Similarly, the weave $\n_i^{\downarrow}(\w_{0,n})$ is the unique weave that coincides with $\n(\w_{0,n})$ at the left, contains only tetravalent and hexagonal vertices, and brings the $i$-th $s_1$-strand of $\w_{0,n}$ to the bottom level, using a minimal number of weave vertices.

Finally, we denote by $\n_i^{\uparrow}(\w_{0,n})^{op}$ and $\n_i^{\downarrow}(\w_{0,n})^{op}$ the weaves obtained by reflecting $\n_i^{\uparrow}(\w_{0,n})$ and $\n_i^{\downarrow}(\w_{0,n})$ along a (disjoint) vertical axis.$\Box$

In Definition <ref>, bringing the $i$-th $s_1$-strand of $\w_{0,n}$ to the top level means considering a horizontal weave which starts at $\n(\w_{0,n})$ on the left hand side and contains a sequence of tetravalent and hexagonal vertices (no trivalent vertices) such that following the $i$-th $s_1$-strand of $\w_{0,n}$ under these vertices (passing through them straight) ends up at the top strand at the right hand side. There are many weaves that verify this property, but by the Zamolodchikov relation proven in [14], they are all equivalent and we might as well take the one with a minimal number of vertices. Figure <ref> illustrates several examples of the weaves $\n_i^{\uparrow}(\w_{0,n})$ and $\n_i^{\downarrow}(\w_{0,n})$ in Definition <ref> in the cases of $n=3,4$. Note that $\n_{n-1}^{\uparrow}(\w_{0,n})=\n_{n-1}^{\uparrow}(\w_{0,n})^{op}=\n_1^{\downarrow}(\w_{0,n})=\n_1^{\downarrow}(\w_{0,n})^{op}=\n(\w_{0,n})$ for any $n\in\N$.

Instances of the weaves $\n_i^{\uparrow}(\w_{0,n})$ and $\n_i^{\downarrow}(\w_{0,n})$, and their opposites, from Definition <ref>, in some of the cases for $n=3,4$.

The weave $\c_i^{\uparrow}(\w_{0,n})$ is the weave obtained by horizontally concatenating the three weaves $\n_i^{\uparrow}(\w_{0,n})$, $\c^{\uparrow}(w_i)$, and $\n_i^{\uparrow}(\w_{0,n})^{op}$, left to right, where $w_i$ denotes the reduced expression for $\w_{0,n}$ found at the right of the weave $\n_i^{\uparrow}(\w_{0,n})$.

Similarly, the weave $\c_i^{\downarrow}(\w_{0,n})$ is the weave obtained by horizontally concatenating the three weaves $\n_i^{\downarrow}(\w_{0,n})$, $\c^{\downarrow}(w_i)$ and $\n_i^{\downarrow}(\w_{0,n})^{op}$, left to right, where $w_i$ denotes the reduced expression for $\w_{0,n}$ found at the right of the weave $\n_i^{\downarrow}(\w_{0,n})$.

Figures <ref> and <ref> illustrate examples of the weaves $\c_i^{\uparrow}(\w_{0,n})$ and $\c_i^{\downarrow}(\w_{0,n})$ in Definitions <ref> and <ref> in the cases of $n=3,4$. For the next definition, recall that we always label the horizontal lines, in a Type 1 and Type 2 column of the GP-graph $\bG$, with consecutive natural numbers, from bottom to top. For the moment, let us assume that these labels are in $[1,n]$.

For $i\in [1,n-1]$, the weave associated to a Type 2 column of a GP-graph $\bG$ whose vertical edge has a white vertex at the $i$th horizontal line and a black vertex at the $(i+1)$st horizontal line is the weave $\c_i^{\uparrow}(\w_{0,n})$, and the weave associated to a Type 2 column of a GP-graph $\bG$ whose vertical edge has a black vertex at the $i$th horizontal line and a white vertex at the $(i+1)$st horizontal line is the weave $\c_{n-i}^{\downarrow}(\w_{0,n})$.

All possible Type 2 columns for $n=3$ and their corresponding weaves. In detail, $\c_{2}^{\uparrow}(\w_{0,3})$ (upper left), $\c_{1}^{\uparrow}(\w_{0,3})$ (upper right), $\c_{1}^{\downarrow}(\w_{0,3})$ (lower left) and $\c_{2}^{\downarrow}(\w_{0,3})$ (lower right).

Some of the Type 2 columns for $n=4$ and their corresponding weaves. In detail, $\c_{3}^{\uparrow}(\w_{0,4})$ (1st row left), $\c_{2}^{\uparrow}(\w_{0,4})$ (1st row right), $\c_{1}^{\downarrow}(\w_{0,4})$ (2nd row left), $\c_{2}^{\downarrow}(\w_{0,4})$ (2nd row right), $\c_{1}^{\uparrow}(\w_{0,4})$ (3rd row) and $\c_{3}^{\downarrow}(\w_{0,4})$ (4th row).

It is possible to horizontally concatenate the weaves $\c_{i}^{\uparrow}(\w_{0,n})$ and $\c_{j}^{\downarrow}(\w_{0,n})$, for any pair $i,j\in[1,n-1]$, in two ways: $\c_{i}^{\uparrow}(\w_{0,n})\c_{j}^{\downarrow}(\w_{0,n})$ and $\c_{j}^{\downarrow}(\w_{0,n})\c_{i}^{\uparrow}(\w_{0,n})$. These resulting two weaves are weave equivalent if and only if $i\neq j$. Furthermore, $\c_{i}^{\uparrow}(\w_{0,n})\c_{i}^{\downarrow}(\w_{0,n})$ and $\c_{i}^{\downarrow}(\w_{0,n})\c_{i}^{\uparrow}(\w_{0,n})$ differ exactly by one weave mutation. This is the weave analogue of one of the mutation rules for double wiring diagrams [25].$\Box$

§.§.§ Local weaves for Type 3 Columns

Let us consider a column of Type 3 with labels $1,2,\ldots,n$ for the horizontal lines on the right (counting from botom to top) and with a white lollipop attached to the $i$th horizontal line with $i\in [1,n]$. The case of a black lollipop is similar, and discussed later.

