1-cycles undergoes upon
performing a square move:

######  Proposition 5.13 ([STWZ19]*Theorem 5.8).

Let $\Lambda_{0},\Lambda_{1}\subset T^{*,\infty}\Sigma$ be alternating
Legendrians that differ by a square move at the null region $F$, and
$L_{0},L_{1}$ the corresponding conjugate Lagrangians. For
$\mathcal{F}_{0}\in\operatorname{Sh}^{pp}_{\Lambda_{0}}(\Sigma)$ an
alternating sheaf of microlocal rank $1$, its image
$\mathcal{F}_{1}\in\operatorname{Sh}^{pp}_{\Lambda_{1}}(\Sigma)$ is an
alternating sheaf of microlocal rank $1$ if and only if
$m_{L_{0},\gamma_{F}}(\mathcal{F}_{0})\neq-1$. Under this assumption, for any
null region $C$,

$m_{L_{1},\gamma_{C}}(\mathcal{F}_{1})=\begin{cases}m_{L_{0},\gamma_{C}}(\mathcal{F}_{0})\left(1+m_{L_{0},\gamma_{F}}(\mathcal{F}_{0})\right)^{\left<\gamma_{F},\gamma_{C}\right>},&C\neq
F,\\\ m_{L_{0},\gamma_{C}}(\mathcal{F}_{0})^{-1},&C=F.\end{cases}$

As we will see in Section 6, this is a cluster $\mathcal{X}$-transformation in
a cluster $\mathcal{X}$-variety defined by the intersection quiver of
$H_{1}(L;\mathbb{Z})$.

###  5.3. Sheaf quantization of Legendrian weaves

Let $\Lambda\subset T^{*,\infty}\Sigma$ be a Legendrian link and $L\subset
T^{*}\Sigma$ a Lagrangian filling of $\Lambda$ whose Legendrian lift is a
Legendrian weave $\widetilde{L}\subset J^{1}\Sigma$. In this context of
Legendrian weaves, the sheaf quantization of $L$ can be obtained explicitly as
follows.

####  5.3.1. Sheaves singularly supported on weaves

Given the Legendrian weave $\widetilde{L}\subset J^{1}\Sigma$, a microlocal
rank $r$ sheaf
$\widetilde{\mathcal{F}}\in\operatorname{Sh}^{pp}_{\widetilde{L}}(\Sigma\times\mathbb{R})$
with singular support on the Legendrian weave can be described by a framed
flag on the associated $n$-graph $G$, generalizing Example 5.3. This is a
consequence of the discussion in [CasalsZas20]*Section 5. The framed flag
moduli for a weave is described as follows. By definition, a flag of local
systems on $X$ is a local system $E\rightarrow X$ with a complete filtration
$E_{\bullet}$ by local systems $E_{k}\rightarrow X$ such that the total
monodromy preserves the filtration. For $U\subset X$, a flag of sub-local
systems $F_{\bullet}$ on $U$ is said to be compatible with $E_{\bullet}$ if
for $\gamma\in\pi_{1}(U,u)$ and $v\in F_{k}$, $i_{k}(\gamma\cdot
v)=i_{k*}\gamma\cdot i_{k}(v)$, where $i_{k}:F_{k}(u)\hookrightarrow
E_{k}(u)$. In [CasalsZas20]*Section 5, a framed flag for a weave (given by an
$n$-graph $G$) is given by the following data:

  1. (1)

a rank $rn$ local system $E\rightarrow\Sigma$;

  2. (2)

for each face $F$, a flag of local systems $0=V_{0}(F)\subset
V_{1}(F)\subset...\subset V_{n}(F)\cong\Bbbk^{rn}$;

  3. (3)

for each pair of adjacent faces $F_{1}$ and $F_{2}$ sharing a common edge
$e\in G_{i}$, only the $i$-th vector space of $V_{\bullet}(F_{1})$ and
$V_{\bullet}(F_{2})$ are transverse, and there are chosen isomorphisms
$V_{j}(F_{1})\cong V_{j}(F_{2})\,(j\neq i)$.

  4. (4)

by gluing, the isomorphisms define local systems in each region, and these
local systems in each region are compatible with $E$.

####  5.3.2. Microlocal holonomies

Given a Legendrian weave $\widetilde{L}\subset J^{1}\Sigma_{\text{cl}}$ with
boundary $\Lambda\subset J^{1}(\partial\Sigma_{\text{cl}})$, and a sheaf
$\widetilde{\mathcal{F}}\in\operatorname{Sh}^{pp}_{\widetilde{L}}(\Sigma_{\text{op}}\times\mathbb{R})$,
the following examples suffice in order to compute microlocal holonomies for
the weaves we need. We follow Section 5.1.1 and Example 5.4 (see also Appendix
B.2). Given a sheaf
$\widetilde{\mathcal{F}}\in\operatorname{Sh}^{pp}_{\widetilde{L}}(\Sigma_{\text{op}}\times\mathbb{R})$
which is the sheaf quantization of a local system on the free Legendrian weave
$\widetilde{L}$, the monodromy of the local system exactly determines the
microlocal monodromy of the sheaf. The computations of unsigned microlocal
holonomies, according to [STZ17]*Section 5 and [CasalsZas20]*Section 7, are as
follows:

######  Example 5.14.

Suppose $\gamma$ is a path passing transversely through an edge $e_{i}$
starting from the $i$-th sheet of the Legendrian front. Then the corresponding
flags in the two adjacent regions have the same $V_{0},...,V_{i-1}$, and also
the same $V_{i+1},...,V_{n}$. Consider the quotient $V:=V_{i+1}/V_{i-1}$. By
taking $V_{i}/V_{i-1}\subset V_{i+1}/V_{i-1}$, we get two lines $l_{a},l_{b}$
(in microlocal rank $r$, to $r$-dimensional subspaces), equipped with volume
forms $\omega_{a}$ and $\omega_{b}$. The parallel transport map is given by
the isomorphism $l_{a}\xrightarrow{\sim}V/l_{b}$, and hence is computed by
$\omega_{a}\wedge\omega_{b}\in\det(V)$.

######  Example 5.15.

Suppose $\gamma$ is a path passing transversely through the edges
$e_{i},e_{i+1},e_{i}$ starting from the $i$-th sheet of the Legendrian front.
Then the corresponding flags in the four regions have the same
$V_{0},...,V_{i-1}$, and also the same $V_{i+2},...,V_{n}$. Consider the
3-dimensional quotient $V:=V_{i+2}/V_{i-1}$. By taking $V_{i}/V_{i-1}\subset
V_{i+1}/V_{i-1}\subset V_{i+2}/V_{i-1}$, we get four pairs of lines and planes

$(l_{a},l_{A}),\,\,(l_{AB},l_{A}),\,\,(l_{AB},l_{B}),\,\,(l_{b},l_{B}),$

where $l_{AB}=l_{A}\cap l_{B}$. The pairs $(l_{a},l_{A}),(l_{b},l_{B})$ are
equipped with volume forms $(\omega_{a},\omega_{A})$ and
$(\omega_{b},\omega_{B})$. The parallel transport map is given by the
composition

$l_{a}\xrightarrow{\sim}l_{A}/l_{AB}\xrightarrow{\sim}V/l_{B},$

which is computed by $\omega_{a}\wedge\omega_{B}\in\det(V)$.

Given the microlocal merodromy computations in Examples 5.14 and 5.15, we can
compute the microlocal monodromy of $\widetilde{\mathcal{F}}$ along (the
Legendrian lift $\widetilde{L}$ of) the 1-cycles $\gamma\in
H_{1}(\widetilde{L};\mathbb{Z})$, see [CasalsZas20]*Section 7. For simplicity,
here we only describe the microlocal monodromy along $\sf I$-cycles and $\sf
Y$-cycles.

######  Example 5.16 ($\sf I$-cycles).

Suppose $\gamma\in H_{1}(\widetilde{L};\mathbb{Z})$ is an $\sf I$-cycle that
locally lies on the $i$ & $(i+1)$-th sheet of
$\pi_{\text{front}}(\widetilde{L})$. Thus the flags in the four regions have
the same $V_{0},...,V_{i-1}$, and also the same $V_{i+1},...,V_{n}$. Consider
$V=V_{i+1}/V_{i-1}$. Then by taking $V_{i}/V_{i-1}\subset V_{i+1}/V_{i-1}$, we
get four lines $l_{a},l_{b},l_{c},l_{d}\subset V$ in the four regions, such
that any two lines in adjacent regions are transverse, as in Figure 57. Then
the microlocal monodromy $m_{\widetilde{L},\gamma}(\widetilde{\mathcal{F}})$
along $\gamma$ is the composition

$l_{a}\xrightarrow{\sim}V/l_{b}\xrightarrow{\sim}c\xrightarrow{\sim}V/l_{d}\xrightarrow{\sim}l_{a}.$

Therefore, the monodromy is the cross ratio which, assigning each $l_{i}$ a
volume form $\omega_{i}$, by Example 5.14, can be written as

$\left<l_{a},l_{b},l_{c},l_{d}\right>=\frac{\omega_{a}\wedge\omega_{b}}{\omega_{b}\wedge\omega_{c}}\frac{\omega_{c}\wedge\omega_{d}}{\omega_{d}\wedge\omega_{a}}.$

$a$$b$$c$$d$$e$ Figure 57. Neighborhood of a monochromatic edge $e$ with the
data determining a constructible sheaf $F$. As we show, the microlocal
monodromy $m_{\Lambda,\gamma}(F)$ along the 1-cycle $\gamma(e)$ is given by
the cross-ratio $\langle l_{a},l_{b},l_{c},l_{d}\rangle$.