By construction, the weaves associated with the two Type 1 columns sandwiching this Type 3 column are $\n(\w_{0,n-1})$ and $\n(\w_{0,n})$, respectively. Hence, the weave we associate to such a Type 3 column must have these boundary conditions. Let us start with the following weave:

The weaves $\i_i^w$ (left) and $\i_i^b$ (right) from Definitions <ref> and <ref>.

The weave $\i^w_i$ is the unique weave with no weave vertices, satisfying:

(i) At its left, $\i^w_i$ coincides with the horizontal weave $\n(\w_{0,n-1})$, and at its right, $\i^w_i$ coincides with the horizontal weave $\n(s_{[n-1,i+1]}^{-1}\w_{0,n-1}s_{[n-1,n-i]})$.

(ii) The weave lines in $\n(s_{[n-1,i+1]}^{-1}\w_{0,n-1}s_{[n-1,n-i]})$ labeled by the transpositions in the reduced expression $s_{[n-1,n-i]}$ diverge upwards to vertical rays.

(iii) The weave lines in $\n(s_{[n-1,i+1]}^{-1}\w_{0,n-1}s_{[n-1,n-i]})$ labeled by the transpositions in the reduced expression $s_{[n-1,n-i]}^{-1}$ diverge downwards to vertical rays.

See Figure <ref> (left) for a depiction of $\i^w_i$, illustrating what is meant by diverging upwards and downwards to vertical rays.

Note that the word $\n(s_{[n-1,i+1]}^{-1}\w_{0,n-1}s_{[n-1,n-i]})$ in Definition <ref> is a reduced expression for the half-twist $w_{0,n}$.

There is a geometric reason behind $\i^w_i$, related to the front $\mathfrak{f}(\bG)$ of $\La(\bG)$. Indeed, the appearance of both $s_{[n-1,i+1]}^{-1}$ and $s_{[n-1,n-i]}$ in $\n(s_{[n-1,i+1]}^{-1}\w_{0,n-1}s_{[n-1,n-i]})$ is caused by the new emerging crossings when one performs the sequence of Reidemeister II moves as indicated in Figure <ref> (left). In this sense, a white lollipop introduces new crossings, which translate into new weave lines.

A sequence of RII moves in the front projection, which explain the incoming (resp. outcoming) strands in a weave when a white (resp. black) lollipop appears. Note that the level at which the crossing occurs dictated the label of the weave strand, according to Definition <ref> (resp. <ref>).

Now, the weaves $\i^w_i$ in Definition <ref> cannot quite be the weaves for the Type 3 column yet because the labeling on the right hand side is not $\w_{0,n}$, but rather $\n(s_{[n-1,i+1]}^{-1}\w_{0,n-1}s_{[n-1,n-i]})$. To fix this, let $\n^w_i$ be any horizontal weave that coincides with $\n(s_{[n-1,n-i+1]}^{-1}\w_{0,n-1}s_{[n-1,i]})$ on the left, coincides with $\n(\w_{0,n})$ on the right, and with no trivalent weave vertices in the middle. Any choice of $\n(\w_{0,n})$ would yield and equivalent weave.

The weave $\l^w_i$ associated to a Type 3 column with a white lollipop at the $i$th horizontal line is the horizontal concatenation of $\i^w_i$ and $\n^w_i$.

Figures <ref> and <ref> illustrate the weaves $\l^w_1,\l^w_2,\l^w_3$, and $\l^w_4$ for $n=4$, with the coloring convention that $s_1$ is blue, $s_2$ is red and $s_3$ is green. The pink boxes in the figures contain the $\i^w_i$ pieces, and the yellow boxes contain the $\n^w_i$ pieces. The figures also draw the corresponding pieces of the fronts $\mathfrak{f}(\bG)$, explaining the contact geometric origin of these weaves.

Weaves $\l^w_i$ associated to a white lollipop with $n=4$, as in Definition <ref>. The first row depicts the case $i=1$ and the second row the case $i=2$. The weaves $\n^w_i$ are drawn within the yellow boxes. The weaves $\i^w_i$, with the incoming weave strands, are depicted within the pink boxes.

Weaves $\l^w_i$ associated to a white lollipop with $n=4$. The first row depicts the case $i=3$ and the second row the case $i=4$. The weaves $\n^w_i$ are drawn within the yellow boxes, and the weaves $\i^w_i$ are depicted in the pink boxes.

The weave $\l^b_2$ associated to a black lollipop when $n=4$, in accordance with Definition <ref>. The weave $\n^b_2$ is drawn in the yellow box, and $\i^b_2$ in the pink box. Note that the departing strands of the weave (in the pink box) are in bijection with the crossings of the front, as indicated.

The case of a column of Type 3 with labels $1,2,\ldots,n$ for the horizontal lines on the right, and a black lollipop at the $i$th horizontal line is similar. The necessary definitions are as follows.

The weave $\i^b_i$ is the unique weave with no weave vertices, satisfying:

(i) At its left, $\i^b_i$ coincides with the horizontal weave $\n(s_{[i-1,1]}^{-1}\w_{0,n-1}s_{[n-i+1,1]})$, and at its right, $\i^b_i$ coincides with the horizontal weave $\n(\w_{0,n-1})$.

(ii) The weave lines in $\n(s_{[i-1,1]}^{-1}\w_{0,n-1}s_{[n-i+1,1]})$ labeled by the transpositions in the reduced expression $s_{[i-1,1]}^{-1}$ diverge downwards to vertical rays.

(iii) The weave lines in $\n(s_{[i-1,1]}^{-1}\w_{0,n-1}s_{[n-i+1,1]})$ labeled by the transpositions in the reduced expression $s_{[n-i+1,1]}$ diverge upwards to vertical rays.

See Figure <ref> (right) for a depiction of $\i^b_i$. Similarly, we denote by $\n^b_i$ any horizontal weave which coincides with the horizontal weave $\n(s_{[i-1,1]}^{-1}\w_{0,n-1}s_{[n-i+1,1]})$ on the right, coincides with $\n(\w_{0,n})$ on the left, and with no trivalent weave vertices in the middle.

The weave $\mathfrak{l}^b_i$ associated to a Type 3 column with a black lollipop at the $i$th horizontal line is the horizontal concatenation of $\mathfrak{n}^b_i$ and $\mathfrak{i}^b_i$.

See Figure <ref> for an example of $\l^b_i$ in the case $i=3$ and $n=4$.

§.§.§ The initial weave
Let $\bG$ be a GP-graph. The previous three Subsections <ref>, <ref>, <ref> have explained how to obtain a weave from each of the three types of elementary columns. Note that each these weaves coincides with $\n(\w_{0,k})$ and with $\n(\w_{0,m})$ for some $k$ and $m$ at its two ends. For Type 1 and Type 2, $k=m$, and for Type 3 $|k-m|=1$. Note that, if we consider two adjacent elementary columns in $\bG$, the associated weaves will coincide at the common side, and thus can be horizontally concatenated.

Let $\bG$ be a GP-graph. The initial weave $\mathfrak{w}(\bG)$ associated weave $\bG$ is the weave obtained by subdividing $\bG$ into elementary columns and then concatenating the weaves associated with each elementary column, in the order dictated by the columns.

A GP-graph for a Legendrian $m(5_2)$ and its associated initial weave.

Consider the GP-graphs in the first row of Figure <ref>, whose Legendrian fronts are depicted in Figure <ref>. Then, Figure <ref> illustrates the initial weave associated to the GP-graph $\bG_1$ for $m(5_2)$ in Figure <ref> (upper left). Note that the braid word associated to this initial weave is $\beta=(\sigma_2\sigma_1^2\sigma_2^2\sigma_1^2\sigma_2\sigma_1\sigma_2)\Delta_2^{-2}=\sigma_1^3\sigma_2\sigma_1^{-1}\sigma_2$, which is indeed a braid expression for the smooth type $m(5_2)$. Observe that, despite $\beta$ being a strongly quasi-positive braid, $m(5_2)$ is not representable by a positive braid. Similarly, Figure <ref> illustrates the initial weave associated to the GP-graph $\bG_2$ for $m(7_2)$ in Figure <ref> (upper right).$\Box$

A GP-graph for a Legendrian $m(7_2)$ and its associated initial weave.

§.§ Topology of the Initial Weave
Let $\bG$ be a GP-graph and $\mathfrak{w}(\bG)\sse\R^2$ its initial weave. In this subsection we show how to obtain a Legendrian link $\La(\bG)\sse(\R^3,\xi_\st)$ and an embedded exact Lagrangian filling $L(\ww)\sse(\R^4,\la_\st)$ from the weave $\ww=\ww(\bG)$.

§.§.§ The braid of an initial weave
Suppose $\ww(\bG)$ is an $N$-weave. Let $K\sse\R^2$ be a compact subset such that $(\R^2\setminus K)\cap\mathfrak{w}(\bG)$ contains no weave vertices. Note that the number of weave lines in $(\R^2\setminus K)\cap\mathfrak{w}(\bG)$ and their labeling is independent of any $K$ with such property. The weave lines are labeled by simple transpositions $s_i\in S_N$, which can be lifted to unique positive generators $\sigma_i$ in the Artin braid group $\Br_N$.

Let $\beta(\bG)$ be the positive braid word obtained by reading the positive braid generators associated with the weave lines of $(\R^2\setminus K)\cap\mathfrak{w}(\bG)$ in a counterclockwise manner, starting at the unique strand the corresponds to the left-most white lollipop in $\bG$.

Part of the usefulness of Definition <ref> is the following lemma, which explains that the braid word $\beta(\bG)$ is actually a braid presentation for the alternating strand diagram of $\La(\bG)$, understood as a Legendrian link in $(\R^3,\xi_\st)$. See <cit.> for details on the $(-1)$-framed closure of a positive braid word.

Let $\bG$ be a GP-graph, $\mathfrak{w}(\bG)\sse\R^2$ its initial weave and $\beta(\bG)$ its positive braid word. Then, the $(-1)$-framed closure of $\beta(\ww(\bG))$ is a front for the Legendrian link $\La(\bG)$.

A front for $\La(\bG)$ is provided in Subsection <ref>, it suffices to argue that it is homotopic to the $(-1)$-framed closure of $\beta(\ww(\bG))$. The desired Legendrian isotopy is obtained by first performing the front homotopies in Figure <ref> at each (internal) cusp corresponding to a lollipop, white or black. Then, we use a sequence of Reidemeister II and Reidemeister III moves to push the cusps to the extremes: the left cusps, coming from white lollipops, are sent to the left of the front, and the right cusps, coming from black lollipops, are sent to the right of the front. Once all cusps are pushed to the corresponding extreme of the front, one obtained the $(-1)$-framed closure of a positive braid. The fact that this positive braid is indeed $\beta(\ww(\bG))$ then follows from the construction of the initial weave in Subsection <ref>.

Lemma <ref> can be phrased as follows. Consider the Legendrian link $\La(\beta(\bG))\sse(\R^3,\xi_\st)$ whose front is the $(-1)$-framed closure of the braid word $\beta(\bG)$. Then the Legendrian links $\La(\beta(\bG))$ and $\La(\bG)$ are Legendrian isotopic in $(\R^3,\xi_\st)$.