######  Example 5.17 ($\sf Y$-cycles).

Suppose $\gamma\in H_{1}(\widetilde{L};\mathbb{Z})$ is an $\sf Y$-cycle that
locally lies on the $(i-1)$, $i$ & $(i+1)$-th sheet of
$\pi_{\text{front}}(\widetilde{L})$. Then consider $V=V_{i+2}/V_{i-1}$. Then
by taking $V_{i}/V_{i-1},V_{i+1}/V_{i-1}\subset V_{i+1}/V_{i-1}$, we get a
couple of flags as in Figure 58. In particular we have 1-dimensional vector
spaces $l_{a},l_{b},l_{c}$ and 2-dimensional vector spaces
$l_{A},l_{B},l_{C}$. Then by assigning each $l_{i}$ a volume form, by Example
5.15 the monodromy is

$\left<(l_{a},l_{A}),(l_{b},l_{B}),(l_{c},l_{C})\right>=\frac{(\omega_{B}\wedge\omega_{a})(\omega_{C}\wedge\omega_{b})(\omega_{A}\wedge\omega_{c})}{(\omega_{B}\wedge\omega_{c})(\omega_{C}\wedge\omega_{a})(\omega_{A}\wedge\omega_{b})}.$

$\bullet$$\bullet$$\bullet$$(a,A)$$(c,C)$$(b,B)$$(a,ab)$$(b,ab)$$(c,bc)$$(b,bc)$$(c,ac)$$(a,ac)$
Figure 58. Neighborhood of a $\sf Y$-cycle with the data determining a
constructible sheaf $F$. As we compute, the microlocal monodromy
$m_{\Lambda,\gamma}(F)$ along the associated 1-cycle $\gamma$ is given by the
triple ratio of the three transverse flags. Here $l_{ab}=l_{a}+l_{b}$, while
$l_{AB}=l_{A}\cap l_{B}$.

We need to fix the signs when computing microlocal holonomies on Legendrian
surfaces (as explained in Appendix B.2). Hence we choose a coherent collection
of sign curves (following Appendix B.2 Definition B.14 and
[CasalsWeng22]*Section 4.5), which accounts for fixing a relative spin
structure on $\widetilde{L}$.

######  Definition 5.18 ([CasalsWeng22]*Definition 4.15).

Let $\widetilde{L}(\mathfrak{w})\subset J^{1}\Sigma_{\text{cl}}$ be a
Legendrian weave. Then a coherent collection of sign curves is a graph $P$
whose vertices are the trivalent vertices in $\mathfrak{w}$ and boundary
points on $\Lambda$, such that each trivalent vertex has degree 1.

For simplicity, consider the free Legendrian weave
$\widetilde{L}(\mathfrak{w}_{\beta})\subset J^{1}\mathbb{D}^{2}$ with boundary
on a Legendrian positive braid closure $\Lambda_{\beta\Delta^{2}}\subset
J^{1}S^{1}$ as in Subsection 2.2.2.232323Following the identification in
Subsection 2.3, we do not distinguish $\Lambda_{\beta\Delta^{2}}$ and
$\Lambda(\beta\Delta^{2})$ used in Subsection 2.1.1. Then we can choose a
coherent collection of sign curves such that each trivalent vertex is
connected to the boundary crossing in $\beta$ as in Figure 59.

Figure 59. The sign curves (in dashed lines) associated to the Legendrian
weave $\widetilde{L}(\mathfrak{w}_{\beta})\subset J^{1}\mathbb{D}^{2}$ with
boundary being the Legendrian cylindrical braid closure
$\Lambda_{\beta\Delta^{2}}\in J^{1}S^{1}$.

In order to compute microlocal merodromies along relative 1-cycles, we need to
recall the following lemma. We follow the notations in [ShenWeng21,
GaoShenWeng20]: a pair of flags $B,B^{\prime}$ are in relative position
$s_{i}\in S_{n}$ if

$B_{i}\neq B^{\prime}_{i};\;\;B_{j}=B^{\prime}_{j},\,\forall\,j\neq i.$

In addition, two decorations $A,A^{\prime}$ over $B,B^{\prime}$, are called
compatible if there exists $v_{i}\neq v_{i}^{\prime}\in\Bbbk^{n}$ such that

$\omega_{i}=\omega_{i-1}\wedge
v_{i},\,\omega^{\prime}_{i}=\omega^{\prime}_{i-1}\wedge
v^{\prime}_{i};\;\;\omega_{i+1}=\omega_{i-1}\wedge v_{i}\wedge
v^{\prime}_{i};\;\;\omega_{j}=\omega^{\prime}_{j},\,\forall\,j\neq i.$

######  Lemma 5.19 ([ShenWeng21]*Lemma 2.10, [GaoShenWeng20]*Lemma 4.14).

Let $w\in S_{n}$ be an element of the symmetric group. Given
$B\xrightarrow{w}B^{\prime}$, for any decoration $A$ over $B$, there exists a
unique decoration $A^{\prime}$ over $B^{\prime}$ such that
$A\xrightarrow{w}A^{\prime}$ is compatible.

As in Proposition 5.8, we associate $n$ base points on each Legendrian link
$\Lambda_{\beta\Delta^{2}}$ on the top right region in Figure 59. By Lemma
5.19, given a system of flags associated to $\beta\Delta^{2}$, we can start
with a decorated flag on the top right region and determine a compatible
decoration on the flags associated to $\Lambda(\beta\Delta^{2})$
counterclockwise. By Proposition 5.8, the compatible decorations are
equivalent to trivialization data of the microstalks along each strand. These
trivializations coincide with the ones determined by the parallel transport
maps along the paths specified above.242424These paths should be viewed as
capping paths, connecting arbitrary points to the base points (on the top
right region of $\Lambda(\beta\Delta^{2})$) as in the computation of the
Legendrian contact dg-algebra.

######  Lemma 5.20.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid and
$\widetilde{L}(\mathfrak{w}_{\beta})\subset J^{1}\mathbb{D}^{2}$ the
associated free Legendrian weave with boundary on a Legendrian positive braid
closure $\Lambda_{\beta\Delta^{2}}\in J^{1}S^{1}$. Consider the choice of
(micro)framings in Proposition 5.8 and the coherent collection of sign curves
such that each trivalent vertex is connected to the boundary crossing in
$\beta$ (Figure 59).

Then the decorations of flags over $\beta$ determined by microlocal parallel
transport maps along $\beta$ starting from the microframing on the right
coincide with the decorations determined by Lemma 5.19 by following the
decoration on the right.

###### Proof.

At a crossing $s_{i}$ in the braid $\beta$, suppose the flag on the right is
$B$ and the flag on the left is $B^{\prime}$. Suppose the decoration on $B$
(determined by microstalks) is

$\omega_{j}=v_{1}\wedge v_{2}\wedge\dots\wedge v_{j},\;\;1\leq j\leq n.$

At the crossing, the strand going up induces the parallel transport from
$v_{i}$ to $v_{i}$, while the strand going down passes through the sign curve,
and thus induces the parallel transport from $v_{i+1}$ to $-v_{i+1}$. Hence
the decoration on $B^{\prime}$ is

$\begin{array}[]{c}\omega_{j}^{\prime}=v_{1}^{\prime}\wedge
v_{2}^{\prime}\wedge\dots\wedge v_{j}^{\prime},\;\;1\leq j\leq n,\\\
v_{j}^{\prime}=v_{j},\,\forall\,j\neq
i,i+1;\;\;v_{i}^{\prime}=-v_{i+1},\;v_{i+1}^{\prime}=v_{i}.\end{array}$

Hence Lemma 5.19 indeed provides a way to compute parallel transport maps and
microlocal merodromies. ∎

In parallel to Proposition 5.13, the effect of a Legendrian mutation is also
readily computed. By direct computation in [STWZ19]*Proposition 5.8 and
[CasalsZas20]*Lemma 7.11, the change of microlocal monodromies under a
Legendrian mutation along an $\sf I$-cycle or $\sf Y$-cycle $\gamma\in
H_{1}(\widetilde{L};\mathbb{Z})$ reads as follows.

######  Proposition 5.21 ([CasalsWeng22, CasalsZas20]).

Let $\widetilde{L}_{0},\widetilde{L}_{1}\subset J^{1}\Sigma$ be Legendrian
weaves that differ by a Legendrian mutation along an $\sf I$-cycle or $\sf
Y$-cycle $\gamma\in H_{1}(\widetilde{L}_{0};\mathbb{Z})$. Then for microlocal
rank $1$ sheaves
$\widetilde{\mathcal{F}}_{0}\in\operatorname{Sh}^{pp}_{\widetilde{L}_{0}}(\Sigma\times\mathbb{R})$
and
$\widetilde{\mathcal{F}}_{1}\in\operatorname{Sh}^{pp}_{\widetilde{L}_{1}}(\Sigma\times\mathbb{R})$
that are identical away from the standard $3$-ball in $\Sigma\times\mathbb{R}$
where Legendrian mutation is applied, we have

$m_{L_{1},\xi}(\widetilde{\mathcal{F}}_{1})=\begin{cases}m_{L_{0},\xi}(\widetilde{\mathcal{F}}_{0})\big{(}1+m_{L_{0},\gamma}(\widetilde{\mathcal{F}}_{0})\big{)}^{\left<\gamma,\xi\right>},&\xi\neq\gamma,\\\
m_{L_{0},\xi}(\widetilde{\mathcal{F}}_{0})^{-1},&\xi=\gamma.\end{cases}$

Again, as we will see in Section 6, this is a cluster
$\mathcal{X}$-transformation in a cluster $\mathcal{X}$-variety defined by the
intersection quiver of $H_{1}(\widetilde{L};\mathbb{Z})$.

####  5.3.3. Sheaves singularly supported on links via weaves

Following the construction (in the proof of) Theorem 5.5, the sheaf
quantization of the Lagrangian projection of the $\widetilde{L}$ in
$\operatorname{Sh}^{pp}_{\Lambda}(\Sigma)$ in the case of Legendrian weaves is
described as follows.