§.§.§ The surface of the initial weave
Let $\ww\sse\R^2$ be a weave and $\Lambda(\ww)\sse(\R^5,\xi_\st)$ the Legendrian represented by its front. By definition, the Lagrangian $L(\ww)\sse(\R^4,\la_\st)$ is the Lagrangian projection of $\Lambda(\ww)$. We refer to <cit.> for details on how weaves yield exact Lagrangian fillings of Legendrian links in $(\R^3,\xi_\st)$, and recall that $\ww$ is said to be free if $L(\ww)\sse(\R^4,\la_\st)$ is embedded.

Let $\bG$ be a GP-graph and suppose its initial weave $\ww=\mathfrak{w}(\bG)$ is an $N$-weave. Then $L=L(\ww)\sse(\R^4,\la_\st)$ is an embedded exact Lagrangian filling of $\La(\bG)$ and
\begin{align*}\chi(L)=&N-\#(\mbox{trivalent vertices of }\mathfrak{w})\\
=&\#(\text{horizontal lines in $\bG$})-\#(\text{vertical edges in $\bG$}).
\end{align*}

where $\chi$ denotes the Euler characteristic.

Note that the first equality for $\chi(L)$ follows from the Riemann-Hurwitz formula, see <cit.>; the second equality follows from the facts that the number of trivalent vertices of $\mathfrak{w}$ equals the number of vertical edges of $\bG$, and that $N$ equals the number of horizontal lines in $\bG$; exactness is guaranteed by virtue of being the Lagrangian projection of a Legendrian surface.

Thus, it remains to verify embeddedness for each of the elementary columns. For that, we note that $\n(\w_{0,n})$ is a free weave, for all $n\in\N$. Indeed, the front for $\n(\w_{0,n})$ can be chosen to have constant slope in the horizontal direction; any slice in the vertical direction yields a front for $\n(\w_{0,n})$ which is a positive braid coinciding with the reduced expression $\w_{0,n}$. Thus, any pair of strands intersects exactly once, and the slopes between the corresponding sheets can be chosen to increase as they move away from the crossing of intersection. In this manner, no Reeb chords are created and $\n(\w_{0,n})$ is free. Since $\n(\w_{0,n})$ is a free weave, for all $n\in\N$, the weaves associated to Type 1 and Type 3 columns are free. For a Type 2 elementary column, we note that adding a trivalent vertex to any strand of a weave $\ww$ which has an open end (i.e. such that the strand goes off to infinity in the plane) gives an embedded Lagrangian cobordism from $\ww$ to a new weave. Thus, the new weave obtained by adding a trivalent vertex is free if $\ww$ is free. Hence, given that a Type 2 column is obtained by adding a trivalent vertex to $\n(\w_{0,n})$, for some $n\in\N$, its associated weave is always free, as required.

The number of boundary components of $L(\ww(\bG))$ is not as easy to read from $\bG$, but it is simply computed from $\beta(\bG)$: it is given by the number of cycles in the cycle decomposition of the Coxeter projection of $\beta(\bG)$. That is, we apply the natural projection map $\Br_N\rightarrow S_N$ to $\beta(\bG)$ and compute the number of cycles associated to the resulting permutation. Note that the Euler characteristic $\chi(L(\ww(\bG)))$ and the number of boundary components of $L(\ww(\bG))$ uniquely determine the smooth topology of $L(\ww(\bG))$ as an abstract surface.[Its embedded topology, especially up to Hamiltonian isotopy, is a different matter. That requires a deeper understanding of the weave $\ww(\bG)$, see [14].]

In upcoming work of the first author with W. Li, we prove that $L(\ww(\bG))$ is Hamiltonian isotopic to an exact Lagrangian representative of the Goncharov-Kenyon conjugate surface [30] associated to the graph $\bG$, embedded as in <cit.>.$\Box$

Finally, a central feature of weaves is the following: it is possible to draw many weaves which coincide with $\ww(\bG)$, outside a large enough compact set $K\sse\R^2$, and which represent embedded exact Lagrangian fillings of $\La(\bG)$. In fact, as explained in Subsection <ref>, there are some local modifications that we can perform to the weave – weave mutations – such that the smooth embedded class of the associated (Lagrangian) surface in $(\R^4,\la_\st)$ remains the same but the Hamiltonian isotopy class typically changes. Square face mutation of a GP-graph $\bG$ is recovered by weave mutations but, importantly, weave mutations allow for more general mutations, including mutations at non-square faces of $\bG$ and sugar-free regions. The result of such weave mutations applied to $\ww=\ww(\bG)$ is again another weave $\mu(\ww)$: it may no longer be of the form $\ww(\bG')$ for any GP-graph $\bG'$, but it is a weave and thus, using the calculus in [14], we can manipulate it efficiently and use it to prove the results in this article. In this process, we need explicit geometric cycles representing generators of the absolute homology $H_1(L(\ww))$. In fact, such geometric cycles lead to the quiver for the initial seed. Thus, we now gear towards understanding how to construct geometric representatives of homology classes using weaves.

§.§ Naive Absolute Cycles in
In this section we explain how to find a set of geometric (absolute) cycles on $L=L(\ww(\bG))$ which generate $H_1(L)$.

Since the genus of an embedded exact Lagrangian filling is determined by the (maximal) Thurston-Bennequin invariant of its Legendrian boundary, all embedded exact Lagrangian fillings of a given Legendrian link are topologically equivalent as abstract surfaces, i.e.  they have the same genus. In the case of a Legendrian link $\La(\bG)$ associated with a GP-graph, it is readily seen that this is the same abstract topological type as that of the Goncharov-Kenyon conjugate surface $S=S(\bG)$ [30].
Since the conjugate surface $S$ deformation retracts back to the GP-graph $\bG$, it follows that the boundaries of the faces of $\bG$ form a basis for the absolute homology groups $H_1(\bG)\cong H_1(S)\cong H_1(L)$. This basis, indexed by the faces of $\bG$, will be referred to as the naive basis of $H_1(L)$.

We emphasize that this set of generating absolute cycles is not good enough in order to construct cluster structures, nor its intersection quiver gives the correct initial quiver. Thus, these cycles will be referred to as the set of naive absolute cycles, and we will perform the necessary corrections in Subsection <ref> below.

In order to proceed geometrically, we would like identify the naive basis elements of $H_1(L)$ as lifts of a specific collection of absolute cycles on the weave front $\Sigma=\Sigma(\ww(\bG))$, ideally a collection of $\sf Y$-cycles on $\ww(\bG)$ (Definition <ref>). Since any GP-graph $\bG$ can be decomposed into elementary columns, we can try to build these absolute cycles by concatenating appropriate relative cycles associated with each elementary column.

§.§.§ Local representatives of naive absolute cycles in a Type 1 Column
In an elementary Type 1 column of $\bG$ with $n$ horizontal lines, there are $n-1$ faces, i.e. gaps, between these $n$ horizontal lines. For each of these $n-1$ gaps, we identify a unique weave line as follows. First, we observe that a cross-section of the weave front $\Sigma$ associated to a Type 1 column is, by construction, the reduced expression $\w_{0,n}$ of $w_{0,n}$. In this reduced expression, the lowest Coxeter generator ($s_i$ with the smallest $i$) appears exactly $(n-1)$ times. Second, there is a geometric bijection between these $(n-1)$ faces and the $(n-1)$ appearances of the lowest Coxeter generator in the reduced expression $\w_{0,k}$. Indeed, for a face $f$ at a Type 1 column, the intersection of $\partial f$ with a Type 1 column has two connected components, which go along two neighboring horizontal lines, say the $j$th and the $(j+1)$st. Since each horizontal line is the deformation retract of a sheet in the weave front $\Sigma$, a natural choice of the local weave line representative will be the intersection of the two corresponding sheets in $\Sigma$, which in turn corresponds to the $j$th appearance of the lowest Coxeter generator. Figure <ref> illustrates a cross-section of the weave front for $n=4$. Figure <ref> illustrates all the possible cases for $n=3,4$.

ıin 1,...,4

(0,-ı/2) node [left] $\i$  – (2,-ı/2);

(0,0) node [left] $1$ – (0.5,0) – (2,1.5) – (3.5,1.5) node [right] $1$;
(0,0.5) node [left] $2$ – (0.5,0.5) – (1,0) –(1.5,0) – (2.5,1) – (3.5,1) node [right] $2$;
(0,1) node [left] $3$ – (1,1) – (2,0) – (2.5,0) – (3,0.5) – (3.5,0.5) node [right] $3$;
(0,1.5) node [left] $4$ – (1.5,1.5) – (3,0) – (3.5,0) node [right] $4$;
at (-1.5,0.75) [] bottom;
at (5,0.75) [] top;
ıin 0,1,2

[fill=blue] (0.75+ı,0.25) circle [radius =0.1];

ıin 0,1

[fill=red] (1.25+ı,0.75) circle [radius=0.1];

[fill=green] (1.75,1.25) circle [radius=0.1];

(Left) An elementary column with four horizontal lines. (Right) The corresponding cross-section for its associated weave surface $\Sigma(\bG)$.

Associating a (piece of a) cycle in the weave for Type 1 columns. The first row depicts the two cases for $n=3$ strands, and the second and third rows depict the three cases for $n=4$, and an example with a union, in the lower right corner. In all cases, the face $F\sse\bG$ is highlighted in yellow, and the associated $s_1$-edge in the weave $\ww(\bG)$ is also highlighted in the same color. In the last case of the union, in the lower right, one of the faces and its cycle are highligthed in green.

§.§.§ Local representatives of naive absolute cycles in a Type 2 column
Consider a Type 2 column with $n$ horizontal lines and a single vertical edge between the $j$th and $(j+1)$st horizontal lines. First, for the faces bounded by any other pair of consecutive horizontal lines, say $k$th and $(k+1)$st with $k\neq j$, the associated naive absolute cycle in the weave is the unique long $\sf I$-cycle connecting the corresponding weave cycles on the two adjacent Type 1 columns. In other words, one starts at the $k$th appearance (counting from below) of the lowest Coxeter generator on the left (see Subsection <ref>) and follows that weave line straight through any hexagonal vertices. By the construction of the weave $\ww$ in Subsection <ref>, this process will go through the weave until it reaches its right hand side at the $k$th appearance of the lowest Coxeter generator. Figure <ref> depicts examples of such faces in purple. Note that, as depicted on the right of the second row in that figure, the $\sf I$-cycle might go through hexagonal vertices but shall always has the $k$th lowest Coxeter generator in $\w_{0,n}$ at the two ends.

Second, for the two faces that involve with the unique vertical edge, the associated absolute cycle in $\ww$ is the unique $\sf I$-cycle that starts with the $j$th appearance of the lowest Coxeter generator at its boundary end (see Subsection <ref>) and has the other end at the unique trivalent vertex of $\ww$.
Figure <ref> depicts examples of such faces in yellow and green. Observe that, in general, these $\sf I$-cycles will also go through hexagonal vertices but always have the $j$th lowest Coxeter generator at its boundary end.

Several examples of faces in Type 2 elementary columns with $n=4$ $\bG$-strands, and their associated $\sf I$-cycle in the weaves $\ww(\bG)$.