First, by Theorem 5.5.(1), there is a unique microlocal rank 1 sheaf
$\widetilde{\mathcal{F}}\in\operatorname{Sh}_{\widetilde{L}}^{pp}(\Sigma\times\mathbb{R})$
whose microlocal monodromy along $\widetilde{L}$ is the prescribed rank 1
local system. Then, by Theorem 5.5.(2), the projection
$\pi_{\Sigma}:\Sigma\times\mathbb{R}\rightarrow\Sigma$ defines the sheaf
quantization $\mathcal{F}$ of the Lagrangian filling $L\subset T^{*}\Sigma$ by

$\mathcal{F}=\pi_{\Sigma!}\widetilde{\mathcal{F}}\in\operatorname{Sh}_{\Lambda}^{pp}(\Sigma).$

One can compute that the stalk of $\mathcal{F}$ is determined by the stalk of
$\widetilde{\mathcal{F}}$ at $M\times\\{+\infty\\}$; see also Theorem
B.20.(2). Hence wherever the projection $\widetilde{L}\rightarrow\Sigma$ or
$L\rightarrow\Sigma$ is a (branched) $k$-fold covering, the rank of the stalk
of $\mathcal{F}$ is $k$. Indeed, $\mathcal{F}$ can be described by the diagram
as in Figure 60. Taking proper push forward by the projection $\pi_{\Sigma}$
does not lose any information of the sheaf, as is promised by the full
faithfulness in Theorem 5.5.

In particular, let $\Lambda_{\beta\Delta^{2}}\subset
T^{*,\infty}\mathbb{D}^{2}$ be a cylindrical positive braid closure,
satellited along the unit outward conormal bundle of a disk. If we view
$\widetilde{L}\subset J^{1}\mathbb{D}^{2}\cong
T^{*,\infty}_{\tau>0}(\mathbb{D}^{2}\times\mathbb{R})$ as a free Legendrian
weave which forms a multiple cover of $\mathbb{D}^{2}$, and
$\widetilde{\mathcal{F}}\in\operatorname{Sh}^{pp}_{\widetilde{L}}(\mathbb{D}^{2}\times\mathbb{R})$,
then
$\mathcal{F}|_{\mathbb{D}^{2}\backslash\\{0\\}}=\widetilde{\mathcal{F}}|_{\partial\mathbb{D}^{2}\times\mathbb{R}}$.

Figure 60. The sheaf quantization of a Legendrian weave on $M\times\mathbb{R}$
and the proper push forward on $M$ following Theorem 5.5.

######  Example 5.22.

Consider the special case of the Legendrian weave associated to an
$n$-triangulation described in Subsections 2.2.2 and 2.3. Let $\Sigma$ be a
triangle with vertices $x_{1},x_{2}$ and $x_{3}$, and $\bigcirc_{n}$ the
inward unit conormal bundle of $n$ concentric sectors at the vertices of the
triangle. Let $A_{0}(x)$ be the sector centered at $x$ bounded by the smallest
concentric arc, and $A_{i}(x)$ the region between the $i$-th and $(i+1)$-th
concentric arc (from inside to outside) centered at $x$. Consider any
microlocal rank $1$ sheaf
$\mathcal{F}\in\operatorname{Sh}^{pp}_{\bigcirc_{n}}(\Sigma)$ whose stalks in
$A_{0}(x_{j})$ are acyclic. The stalks of $\mathcal{F}$ in
$A_{0}(x_{j}),A_{1}(x_{j}),\dots,A_{n}(x_{j})$ define a flag

$F_{\bullet}(x_{j}):\,\,0=F_{0}(x_{j})\subset F_{1}(x_{j})\subset...\subset
F_{n}(x_{j})\cong\Bbbk^{n}.$

Then, by [CasalsZas20, Theorem 8.1], $\mathcal{F}$ is the image of a rank $1$
local system on the Legendrian weave $\widetilde{L}(\mathfrak{w}_{n}^{*})$
associated to an $n$-triangulation under the Jin-Treumann functor (Theorem
5.5)

$\operatorname{Loc}^{pp}(\widetilde{L}(\mathfrak{w}_{n}^{*}))\hookrightarrow\operatorname{Sh}^{pp}_{\widetilde{L}(\mathfrak{w}_{n}^{*})}(\Sigma\times\mathbb{R})\hookrightarrow\operatorname{Sh}^{pp}_{\bigcirc_{n}}(\Sigma)$

if and only if $F_{\bullet}(x_{1}),F_{\bullet}(x_{2})$ and
$F_{\bullet}(x_{3})$ are transverse flags.

##  6\. Comparison of Cluster Coordinates and Related Applications

By using the results from Section 5, a Lagrangian filling $L$ of a Legendrian
link $\Lambda\subset T^{*,\infty}\Sigma$ determines fully faithful embeddings

$\operatorname{Loc}^{1}(\widetilde{L})\hookrightarrow\operatorname{Sh}^{1}_{\widetilde{L}}(\Sigma\times\mathbb{R})\hookrightarrow\operatorname{Sh}^{1}_{\Lambda}(\Sigma).$

These result in the following open embeddings of moduli spaces

$H^{1}(L;\Bbbk^{\times})\hookrightarrow\mathcal{M}_{1}(\Sigma,\Lambda)_{0},\;\text{and}\;H^{1}(L\backslash
T,\Lambda\backslash
T;\Bbbk^{\times})\hookrightarrow\mathcal{M}_{1}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}.$

By [CasalsWeng22], microlocal holonomies along certain curves on the
Lagrangian fillings give rise to cluster $\mathcal{X}$-coordinates on
$\mathcal{M}_{1}(\Sigma,\Lambda)_{0}$, and microlocal merodromies give rise to
cluster $\mathcal{X}$-coordinates on
$\mathcal{M}_{1}^{\mu,\textit{fr}}(\Sigma,\Lambda)_{0}$. With the Hamiltonian
isotopies in Section 4, between conjugate Lagrangian surfaces and (Lagrangian
projections of) Legendrian weaves, we can relate cluster structures that arise
from different combinatoric data, either plabic graphs or weaves, and identify
the different cluster coordinates.

First, in Section 6.1, we compare the cluster $\mathcal{A}$-structures on the
double Bott-Samelson cell $\mathrm{Conf}^{e}_{\beta}(\mathcal{C})$ defined in
[ShenWeng21, GaoShenWeng20] using plabic graphs and genralized minors, and the
framed moduli of sheaves
$\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}$,
defined in [CasalsWeng22] using microlocal merodromies along Legendrian
weaves, and hence prove Theorem 1.2. As a corollary, we explain in Section
6.1.3 how to deduce the identifications between cluster
$\mathcal{X}$-coordinates.

Second, in Section 6.2, we compare the cluster $\mathcal{X}$-structures on the
moduli space of framed local systems on punctured surfaces
$\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma)$, comparing the cluster
coordinates defined in [FockGon06a] using plabic graphs, the ones defined in
[GMN14] using non-abelianization maps, relating them to weaves, and the ones
defined using microlocal monodromies of sheaves [STWZ19, CasalsZas20]. In
particular, this proves Theorem 1.3.

###  6.1. Cluster coordinates on flag moduli over $\mathbb{D}^{2}$ and Theorem
1.2

Consider the moduli space of microlocal sheaves for a positive cylindrical
braid closure $\Lambda_{\beta\Delta^{2}}\in T^{*,\infty}\mathbb{D}^{2}$ for
$\beta\in\mbox{Br}_{n}^{+}$.252525Following the convention in Subsection
2.1.1, the strands in the braids are labeled from bottom to top, i.e. the
crossing $s_{1}$ is on the bottom and $s_{n-1}$ is on the top.. Let us study
the cluster $\mathcal{A}$-coordinates on the moduli space of (micro)framed
microlocal rank 1 sheaves
$\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}$,
which, according to [BFZ05], also determine the cluster
$\mathcal{X}$-variables on the moduli of (unframed) sheaves
$\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}$.

####  6.1.1. Flag moduli over $\mathbb{D}^{2}$ and double Bott-Samelson cells

Proposition 5.8, see also Appendix B.4 (and Proposition B.35), describes the
moduli space of framed microlocal rank 1 sheaves in
$\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$
for the Legendrian rainbow closure of a positive $n$-braid $\beta$, see
[ShenWeng21]. The moduli of (unframed) microlocal rank 1 sheaves
$\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$ has been studied
in [ShenWeng21] as double Bott-Samelson cells. Denote by $B_{0}$ a flag

$B_{0}:0=V_{0}\subset V_{1}\subset\dots\subset V_{n}\cong\Bbbk^{n},$

and denote by $B^{0}$ a flag

$B^{0}:\Bbbk^{n}\cong V^{\prime}_{n}\twoheadrightarrow\dots\twoheadrightarrow
V^{\prime}_{1}\twoheadrightarrow V^{\prime}_{0}=0.$

They are said to be in opposite positions if $V_{j}=V_{j}^{\prime}$.

######  Definition 6.1.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive $n$-braid. The double Bott-
Samelson cell $\mathrm{Conf}_{\beta}^{e}(\mathcal{B})$ is the moduli stack of
sequences of $\operatorname{PGL}_{n}$-flags

$\textstyle{B^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{1}}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{2}}}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{3}}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{k}}}$$\textstyle{B_{k}.}$

where $B_{0},B_{1},\dots,B_{k}$ are $\operatorname{PGL}_{n}$-flags such that
$B_{j-1},B_{j}$ are in relative position $s_{i_{j}}\in S_{n}$, and
$B_{0},B^{0}$ are in opposite positions.