§.§.§ Local representatives of naive absolute cycles in a Type 3 column
In a Type 3 column, the majority of faces are similar to a face in a Type 1 column. In the weave, their boundaries are represented by weave lines going across the weave as the unique long $\sf I$-cycles with the correct boundary conditions. The only exceptional face in a Type 3 column is the face $f$ which contains a lollipop,
which we will discuss in detail in this subsection.

Let us first consider the case of a white lollipop attaching to the $j$th horizontal line on the right. For simplicity let us assume that the horizontal lines on the right are indexed by $1,2,\dots, n$ starting from the bottom. Following Subsection <ref>, the leftmost and rightmost ends of the cycle $\gamma_f$ are determined by the Type 1 rules. Namely, given that the face $f$ restricts to one gap on the left and two gaps, on the right, the ends of the cycle $\gamma_f$ in $\ww$ must be the unique $\sf I$-cycle associated to those gaps. Thus, the cycle $\gamma_f$ will start at a blue $s_1$ edge of the weave on the left and finish at two blue $s_1$ edges on the right. Now, in general, there does not exist a $\sf I$-cycle (nor a $\sf Y$-cycle) with these boundary conditions in $\ww$. This requires introducing a bident, as follows. Consider the middle slice of $\ww$ where all the newly emerged weave line have become horizontal, i.e., the right boundary of the weave building block $\i_j^w$ (Definition <ref>). Reading the weave lines from bottom to top at this slice yields an expression for the half-twist $w_{0,n}\in S_n$ (note that it is not $\w_{0,n}$). Let us draw the weave slice as the positive braid $s_{[n-1,i+1]}^{-1}\w_{0,n-1}s_{n-1,n-i}$ and mark the $s_1$ edge in $\w_{0,n-1}$ that corresponds to the gap on the left within which the white lollipop emerges. Then, starting at this marked $s_1$ edge, we go along the upper-left and upper-right strands until we reach the highest (and last) possible crossing in each of the strands. These two crossings correspond to two weave lines on the right boundary of $\i_j^w$. These two weave lines are said to be obtained from the (left) $s_1$ edge by a bifurcation.

[fill=lime] (0,1) rectangle (2,2);
(0,0) – (2,0);
(0,0.5) – (2,0.5);
(1,1.5) – (2,1.5);
(0,1) – (2,1);
(0,2) – (2,2);
[fill=white] (1,1.5) circle [radius=0.1];
at (1,-1) [] Type 3 Column;

[lime, line width = 8] (1.25,1.75) – (2.75,0.25) – (3.25,0.75);
(0,0) – (0.5,0) – (2.5,2) – (4,2);
(0,0.5) – (0.5,0.5) – (1,0) –(1.5,0) – (3,1.5) – (4,1.5);
(0,1) – (1,1) – (2,0) – (2.5,0) – (3.5,1) – (4,1);
(0,1.5) node [left] new sheet – (1,1.5) – (1.5,2) – (2,2) – (3.5,0.5) – (4,0.5);
(0,2) – (1,2) – (3,0) – (4,0);

at (1.75,-1) [] Cross-section of weave front $\Sigma$;
ıin 0,1,2

[fill=blue] (0.75+ı,0.25) circle [radius =0.1];
[fill=red] (1.25+ı,0.75) circle [radius=0.1];

ıin 0,1

[fill=green] (1.75+ı,1.25) circle [radius=0.1];
[fill=brown] (1.25+ı,1.75) circle [radius=0.1];

at (3.5,3) [] ;
[lime, line width = 5] (-1,1.5) – (1.25,1.5) – (1.25,2.4) – (5,2.4) – (7,1.2) – (8,1.2);
[lime, line width = 5] (1.25,1.5) – (1.25,-0.3) – (1.5,-0.3) – (6,2.4) – (8,2.4);
[blue] (-1,0) – (1.5,0) – (2,-0.3) – (8,-0.3);
[red] (-1,0.3) – (2,0.3) – (2.5,0) – (8,0);
[blue] (-1,0.6) – (2.5,0.6) – (3,0.3) – (8,0.3);
[green] (-1,0.9) – (3.5,0.9) – (4,0.6) – (8,0.6);
[brown] (1,-0.9) – (1,-0.3) – (1.5,-0.3) – (3.5,0.9) – (5,0.9) – (5.5,1.2) – (6.5,1.2) – (7,1.5) – (8,1.5);
[red] (-1,1.2) – (2.5,1.2) – (3,1.5) – (4.5,1.5) – (5.5,0.9) – (8,0.9);
[blue] (-1,1.5) – (2,1.5) – (2.5,1.8) – (3.5,1.8) – (4,2.1) – (5.5,2.1) – (6,2.4) – (8,2.4);
[brown] (0,2.7) – (0,1.8) – (2,1.8) – (3.5,0.9);
[green] (0.5,2.7) – (0.5,2.1) – (3.5,2.1) – (4.5,1.5) – (3.5,0.9);
[green] (4.5,1.5) – (6,1.5) – (6.5,1.8) – (8,1.8);
[red] (1,2.7) – (1,2.4) – (5,2.4) – (5.5,2.1) – (8,2.1);
[red] (4.5,1.5) – (5.5,2.1);
[blue] (5.5,2.1) – (7,1.2) – (8,1.2);
at (3.5,-1) [] Weave associated to this column;

The top left is an elementary Type 3 column with a white lollipop. Its associated weave $\ww$ is depicted on the second row. At the top right, a vertical slice of the weave front highlighting the two directions of bifurcation that come up from the $s_1$-crossing (in blue) at the bottom.

A bident is an PL-embedding of a $T$-shape domain into the plane containing the weave such that on the left it coincides with an $s_1$-edge and on the right it coincides with the two crossings obtained by bifurcation on this $s_1$-edge.