By definition, the half-decorated double Bott-Samelson cell
$\mathrm{Conf}_{\beta}^{e}(\mathcal{C})$ is the moduli stack of sequences of
$\operatorname{PGL}_{n}$-flags

$\textstyle{B^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{1}}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{2}}}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{3}}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{k}}}$$\textstyle{B_{k}.}$

together with a decoration $A_{k}$ over $B_{k}$, where
$B_{0},B_{1},\dots,B_{k}$ are $\operatorname{PGL}_{n}$-flags such that
$B_{j-1},B_{j}$ are in relative position $s_{i_{j}}\in S_{n}$, and
$B_{0},B^{0}$ are in opposite positions.

######  Theorem 6.2 ([ShenWeng21]*Theorem 6.11).

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid and
$\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$ the moduli of
microlocal rank 1 sheaves on the Legendrian rainbow closure
$\Lambda_{\beta}^{\prec}$. Then
$\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$ is isomorphic to
the double Bott-Samelson cell $\mathrm{Conf}_{\beta}^{e}(\mathcal{B})$.

The framed version is as follows. Consider the (micro)framed moduli of
sheaves, following Section 5.1.3 and Appendix B.4 (see Definition B.27),
parametrizing sheaves with fixed trivializations of microstalks at base
points. We fix the framing data consisting of $n$ base points
$T=\\{p_{1},\dots,p_{n}\\}$ on the right of the braid $\beta$, with one point
in each strand, as in Proposition 5.8 (Figure 56). By Proposition 5.8, the
(micro)framing data is equivalent to a decoration on the flags; these are
oftentimes called affine flags, decorated flags or principal flags as well.

######  Corollary 6.3.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid and
$\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$
the framed moduli space of microlocal rank $1$ sheaves on the Legendrian
rainbow closure $\Lambda_{\beta}^{\prec}$, with framing data determined by $n$
base points $T=\\{p_{1},\dots,p_{n}\\}$. Then
$\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$
is isomorphic to the half decorated double Bott-Samelson cell
$\mathrm{Conf}_{\beta}^{e}(\mathcal{C})$.

######  Remark 6.4.

Note that our convention of half decorated Bott-Samelson cells is different
from [GaoShenWeng20], as in our case the decorated flags below the cusps lie
in $\mathcal{L}_{n}/\mathrm{U}_{+}$ instead of
$\mathcal{L}_{n}/\mathrm{U}_{-}$, where $\mathrm{U}_{\pm}$ are the unipotent
subgroup of upper/lower triangular matrices. It can be shown that different
conventions actually give isomorphic moduli spaces.

Let us identify the sheaf moduli for the rainbow closure
$\Lambda_{\beta}^{\prec}$ and the cylindrical closure
$\Lambda_{\beta\Delta^{2}}$ where $\Delta$ is the half twist, we will compute
microlocal holonomies over the latter moduli space by considering Legendrian
weaves.

######  Proposition 6.5.

Let $\beta\in Br_{n}^{+}$ be a positive braid, $\Delta$ be the half twist,
$\Lambda_{\beta\Delta^{2}}\subset T^{*,\infty}\mathbb{D}^{2}$ the Legendrian
cylindrical braid closure of $\beta\Delta^{2}$ and
$\Lambda_{\beta}^{\prec}\subset T^{*,\infty}_{\eta<0}\mathbb{D}^{2}$ the
Legendrian rainbow braid closure of $\beta$. Then there is an isomorphism

$\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}\cong\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}.$

###### Proof.

Consider
$\mathcal{F}_{rain}\in\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$.
In the leftmost region, below the cusps we have a complete flag

$B_{0}:\,\,0=V_{0}\hookrightarrow V_{1}\hookrightarrow
V_{2}\hookrightarrow\dots\hookrightarrow V_{n}\cong\Bbbk^{n},$

while above the cusps we have a complete flag

$B^{0}:\,\,\Bbbk^{n}\cong V_{n}\twoheadrightarrow\dots\twoheadrightarrow
V_{2}\twoheadrightarrow V_{1}\twoheadrightarrow V_{0}=0,$

such that compositions $V_{i}\hookrightarrow\dots\hookrightarrow
V_{n}\twoheadrightarrow\dots\twoheadrightarrow V_{i}$ is the identity. Focus
only on the $n$ lower half strands below the cusps. Applying Reidemeister II
moves (as in Subsection 2.1.1 Figure 6) we introduce extra crossings between
$n$ lower half strands that form a half twist $\Delta$. When pulling out the
2nd lowest strand by a Reidemeister II move, we introduce a new region in the
stratification by the front projection with stalk $\ker(V_{2}\rightarrow
V_{1})$. After pulling out the $j$-th lowest strand, we introduce $j-1$ new
regions in the stratification with stalks

$\ker(V_{j}\rightarrow
V_{j-1})\hookrightarrow\dots\hookrightarrow\ker(V_{j}\rightarrow
V_{2})\hookrightarrow\ker(V_{j}\rightarrow V_{1})\hookrightarrow V_{j}.$

Hence after pulling out all strands via Reidemeister II moves, on the left of
the half twist $\Delta$, we now have the complete flag 262626We denote it by
$B_{\text{bot}}$ following the convention in Subsection 2.1.1 and 2.2.2, that
$\Lambda_{\beta}^{\prec}$ has the braid $\beta$ on the bottom while the
cylindrical closure $\Lambda_{\beta\Delta^{2}}$ has to be flipped upside down
with the braid $\beta$ on the top. Indeed, the flag $B_{\text{bot}}$ will
correspond to the sheaf $\mathcal{F}_{cyl}$ in the region on the bottom of
$\Lambda_{\beta}$.

$B_{\text{bot}}:\,\,0\hookrightarrow\ker(V_{n}\rightarrow
V_{n-1})\hookrightarrow\dots\hookrightarrow\ker(V_{n}\rightarrow
V_{2})\hookrightarrow\ker(V_{n}\rightarrow V_{1})\hookrightarrow
V_{n}\cong\Bbbk^{n}.$

With the Reidemeister II moves, we have introduced a half twist $\Delta$ on
both sides of $\Lambda_{\beta}^{\prec}$ and a corresponding system of flags.
In addition, the flag in the leftmost region and in the rightmost region are
both $B_{\text{bot}}$ because of the conditions coming from cusps on both
sides. Therefore, we get a sheaf
$\mathcal{F}_{cyl}\in\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}$
determined by the system of flags, as on the right of Figure 61.

Figure 61. The identification between sheaves with singular support in the
rainbow closure $\Lambda_{\beta}^{\prec}$ (left) and the cylindrical closure
$\Lambda_{\beta\Delta^{2}}$ before flipped upside down (right). The Legendrian
in the middle is obtained by applying Reidemeister II moves to the Legendrian
rainbow closure on the left. The grey regions are where the half twist
$\Delta$ is introduced.

Conversely, given
$\mathcal{F}_{cyl}\in\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}$
as on the right of Figure 61, we have a sequence of flags in the regions
formed by the braid $\Delta\beta\Delta$. Hence we can define a corresponding
sheaf $\mathcal{F}_{rain}^{\prime}$ in the regions below all the upper half
strands in the middle of Figure 61, which are formed by the braid
$\Delta\beta\Delta$. In order to define the sheaf
$\mathcal{F}_{\text{rain}}^{\prime}$ on the whole plane in the middle of
Figure 61 we just need to note that the stalks in all upper strata are
determined by the given data below all the upper half strands. Indeed, suppose
the stalks of the sheaf $\mathcal{F}_{cyl}$ on the top region is determined by
the flag

$B_{\text{bot}}:\,\,0\hookrightarrow W_{1}\hookrightarrow
W_{2}\hookrightarrow\dots\hookrightarrow W_{n}\cong\Bbbk^{n}.$

Then the stalks of $\mathcal{F}^{\prime}_{rain}$ on the leftmost region of
$\Delta$, i.e. in the $n$ strata right below the lowest upper half strand, are
determined by the same flag $B_{\text{bot}}$. The stalks of the sheaf in the
$n-1$ regions (from bottom to top) in between the lowest and 2nd lowest upper
half strands have to be the flag $W_{2,\bullet}:0\hookrightarrow
W_{21}\hookrightarrow W_{22}\hookrightarrow\dots\hookrightarrow W_{2,n-1}$
where

$\begin{split}W_{21}=\mathrm{coker}(W_{1}&\rightarrow
W_{2}),W_{22}=\mathrm{coker}(W_{2}\rightarrow W_{3}\oplus W_{21}),\dots,\\\
W_{2,n-1}&=\mathrm{coker}(W_{n-1}\rightarrow W_{n}\oplus
W_{2,n-2}).\end{split}$

Suppose the stalks in between the $(j-1)$-th and $j$-th lowest upper half
strands are determined by the flag $W_{j-1,\bullet}$. For the stalks in
between the $j$-th and $(j+1)$-th lowest upper half strands have to be the
flag $W_{j,\bullet}$ where

$\begin{split}W_{j1}=\mathrm{coker}(W_{j-1,1}&\rightarrow
W_{j-1,2}),W_{j2}=\mathrm{coker}(W_{j-1,2}\rightarrow W_{j-1,3}\oplus
W_{j1}),\dots,\\\ W_{j,n-j+1}=&\,\mathrm{coker}(W_{j-1,n-j+1}\rightarrow
W_{j-1,n-j+2}\oplus W_{j,n-j}).\end{split}$

Therefore inductively we are able to define the sheaf
$\mathcal{F}_{rain}^{\prime}$ on the whole plane in the middle of Figure 61.
By Reidemeitser II moves we get a sheaf
$\mathcal{F}_{rain}\in\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}$
on the left of Figure 61, and the stalks in the region stratified by $\beta$
are the same as $\mathcal{F}_{cyl}$. Hence in particular we know that this is
inverse to the previous assignment. ∎

Since the framing data is simply a decoration on one of the flags, it is
preserved in the above bijection. Hence, similar to Corollary 6.3, we can show
the following corollary from Proposition 6.5.

######  Corollary 6.6.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid, $\Delta$ be the half
twist, $\Lambda_{\beta\Delta^{2}}\subset T^{*,\infty}\mathbb{D}^{2}$ the
Legendrian cylindrical braid closure of ${\beta\Delta^{2}}$ and
$\Lambda_{\beta}^{\prec}\subset T^{*,\infty}\mathbb{D}^{2}$ the Legendrian
rainbow braid closure of $\beta$. Then there is an isomorphism

$\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta}^{\prec})_{0}\cong\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}.$

######  Remark 6.7.

More conceptually, one can view
$\Lambda_{\beta\Delta^{2}},\Lambda_{\beta}^{\prec}$ as Legendrians in the
contact boundary $S^{3}$ of the Weinstein manifold $\mathbb{D}^{4}$, and adapt
the proof in [GPS18a, GPS18b] or [NadShen20] by translating the sheaf category
with singular support condition into the microlocal sheaf category over the
relative Lagrangian skeleton of $(\mathbb{D}^{4},\Lambda)$ and apply Weinstein
deformation invariance, keeping track of the microlocal rank. However, the
proof above gives an explicit bijection that could be of independent interest
and makes the following concrete computation possible.

Moreover, we can choose relative spin structures by assigning sign points and
sign curves as in Definition 5.18 (Appendix B.2 Definition B.12 and B.14) on
$\Lambda_{\beta\Delta^{2}}$ and $\Lambda_{\beta}^{\prec}$ as in Lemma 5.19,
and consider the decoration of the flags.

Note that a complete flag of vector spaces is uniquely determined by a coset
$x\mathrm{B}_{+}\in\operatorname{PGL}_{n}/\mathrm{B}_{+}$, where
$\mathrm{B}_{+}$ is a Borel subgroup; e.g. the subgroup of all upper
triangular invertible matrices. A decorated complete flag of vector spaces
with a common ambient volume form is uniquely determined by a coset
$x\mathrm{U}_{+}\in\mathcal{L}_{n}/\mathrm{U}_{+}$, where $\mathrm{U}_{+}$ is
a unipotent subgroup, e.g. the subgroup of all strictly upper triangular
matrices.

######  Lemma 6.8.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid, $\Delta$ be the half
twist, $\Lambda_{\beta\Delta^{2}}\subset T^{*,\infty}\mathbb{D}^{2}$ the
Legendrian cylindrical braid closure of $\beta\Delta^{2}$ and
$\Lambda_{\beta}^{\prec}\subset T^{*,\infty}\mathbb{D}^{2}$ the Legendrian
rainbow braid closure of $\beta$. For each crossing in $\beta$, assign a sign
point on the lower right half strand as in Definition 5.18. Suppose that the
unique compatible decorations of flags over $\beta$ in Lemma 5.19 is given by

$U_{+}\xrightarrow{s_{i_{1}}}x_{1}U_{+}\xrightarrow{s_{i_{2}}}\dots\xrightarrow{s_{i_{k-1}}}x_{k-1}U_{+}\xrightarrow{s_{i_{k}}}x_{k}U_{+}.$

Then the decoration of the flag $B_{\text{bot}}$ in the bottom region of
$\Lambda_{\beta}$272727The convention follows the one in Subsection 2.1.1 and
2.2.2. determined by microlocal parallel transport maps is

$\omega_{\text{bot},j}=e_{n}\wedge e_{n-1}\wedge\dots\wedge
e_{n-j+1},\;\;1\leq j\leq n.$

###### Proof.

Lemma 5.20 shows that the decorations for the flags over $\beta$ determined by
compatible decorations coincide with the decorations determined by microlocal
parallel transport maps. Therefore we may assume that decoration of the flag
$B_{0}$ on the leftmost region of $\beta$ is determined by volume forms

$\omega_{0,j}=e_{1}\wedge e_{2}\wedge\dots\wedge e_{j},\;\;1\leq j\leq n,$

or equivalently, by Proposition 5.8 the microstalks on the $n$ strands in the
leftmost region are $e_{1},e_{2},\dots,e_{n}$. Then the microlocal parallel
transport maps in the region $\Delta$ determines the microstalks on the top by
$e_{n},e_{n-1},\dots,e_{1}$. Therefore, Proposition 5.8 shows that the
decoration of $B_{bot}$ is determined by

$\omega_{\text{bot},j}=e_{n}\wedge e_{n-1}\wedge\dots\wedge
e_{n-j+1},\;\;1\leq j\leq n.$

This completes the proof. ∎

####  6.1.2. Cluster $\mathcal{A}$-coordinates on double Bott-Samelson cells

The cluster $\mathcal{A}$-coordinates on half decorated double Bott-Samelson
cells $\mathrm{Conf}_{\beta}^{e}(\mathcal{C})$ constructed in [ShenWeng21,
GaoShenWeng20] are computed by generalized minors. Let us show that these
cluster coordinates coincide with certain microlocal merodromies of framed
sheaves (so as to compare with [CasalsWeng22, CasalsZas20]).

Following Lemma 5.19, given any point in the half-decorated Bott-Samelson
cell, we can uniquely determine compatible decorations clockwise from right to
left and then to the top as in Gao-Shen-Weng [GaoShenWeng20]. In particular,
we get a unique decorated sequence of flags

$\textstyle{A^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{1}}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{2}}}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{3}}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{k}}}$$\textstyle{A_{k}.}$

(Note that by Lemma 5.19, the decorations from $A_{0}$ to $A_{k}$ are indeed
the ones determined by microlocal parallel trasnport maps.) Since decorated
flags with a common ambient volume form can be characterized by cosets
$x\mathrm{U}_{+}\in\mathcal{L}_{n}/\mathrm{U}_{+}$, we can fix a
trivialization for the decorated flag $A_{0}$ and write the sequence of flags
as

$\textstyle{U_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{1}}}$$\textstyle{x_{1}U_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{2}}}$$\textstyle{x_{2}U_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{3}}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{k}}}$$\textstyle{x_{k}U_{+}.}$

Consider the triangulation associated to the pair $(e,\beta)$ on top of Figure
62. On top of the triangulation, we draw $n-1$ parallel lines, labeled from
top to bottom.282828Following the conventions in Subsection 2.1.1, in the
string diagram and the plabic fence, the strands are labeled from top to
bottom, (i.e. $s_{1}$ is on the top while $s_{n-1}$ is on the bottom), which
is opposite to the labeling of the crossings in the Legendrian braid closure.
Depending on the labeling of the base, each triangle places a node at one of
the lines, cutting it into segments called strings. For each string $a$ on the
$i$-th line, it will at least cross one diagonal in the triangulation, say,
the $j$-th diagonal connecting $\mathrm{U}_{-}$ and $x_{j}\mathrm{U}_{+}$.
Then we define the cluster $\mathcal{A}$-coordinate associated to $a$ as the
$i$-th principal minor of $x_{j}$, i.e.

$A_{a}=\Delta_{i}(x_{j}).$

(Note that there may be more than one choice for the diagonal $1\leq j\leq k$,
but it turns out that any diagonal that intersects the string $a$ will define
the same function $A_{a}$, see [ShenWeng21]*Proposition 3.21.)

Figure 62. The diagram of flags on the top illustrates a point in
$\mathrm{Conf}^{e}_{\beta}(\mathcal{C})$ for the positive braid
$\beta=s_{1}s_{2}s_{1}^{2}s_{2}$. The decoration of the flags are determined
clockwise from right to left and then to the top. The diagram in the middle is
the string diagram (consisting of all red strings, either closed or extendable
to the right) associated to the grey triangulation. The one on the bottom is
the plabic fence $\mathbb{G}_{\beta}$, where closed faces correspond to closed
strings, and half-open faces correspond to the half-open strings.

In order to interpret the geometric meaning of the above minors, we first
identify the set of strings with the set of null regions (i.e. faces) in the
corresponding plabic fence associated to the braid $\mathbb{G}_{\beta}$. The
following lemma is immediate from Figure 62.

######  Lemma 6.9.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid. Then the set of closed
strings $I_{\text{str}}^{\text{cl}}$ in the string diagram of the
triangulation associated to $(e,\beta)$ corresponds bijectively to the set of
closed null regions $I_{\text{fence}}^{\text{cl}}$ in the plabic fence
associated to $\beta$.

As a result, the set of closed strings corresponds bijectively to the basis of
$H_{1}(L(\mathbb{G}_{\beta});\mathbb{Z})$ for the conjugate Lagrangian in
Subsection 2.1.2, and by Corollary 4.1, is identified with the basis of
$H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})$ for the Legendrian weave
$\mathfrak{w}_{\beta}$ by the corresponding long $\sf I$-cycles in Subsection
2.2.1292929In this section, abusing notations, we write $\mathfrak{w}_{\beta}$
for the weave instead of $\widetilde{L}(\mathfrak{w}_{\beta})$.. Moreover, the
half open strings are relative 1-cycles on the conjugate Lagrangian, and by
Corollary 4.1 they are identified with the relative $\sf I$-cycles in the
weave.

Based on the string diagram and the plabic fence, we now explain a choice of
dual relative 1-cycles in
$(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*})$ corresponding to
all strings in order to define cluster $\mathcal{A}$-coordinates from sheaves
on $\mathfrak{w}_{\beta}$, where

$\mathfrak{w}_{\beta}^{*}=\mathfrak{w}_{\beta}\backslash
T=\mathfrak{w}_{\beta}\backslash\\{p_{1},\dots,p_{n}\\},\;\;\Lambda_{\beta\Delta^{2}}^{*}=\Lambda_{\beta\Delta^{2}}\backslash
T=\Lambda_{\beta\Delta^{2}}\backslash\\{p_{1},\dots,p_{n}\\}.$

The $k$ grey edges in the triangulation in Figure 62 connecting the top
decorated flag $\mathrm{U}_{-}$ and $x\mathrm{U}_{+}$ determine $n$ vertical
slices that split the front projection of $\mathfrak{w}_{\beta}$ (in
particular, we choose the $k$ slices so that they avoid all the candy twists
in the Legendrian weave), as depicted in Figure 63. On each slice we have the
Legendrian braid $\Delta$ connecting $n$ points on both ends.