(Left) Examples of faces in Type 3 elementary columns with $n=3,4$ $\bG$-strands. (Right) The associated naive cycles, with the bidents, in the corresponding weaves $\ww(\bG)$.

In line with the techniques used in <cit.>, such a bident represents a relative cycle in $L=L(\ww)$ for the Type 3 column weave $\ww$. This relative cycle can be described by picturing its intersection with vertical slices of the weave front $\Sigma=\Sigma(\ww)$. Figure <ref> depicts such slices in two cases, one with $n=3$ and the other with $n=4$. Indeed, all that occurs as the vertical slices sweeps from left to right is that a pair of points is created in a strand of the braid (in the vertical slice), and then they move along the strands of the braid until they reach the desired position near the two crossings determined by the bifurcation. Meanwhile, the initial two points that originated from the left $s_1$ edge also move near those two crossings. Once the points have all moved near the two bifurcation crossings, this cycle can continued to be represented by two $\sf I$-cycles in the weave. This description allows for the computation of the homology class of this relative cycle and thus the computation of its intersections with other weave cycles.

The dissection of the bident in two cases. On the left, two weaves $\ww$ are depicted (associated with Type 3 columns), with a piece of a cycle drawn in yellow, and purple dashed vertical lines indicate two locations, the initial (i) and the final (f), where we draw slices. On the right, a movie of slices from (i) to (f) where the cycle in yellow intersects the slices in the star-shaped orange marks. The pink boxes on the right indicate which pair of star-shaped orange marks have been created in the bident process.

Finally, the case of an elementary column of Type 3 with a black lollipop is treated in exactly the same manner as for a white lollipop, with the roles vertically reversed. For a face $f$ containing a black lollipop in a Type 3 column, the boundary conditions being $\sf I$-cycles on the weave and having a unique bident determine the cycle $\gamma_f\sse\R^2$ in the same manner as in the white lollipop case, except now the bident is left-pointing.

[fill=lime] (0,0.5) rectangle (2,1.5);
ıin 0,1,3

(0,1) – (1,1);
[fill=black] (1,1) circle [radius=0.1];
at (1,-1) [] A Type 3 column $\bG$;

[lime, line width = 5] (0,0.6) – (1,0.6) – (2,0) – (2.25,0) – (2.25,1.2);
[lime, line width = 5] (0,1.5) – (2.25,1.5) – (2.25,1.2) – (5,1.2);
[blue] (0,0) – (1,0) – (1.5,0.3) – (3,0.3) – (3,-0.6);
[red] (0,0.3) – (1.5,0.3) – (2,0) – (2.5,0) – (2.5,-0.6);
[blue] (0,0.6) – (1,0.6) – (1.5,0.3);
[red] (1.5,0.3) – (2,0.6) – (5,0.6);
[green] (0,0.9) – (5,0.9);
[red] (0,1.2) – (5,1.2);
[blue] (0,1.5) – (2.5,1.5) – (2.5,2.1);
at (2.5,-1) [] Associated weave $\ww(\bG)$;

A case for the naive absolute cycle in a Type 3 column with a black lollipop. The face $f\in\bG$ and its associated cycle $\gamma_f\sse\R^2$ are both highlighted in light green.

Figure <ref> (upper left) depicts the weave $\ww(\bG)$ associated to the GP-graph $\bG$ given by Figure <ref> (upper left), with its two naive absolute cycles $\gamma_1,\gamma_2$. Geometrically, this weave $\ww(\bG)$ is an embedded exact Lagrangian once-punctured torus bounding one of the two max-tb Legendrian representatives of the $m(5_2)$ twist knot. Note that, since both faces of $\bG$ contain lollipops, both cycles $\gamma_1,\gamma_2$ are drawn with bidents. The upper right and the second row of Figure <ref> depict the weaves and their naive absolute cycles associated to two plabic fences, which correspond to rainbow closures of braid words $\beta=\sigma_1^5\in\mbox{Br}_2^+$ and $\beta=(\sigma_1\sigma_2)^4\in\mbox{Br}_3^+$, respectively. These plabic fences were drawn in Figure <ref>. Note that the naive absolute cycles associated to (the weave of) a plabic fence are all long $\sf I$-cycles.

(Upper left) The initial weave for the GP-graph in Figure <ref> (upper left) and its naive absolute cycles. (Upper right) The initial weave and naive absolute cycles for the plabic fence for the case $n=5$ for Figure <ref> (left). (Second Row) The initial weave and its naive absolute cycles for the plabic fence for the GP-graph drawn in Figure <ref> (right).

Due to the possible existence of bidents, it is hard to tell whether a naive absolute cycle has self intersections (and hence it is not an embedded absolute cycle or $\mathbb{L}$-compressible) or not. It is easier if we can represent the naive absolute cycles are $\sf Y$-cycles (Definition <ref>). Thus, we prove the following:

Let $\bG\sse\R^2$ be a GP-graph and $\ww=\ww(\bG)$ its associated weave. Then, there exist a weave $\ww'$ and an equivalence $\ww'\sim \ww$ such that, under the isotopy[This isotopy naturally induces to a Hamiltonian isotopy between $L(\ww)$ and $L(\ww')$.] between $\Sigma(\ww)$ and $\Sigma(\ww')$, the image of each naive absolute cycle on $L(\ww)$ is homologous to a $\sf Y$-cycle on $L(\ww')$.

For Type 1 and Type 2 elementary columns, the associated (pieces of) naive absolute cycles are already $\sf I$-cycles, and hence $\sf Y$-cycles. In particular, for $\bG$ with no (internal) lollipops, we can take $\ww'=\ww$. It thus suffices to study the case of a Type 3 column, where a bident appears: it suffices to show that there exists a weave equivalence that allows us to replace a bident by a $\sf Y$-cycle.