For a string $a$ on the $i$-th level that intersects the $j$-th diagonal, we
consider the $j$-th vertical slice in the Legendrian weave, whose front
consists of $n$ strands. Denote by $\delta_{a,j,r}$ the strand that starts
from the $r$-th level on the top of the slice to the $(n-r)$-the level on the
bottom of the slice in Figure 63. Define

$\eta_{a,j}=\sum_{r=1}^{i}\xi_{a,j,r}\in
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z}).$

From the definition, if a string $a$ intersects the $j$-th and $j^{\prime}$-th
diagonal in the triangulation, then $\eta_{a,j}$ and $\eta_{a,j^{\prime}}$ are
isotopic relative to $\Lambda_{\beta\Delta^{2}}^{*}$.

Figure 63. The vertical slices in the Legendrian weave $\mathfrak{w}_{\beta}$
in grey and the strands $\xi_{a,j,r}$ on each vertical slice. Two closed
$1$-cycles on $\mathfrak{w}_{\beta}$ corresponding to the closed strings on
the 1st and 2nd levels are respectively drawn in orange and yellow.

Denote by $\eta_{a}$ the homology class in
$H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$
associated to the string $a$.

######  Lemma 6.10.

Let $\beta\in\mbox{Br}_{n}^{+}$ be a positive braid and $\Delta$ be the half
twist. Then $\\{\eta_{a}\,|\,a\in I_{\text{str}}\\}$ forms a basis of
$H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$,
and the subset $\\{\eta_{a}|a\in I_{\text{str}}^{\text{cl}}\\}$ forms a basis
of $H_{1}(\mathfrak{w}_{\beta},\Lambda_{\beta\Delta^{2}};\mathbb{Z})$ dual to
$\\{\gamma_{a}|a\in I_{\text{str}}^{\text{cl}}\\}$ in
$H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})$ via the intersection product.

###### Proof.

We first check that $\\{\eta_{a}|a\in I_{\text{str}}^{\text{cl}}\\}$ forms a
dual basis of $\\{\gamma_{a}|a\in I_{\text{str}}^{\text{cl}}\\}$ via the
intersection product. We need to check that

$\left<\eta_{a},\gamma_{c}\right>=\delta_{ac}.$

First, suppose $a\in I_{\text{str}}^{\text{cl}}$ is on the $i$-th level of the
string diagram, and $c\in I_{\text{str}}^{\text{cl}}$ on the $i^{\prime}$-th
level. Note that according to the description of the 1-cycles
$\\{\gamma_{c}|c\in I_{\text{str}}^{\text{cl}}\\}$, such a cycle $\gamma_{c}$
will intersect the slice of the weave at 2 points near the $i^{\prime}$-th
blue crossing with opposite signs as in Figure 63 and 64. Hence

$\left<\xi_{a,j,i},\gamma_{c}\right>=1,\,\,\,\left<\xi_{a,j,i+1},\gamma_{c}\right>=-1,\,\,\,\left<\xi_{a,j,r},\gamma_{c}\right>=0,\,\forall\,r\neq
i,i+1.$

This shows that indeed $\left<\eta_{a},\gamma_{c}\right>=\delta_{ac}$. Hence
$\\{\eta_{a}|a\in I_{\text{str}}^{\text{cl}}\\}$ forms a dual basis of
$\\{\gamma_{a}|a\in I_{\text{str}}^{\text{cl}}\\}$.

Figure 64. The closed 1-cycles on $\mathfrak{w}_{\beta}$ that start from a
trivalent vertex in $G_{i}\,(1\leq i\leq n-1)$, which correspond to the closed
1-cycles on the plabic fence $\mathbb{G}_{\beta}$ on the $i$-the level,
intersect the vertical slices at the $i$-th blue crossing.

Now we show that $\\{\eta_{a}|a\in I_{\text{str}}\\}$ is a basis of
$H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$.
Consider the exact sequence

$0\rightarrow
H_{1}(\mathfrak{w}_{\beta},\Lambda_{\beta\Delta^{2}};\mathbb{Z})\rightarrow
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})\rightarrow\widetilde{H}_{0}(T;\mathbb{Z})\rightarrow
0.$

We have shown that $\\{\eta_{a}|\,a\in I_{\text{str}}^{\text{cl}}\\}$ forms a
dual basis of $\\{\gamma_{a}|\,a\in I_{\text{str}}^{\text{cl}}\\}$, and thus
it suffices to show that there is a backward map

$\widetilde{H}_{0}(T;\mathbb{Z})\rightarrow
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$

that splits the exact sequence, and send generators to the set
$\\{\eta_{a}|\,a\in I_{\text{str}}\backslash I_{\text{str}}^{\text{cl}}\\}$.
Indeed, consider $s_{i}^{\prime}$ to be to be the semicircle in
$\mathfrak{w}_{\beta}$ around the base point
$p_{i}\in\Lambda_{\beta\Delta^{2}}$. This defines such a backward map, and
clearly

$\xi_{i}^{\prime}\in\mathrm{coker}(H_{1}(\mathfrak{w}_{\beta},\Lambda_{\beta\Delta^{2}};\mathbb{Z})\rightarrow
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})).$

Note that $\xi_{i}^{\prime}$ is isotopic to the strand $\xi_{i}$ on the
rightmost slice of the weave relative to $\Lambda_{\beta\Delta^{2}}^{*}$. For
$a\in I_{\text{str}}\backslash I_{\text{str}}^{\text{cl}}$ the unique half
open string on the $i$-th level of the string diagram, since
$\eta_{a}=\sum_{r=1}^{i}\xi_{i}$, we conclude that they form a basis of
$\widetilde{H}_{0}(T;\mathbb{Z})\hookrightarrow
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$. ∎

Finally, under the isomorphism in (the proof of) Corollary 6.6, we show how to
identify the cluster $\mathcal{A}$-coordinates in
$\mathrm{Conf}_{\beta}^{e}(\mathcal{C})$ defined in [ShenWeng21] with the
microlocal merodromies along relative 1-cycles in
$H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$
specified in the above construction.

###### Proof of Theorem 1.2.

Consider a framed sheaf
$\mathcal{F}\in\mathcal{M}_{1}^{\mu,\textit{fr}}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}$
that is the sheaf quantization of a framed local system on
$\mathfrak{w}_{\beta}$. By Corollary 6.6, it is represented by the decorated
system of flags

$\textstyle{U_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{1}}}$$\textstyle{x_{1}U_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{2}}}$$\textstyle{x_{2}U_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{3}}}$$\textstyle{\dots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i_{k}}}$$\textstyle{x_{k}U_{+},}$

We show that for a given string $a\in I_{\text{str}}$ on the $i$-th level of
the string diagram that intersects the $j$-th diagonal, the microlocal
merodromy

$m_{L(\mathfrak{w}_{\beta}),\eta_{a,j}}(\widetilde{\mathcal{F}})=\Delta_{i}(x_{j}).$

Consider the $j$-th vertical slice of the Legendrian weave
$\mathfrak{w}_{\beta}$ as in Figure 63. On the bottom of the slice, by
Proposition 6.5, we have a decorated flag represented by

$(e_{n},e_{n-1},\dots,e_{2},e_{1}).$

the trivialization data of microlocal stalks from top to bottom are given by
$\Bbbk e_{n},\Bbbk e_{n-1},\dots,$ $\Bbbk e_{2},\Bbbk e_{1}$. On the top of
the slice, we have a decorated flag represented by the matrix

$x_{j}=\left(v_{j1},v_{j2},\dots,v_{jn}\right),$

i.e. the trivialization data of microlocal stalks from top to bottom are given
by $\Bbbk v_{j1},\Bbbk v_{j2},\dots,$ $\Bbbk v_{j,n-1},\Bbbk v_{jn}$. Note
that $\eta_{a,j}=\sum_{r=1}^{i}\xi_{a,j,r}$. Along the strand $\xi_{a,j,r}$,
the parallel transport map is the isomorphism

$t_{a,j,r}:\Bbbk\,v_{jr}\xrightarrow{\sim}\Bbbk\,e_{r}.$

Whenever there is a crossing where on the left the assigned vectors associated
to the flag are $v_{1},v_{2}$ and one the right the vectors are $u_{2},u_{1}$,
we may always assume that the isomorphism
$\Bbbk\left<v_{1},v_{2}\right>\xrightarrow{\sim}\Bbbk\left<u_{2},u_{1}\right>$
are given by an upper triangular matrix $R\in\operatorname{GL}_{2}(\Bbbk)$
with

$R(v_{1})=p_{1}u_{1},\,\,\,R(v_{2})=q_{1}u_{1}+q_{2}u_{2}.$

Therefore, under the above convention of trivializations, we may assume by
induction that the isomorphism
$R_{a,j,12...i-1,i}:\Bbbk\left<v_{j1},v_{j2},\dots,v_{jn}\right>\xrightarrow{\sim}\Bbbk\left<e_{n},e_{n-1},\dots,e_{1}\right>$
satisfies

$R_{a,j,12\dots
i-1,i}(v_{jr})=\sum_{s=1}^{r}t_{a,j,sr}e_{s},\,\,\,t_{a,j,rr}=t_{a,j,r},\,\,\,1\leq
r\leq n.$

The microlocal merodromy along $\eta_{a,j}$ is $t_{a,j,12\dots
i-1,i}=\prod_{r=1}^{i}t_{a,j,r}$, so since $R_{a,j,12\dots i-1,i}$ is an upper
triangular matrix, we can conclude that $t_{a,j,12\dots i-1,i}$ is indeed the
isomorphism

$t_{a,j,12\dots i-1,i}=\det(R_{a,j,12\dots i-1,i}):\Bbbk\,v_{j1}\wedge
v_{j2}\wedge\dots\wedge v_{ji}\xrightarrow{\sim}\Bbbk\,e_{1}\wedge
e_{2}\wedge\dots\wedge e_{i}.$

Write $t_{12\dots i-1,i}=t_{a,j,12\dots i-1,i}$ for short. Write $v_{j1}\wedge
v_{j2}\wedge\dots\wedge
v_{ji}=\sum_{k_{1}<k_{2}<\dots<k_{i}}t_{k_{1}k_{2}\dots k_{i}}e_{k_{1}}\wedge
e_{k_{2}}\wedge\dots\wedge e_{k_{i}}$. Take the wedge with
$e_{i+1}\wedge\dots\wedge e_{n}$, we can conclude that

$v_{j1}\wedge v_{j2}\wedge\dots\wedge v_{ji}\wedge e_{i+1}\wedge\dots\wedge
e_{n}=t_{12\dots i-1,i}e_{1}\wedge e_{2}\wedge\dots\wedge e_{n}=t_{12\dots
i-1,i}.$

However, the left hand side is exactly the $i$-th principal minor of the
matrix $x_{j}$, i.e. $\Delta_{i}(x_{j})$. Hence this shows that the microlocal
merodromy is the cluster $\mathcal{A}$-coordinates defined by principal minors
of matrices. ∎

####  6.1.3. Cluster $\mathcal{X}$-coordinates on Bott-Samelson cells

Following Section 6.1, we discuss the cluster $\mathcal{X}$-variables on
$\mathcal{M}_{1}(\mathbb{D}^{2},\Lambda_{\beta\Delta^{2}})_{0}$ and show that
they are exactly the microlocal monodromies along the 1-cycles in
$H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})$. In [ShenWeng21], cluster
$\mathcal{X}$-variables on the double Bott-Samelson cell
$\mathrm{Conf}_{\beta}^{e}(\mathcal{B})$ are defined independently and they
showed in [ShenWeng21, Proposition 3.43] that the cluster $\mathcal{X}$ and
$\mathcal{A}$-variables satisfy the standard duality relation:

######  Theorem 6.11 ([ShenWeng21]).

Let
$(I_{\text{str}},I_{\text{str}}^{\text{cl}},\left<-,-\right>_{\text{str}})$ be
the intersection quiver associated to the string diagram of $\beta$. Let
$\\{A_{a}\\}_{a\in I_{\text{str}}}$ be the set of cluster
$\mathcal{A}$-variables on $\mathrm{Conf}_{\beta}^{e}(\mathcal{C})$. Then the
set of cluster $\mathcal{X}$-variables $\\{X_{a}\\}_{a\in I_{\text{str}}}$
satisfy

$X_{a}=\prod_{c\in I_{\text{str}}^{\text{cl}}}A_{c}^{\left<a,c\right>}.$

Our goal is to show that $X_{a}$ is the microlocal monodromy along
$\gamma_{a}\in H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})$. It follows from
Theorem 1.2 that $A_{a}$ is the microlocal merodromy along $\delta_{a}\in
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$.
Hence we will prove the theorem by determining the relation between the
corresponding 1-cycles. By Lemma 6.10, $\\{\delta_{a}\\}_{a\in
I_{\text{str}}^{\text{cl}}}$ form a basis of
$H_{1}(\mathfrak{w}_{\beta},\Lambda_{\beta\Delta^{2}};\mathbb{Z})\subset
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$.
Consider the exact sequence

$0\rightarrow H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})\rightarrow
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})\rightarrow\widetilde{H}_{0}(\Lambda_{\beta\Delta^{2}}\backslash
T;\mathbb{Z})\rightarrow 0.$

We can view $H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})$ as a submodule in
$H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$.
Then the relation between $\\{\gamma_{a}\\}$ and $\\{\delta_{c}\\}$ is as
follows:

######  Lemma 6.12.

Under the identification $H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})\subset
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$, we
have

$\gamma_{a}=\sum_{c\in I_{\text{str}}^{\text{cl}}}\left<a,c\right>\delta_{c}.$

###### Proof.

Similar to the proof of Lemma 6.10, we can show that the exact sequence

$0\rightarrow H_{1}(\mathfrak{w}_{\beta};\mathbb{Z})\rightarrow
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})\rightarrow\widetilde{H}_{0}(\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})\rightarrow
0$

splits, where the backward map
$\widetilde{H}_{0}(\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})\rightarrow
H_{1}(\mathfrak{w}_{\beta}^{*},\Lambda_{\beta\Delta^{2}}^{*};\mathbb{Z})$ maps
each connected component $b$ of $\Lambda_{\beta\Delta^{2}}^{*}$ to a path
connecting $b$ and a fixed component $b_{0}$ of
$\Lambda_{\beta\Delta^{2}}^{*}$.

Denote the image of $b\in\pi_{0}(\Lambda_{\beta\Delta^{2}}^{*})$ by
$\gamma_{b}$. Then it is obvious that $\left<\gamma_{a},\gamma_{b}\right>=0$
for any $a\in I_{\text{str}}^{\text{cl}}$ and
$b\in\pi_{0}(\Lambda_{\beta\Delta^{2}}^{*})$. On the other hand, for any $b\in
I_{\text{str}}^{\text{cl}}$, since $\\{\eta_{c}|c\in
I_{\text{str}}^{\text{cl}}\\}$ forms a dual basis of $\\{\gamma_{a}|a\in
I_{\text{str}}^{\text{cl}}\\}$ via the intersection product,

$\sum_{c\in
I_{\text{str}}^{\text{cl}}}\left<\left<a,c\right>\eta_{c},\gamma_{b}\right>=\sum_{c\in
I_{\text{str}}^{\text{cl}}}\left<a,c\right>\delta_{cb}=\left<\gamma_{a},\gamma_{b}\right>.$

This proves the lemma. ∎

In consequences, it follows from Lemma 6.12, Theorem 1.2, and Theorem 6.11
above, that the cluster $\mathcal{X}$-variable $X_{a}$ coincides with the
microlocal monodromy functions of the microlocal rank 1 sheaf
$\widetilde{\mathcal{F}}\in\operatorname{Sh}^{1}_{\mathfrak{w}_{\beta}}(\mathbb{D}^{2}\times\mathbb{R})$
along $\gamma_{a}$.

###  6.2. Cluster coordinates on the moduli of framed local systems via ideal
triangulations and Theorem 1.3

Let $\Sigma$ be a smooth surface with marked points and consider the moduli
space of sheaves $\mathcal{M}_{1}(\Sigma,\bigcirc_{n})_{0}$ for the trivial
Legendrian $n$-satellite $\bigcirc_{n}\subset T^{*,\infty}\Sigma$ of outward
unit conormals around marked points, with acyclic stalks at the marked points.
The moduli space of sheaves $\mathcal{M}_{1}(\Sigma,\bigcirc_{n})_{0}$ is
isomorphic to the cluster varieties
$\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma)$ of rank $n$ framed local systems
on $\Sigma$, studied by Fock-Goncharov [FockGon06a, Gon17Web].

First, let us review two classes of cluster $\mathcal{X}$-variables associated
to an ideal $n$-triangulation, those from bipartite graphs considered in
[GonKen13, Gon17Web], from non-abelianization maps considered in [GMN13,
GMN14], and those from sheaf quantizations of conjugate Lagrangians [STWZ19].
Then we prove Theorem 1.3, comparing these two classes of
$\mathcal{X}$-variables, at the end of the section.

######  Remark 6.13.

Note that [FockGon06a] considered cluster coordinates on the variety
$\mathcal{X}_{\operatorname{PGL}_{n}}(\Sigma)$ of rank $n$
$\operatorname{PGL}_{n}$-local systems on $\Sigma$. However, we focus on
$\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma)$, as cluster
$\mathcal{X}$-variables there determine the $\mathcal{X}$-variables on
$\mathcal{X}_{\operatorname{PGL}_{n}}(\Sigma)$ via a projection map.303030In
[FockGon06a], there is also a corresponding cluster $\mathcal{A}$-variety
$\mathcal{A}_{\operatorname{GL}_{n}}(\Sigma)$, resp.
$\mathcal{A}_{\mathcal{L}_{n}}(\Sigma)$, whose coordinates determine the
cluster $\mathcal{X}$-coordinates of
$\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma)$, resp.
$\mathcal{X}_{\operatorname{PGL}_{n}}(\Sigma)$. A similar construction with
microlocal holonomies, following in Section 6.1 should likely exist but it has
not yet been constructed in the literature. We thus only focus on the cluster
$\mathcal{X}$-structure.

####  6.2.1. Framed local systems on a surface

The moduli space of sheaves $\mathcal{M}_{1}(\Sigma,\bigcirc_{n})_{0}$, see
Appendix B.4, can be identified with the following moduli space of framed
local systems:

######  Definition 6.14 ([FockGon06a]*Definition 1.2).

Let $\Sigma$ be a surface with marked points $\\{x_{1},...,x_{m}\\}$,
$\Sigma_{\text{op}}=\Sigma\backslash\\{x_{1},...,x_{m}\\}$ and
$\Sigma_{\text{cl}}=\Sigma\backslash\bigcup_{i=1}^{m}B_{\epsilon}(x_{i})$. A
framed $\operatorname{GL}_{n}$-local system $(\mathcal{L},\sigma)$ on
$\Sigma_{\text{op}}$ is a $\operatorname{GL}_{n}$-local system $\mathcal{L}$
with a flat section
$\sigma:\partial\Sigma_{\text{cl}}\rightarrow\mathcal{L}/\mathrm{B}_{+}$. The
space $\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma)$ is the moduli space of
framed local systems on $\Sigma_{\text{op}}$.

A framed $\operatorname{PGL}_{n}$-local system $(\mathcal{L},\sigma)$ on
$\Sigma_{\text{op}}$ is a $\operatorname{PGL}_{n}$-local system $\mathcal{L}$
with a flat section
$\sigma:\partial\Sigma_{\text{cl}}\rightarrow\mathcal{L}/\mathrm{B}_{+}.$ The
space $\mathcal{X}_{\operatorname{PGL}_{n}}(\Sigma)$ is the moduli space of
framed local systems on $\Sigma_{\text{op}}$.

######  Remark 6.15.

There exists a natural projection map
$\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma)\rightarrow\mathcal{X}_{\operatorname{PGL}_{n}}(\Sigma)$
with fiber $(\Bbbk^{\times})^{m}$. The fiber over
$(\mathcal{L}_{0},\sigma_{0})$ parametrizes framed
$\operatorname{GL}_{n}$-local systems $(\mathcal{L},\sigma)$ where the flat
section
$\sigma:\partial\Sigma_{\text{cl}}\rightarrow\mathcal{L}/\mathrm{B}_{+}$ is
reduce to $\sigma_{0}$ under the quotient map
$\operatorname{GL}_{n}\rightarrow\operatorname{PGL}_{n}$. Note that the flat
section lives in the Cartan subgroup, and
$\operatorname{GL}_{n}\rightarrow\operatorname{PGL}_{n}$ factors through the
corresponding Cartan subgroups
$(\Bbbk^{\times})^{n}\rightarrow(\Bbbk^{\times})^{n-1}$. Remarks 6.20 and 6.24
explain that, from the point of view of cluster varieties, the projection map
is exactly defined by forgetting the frozen variables.

The following Proposition 6.16 is folklore, see [GonKon21]*Lemma 8.11 for a
similar result in a slightly different setting, and we include its proof so it
is explicitly available:

######  Proposition 6.16.

Let $\Sigma$ be a surface with marked points $\\{x_{1},...,x_{m}\\}$. Let
$\bigcirc_{n}$ be the union of the inward conormal bundle of $n$ concentric
circles around $x_{1},...,x_{m}$. Let
$\mathcal{M}_{1}(\Sigma,\bigcirc_{n})_{0}$ be the moduli space of microlocal
rank 1 sheaves in $\operatorname{Sh}^{1}_{\bigcirc_{n}}(\Sigma)$ with acyclic
stalks at $x_{1},...,x_{m}$. Then there is an equivalence of moduli spaces

$\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma)\xrightarrow{\sim}\mathcal{M}_{1}(\Sigma,\bigcirc_{n})_{0}.$

###### Proof.

We fix the following notations. Consider the stratfication of $\Sigma$ defined
by $\pi(\Lambda_{n})\subset\Sigma$. Write $A_{0}(x_{j})$ for the closed disks
inside the smallest concentric circle around $x_{j}\,(1\leq j\leq m)$. Write
$A_{i}(x_{j})$ for the annulus between the $i$-th and $(i+1)$-th concentric
circles (from inside to outside) around $x_{j}\,(1\leq j\leq m)$ containing
the $i$-th concentric circle. Without loss of generality, let
$B_{\epsilon}(x_{j})=\bigcup_{i=0}^{n-1}A_{i}(x_{j})$ and
$\Sigma_{\text{cl}}=\Sigma\backslash\bigcup_{j=1}^{m}B_{\epsilon}(x_{j})$.

For a $GL_{n}$-local system $\mathcal{L}$, a section $\sigma$ near
$\partial\Sigma_{\text{cl}}$ determines a framing $\\{s_{1},...,s_{n}\\}$ near
$\mathcal{L}|_{\partial B_{\epsilon}(x_{i})}$ for each marked point
$x_{i}\,(1\leq i\leq m)$. Therefore, we can consider the flag of $S^{1}$-local
systems near each marked point $x_{i}\,(1\leq i\leq m)$

$0_{S^{0}}=\mathcal{L}_{0}\subset\mathcal{L}_{1}\subset\mathcal{L}_{2}\subset...\subset\mathcal{L}_{n}\cong\mathcal{L}|_{\partial
B_{\epsilon}(x_{j})}.$

Therefore we can define a sheaf
$\mathcal{F}\in\operatorname{Sh}^{1}_{\bigcirc_{n}}(\Sigma)$ such that (1) in
each small disk $A_{0}(x_{j})$,
$\mathcal{F}|_{A_{0}(x_{j})}=0_{A_{0}(x_{j})}$; (2) in the annulus
$A_{i}(x_{j})$, the restriction of $\mathcal{F}|_{A_{i}(x_{j})}$ is a local
system determined by (the pullback of) a $S^{1}$-local system
$\mathcal{L}_{i}$ to $A_{i}(x_{j})$; (3) finally in $\Sigma_{\text{cl}}$, the
restriction
$\mathcal{F}|_{\Sigma_{\text{cl}}}\simeq\mathcal{L}|_{\Sigma_{\text{cl}}}$.
Locally when we pass the $i$-th concentric circle, we assign the parallel
transport map (see Example 5.3) from $\mathcal{F}|_{A_{i-1}(x_{j})}$ to
$\mathcal{F}|_{A_{i}(x_{j})}$ to be the natural inclusion

$\mathcal{L}_{i-1}\hookrightarrow\mathcal{L}_{i}.$

Since along the $i$-th concentric circle around $x_{j}\,(1\leq j\leq m)$, the
microlocal monodromy is a rank 1 local system
$\mathcal{L}_{i}/\mathcal{L}_{i-1}$. Therefore the sheaf $\mathcal{F}$ lies in
$\mathcal{M}(\Sigma,\bigcirc_{n})_{0}$.

Conversely, consider any sheaf
$\mathcal{F}\in\operatorname{Sh}^{1}_{\bigcirc_{n}}(\Sigma)$ with acyclic
stalks at $x_{1},...,x_{m}$. Then define
$\mathcal{L}=\mathcal{F}|_{\Sigma_{\text{cl}}^{\circ}}$ where
$\Sigma_{\text{cl}}^{\circ}$ is the interior of $\Sigma_{\text{cl}}$,
diffeomorphic to $\Sigma_{\text{op}}$. We now try to define a flag of
$S^{1}$-local systems

$0_{S^{1}}=\mathcal{L}_{0}\subset\mathcal{L}_{1}\subset\mathcal{L}_{2}\subset...\subset\mathcal{L}_{n}\cong\mathcal{L}|_{\partial
B_{\epsilon}(x_{i})}$

along $\mathcal{L}|_{\partial B_{\epsilon}(x_{j})}$ for each marked point
$x_{j}\,(1\leq j\leq m)$. Note that for any $1\leq i\leq n-1$,
$\mathcal{F}|_{A_{i}(x_{j})}$ is a locally constant sheaf, and thus come from
the pullback of an $S^{1}$-local system $\mathcal{L}_{i}$. We can check that
$\mathcal{L}_{i}\,(1\leq i\leq n-1)$ defines a flag of $S^{1}$-local systems.
Note that along the $i$-th concentric circle, the microlocal monodromy

$\mathrm{Cone}(\mathcal{L}_{i-1}\rightarrow\mathcal{L}_{i})$

is a rank 1 local system. When $i=0$, $\mathcal{L}_{0}\cong 0_{S^{1}}$, using
this fact we can tell that $\mathcal{L}_{1}$ is a rank 1 local system (in
degree 0) and $\mathcal{L}_{0}\rightarrow\mathcal{L}_{1}$ is an injection.
Suppose $\mathcal{L}_{i-1}$ is a rank $(i-1)$ local system (in degree 0), then
using the same fact we can tell that $\mathcal{L}_{i}$ is a rank $i$ local
system (in degree 0), and $\mathcal{L}_{i-1}\rightarrow\mathcal{L}_{i}$ is an
injection. Therefore we can get a flag of $S^{1}$-local systems, which
determines the framing around $\mathcal{L}|_{\partial
B_{\epsilon}(x_{j})}\,(1\leq j\leq m)$.

These two constructions from framed local systems to microlocal sheaves and
back are inverses to each other, and thus we have completed the proof. ∎

######  Remark 6.17.

Similarly, one can show that given a collection of braids
$\beta=\\{\beta_{1},...,\beta_{m}\\}$ around each puncture, the moduli stack
of $\beta$-framed local systems
$\mathcal{X}_{\operatorname{GL}_{n}}(\Sigma,\beta)$ introduced in
[GonKon21]*Section 9 is isomorphic to
$\mathcal{M}_{1}(\Sigma,\Lambda_{\beta})_{0}$, which is also referred to as
the moduli of $\beta$-filtered local system in [STWZ19]*Section 3.1.

####  6.2.2. Non-abelianization map and sheaves over weaves

As has been noticed in [CasalsZas20]*Section 3.1, the Legendrian $n$-weave of
an $n$-triangle is closedly related to spectral covers introduced by D.
Gaiotto, G. Moore and A. Neitzke [GMN13, GMN14], at least on the smooth level.
Here we interpret the cluster $\mathcal{X}$-variables on the moduli of framed
local systems coming from non-abelianization [GMN13] using the language of
Legendrian weaves and microlocal sheaves.

When defining cluster $\mathcal{X}$-coordinates, [GMN13, GMN14] only use the
information of the Lagrangian as smooth branched covers and define the
coordinates as monodromies of local systems on that Lagrangian spectral curve.
The spectral covers they considered are Lagrangian branched covers in the