[lime, line width = 8] (0,1) – (1.5,1);
[lime, line width = 8] (4,0.5) – (3,0.5) – (2,1.5) – (1.5,1.5) – (1.5,0.5) – (2,0.5) – (3,1.5) – (4,1.5);
[blue] (0,1) – (2.5,1);
[red] (1,0) – (1,0.5) – (2,0.5) – (2.5,1);
[red] (1,2) – (1,1.5) – (2,1.5) – (2.5,1);
[blue] (2.5,1) – (3,1.5) – (4,1.5);
[blue] (2.5,1) – (3,0.5) – (4,0.5);
[red] (2.5,1) – (4,1);

(1,0.5) – (0,0.5) – (1.5,1) – (6.5,1) – (5.75,0.75) – (6.5,0) – (1.5,0) – (0,1.5) – (5,1.5) – (5.5,1);
(2.5,1) – (2,1.5) – (3.5,1) – (4.5,0);
[dashed] (1,0.5) – (5,0.5) – (5.75,0.75) – (5.5,1);
[dashed] (2.5,1) – (3,0.5) – (4.5,0);
[blue] (0.75,0.75) – (3.25,0.75);
[blue] (3.25,0.75) – (4.5,0);
[blue] (3.25,0.75) – (3,0.5);
[red] (3.5,1) – (3.25,0.75);
[red] (2,1.5) – (3.25,0.75);
[red] (3.25,0.75) – (5.75,0.75);
[thick,lime, decoration=markings,mark=at position 0.5 with stealth,postaction=decorate] (0.95,0.8) to [out=0,in=180] (3.3,0.9) to [out=0,in=45] (3.2,0.75) – (3,0.55);
[thick,lime,decoration=markings,mark=at position 0.7 with stealth,postaction=decorate] (4.45,0) – (3.2,0.75) to [out=155,in=0](1.7,1.4) to [out=180,in=0] (1,0.9)–(0.65,0.9);
[thick,lime,smooth,decoration=markings,mark=at position 0.3 with stealth,postaction=decorate] (3.1,0.55) – (3.5,0.95) – (2.3,1.35) – (4.4,0.1);

[lime, line width = 8] (0,1) – (2.5,1);
[lime, line width = 8] (4,0.5) – (3,0.5) – (2.5,1) – (3,1.5) – (4,1.5);
[blue] (0,1) – (2.5,1);
[red] (1,0) – (1,0.5) – (2,0.5) – (2.5,1);
[red] (1,2) – (1,1.5) – (2,1.5) – (2.5,1);
[blue] (2.5,1) – (3,1.5) – (4,1.5);
[blue] (2.5,1) – (3,0.5) – (4,0.5);
[red] (2.5,1) – (4,1);

(1,0.5) – (0,0.5) – (1.5,1) – (6.5,1) – (5.75,0.75) – (6.5,0) – (1.5,0) – (0,1.5) – (5,1.5) – (5.5,1);
(2.5,1) – (2,1.5) – (3.5,1) – (4.5,0);
[dashed] (1,0.5) – (5,0.5) – (5.75,0.75) – (5.5,1);
[dashed] (2.5,1) – (3,0.5) – (4.5,0);
[blue] (0.75,0.75) – (3.25,0.75);
[blue] (3.25,0.75) – (4.5,0);
[blue] (3.25,0.75) – (3,0.5);
[red] (3.5,1) – (3.25,0.75);
[red] (2,1.5) – (3.25,0.75);
[red] (3.25,0.75) – (5.75,0.75);
[thick,lime, decoration=markings,mark=at position 0.5 with stealth,postaction=decorate] (0.95,0.8) – (3.25,0.8) – (3,0.55);
[thick,lime,decoration=markings,mark=at position 0.7 with stealth,postaction=decorate](4.4,0.1)  – (3.25,0.9) –(0.65,0.9);
[thick,lime,smooth,decoration=markings,mark=at position 0.3 with stealth,postaction=decorate] (3.1,0.55) – (3.25,0.7) – (4.45,0) ;

Two homologous cycles $\gamma_f$ and $\wt\gamma_f$ depicted on the left. The upper left cycle $\gamma_f$ contains a bident and is not a $\sf Y$-cycle, the lower left cycle $\wt\gamma_f$ is $\sf Y$-cycle. The right hand side of each row depicts these cycles in their spatial Legendrian fronts.

For a Type 3 column with a face $f$ containing a white lollipop, this is done as follows. Consider the two weave lines to the right of the bident where the cycle $\gamma_f$ propagates. By construction, these two weave lines intersect (to the right) at a unique hexagonal weave vertex of $\ww(\bG)$. In addition, the horizontal weave line entering from the left at this hexagonal vertex connects with an $\sf I$-cycle to the $s_1$-edge on the left of the bident where $\gamma_f$ starts. Therefore, we can consider the $\sf Y$-cycle $\wt\gamma_f$ which starts with this $s_1$-edge at the left, propagates to the left (as an $\sf I$-cycle) until the hexagonal vertex and then contains a unique $\sf Y$-vertex at the hexagonal vertex. Figure <ref> depicts both cycles $\gamma_f$, at the left of the first row, and $\wt\gamma_f$, at the left of the second row, in the case of the Type 3 elementary column draw in Figure <ref> (upper left). By considering the description of $\gamma_f$ via vertical slices – as in Figure <ref> – it is readily seen that $\gamma_f$ is homologous to $\wt \gamma_f$; alternatively, at the right of each row, Figure <ref> depcits the cycles in the spatial front and one can visualize the homology between $\gamma_f$ and $\wt \gamma_f$ there as well. In consequence, in the case of a white lollipop we can consider the same weave $\ww'=\ww$ and have the naive absolute cycle $\gamma_f$ with a bident be homologous to the $\sf Y$-cycle $\wt \gamma_f$.

For a Type 3 column $\bG$ with a black lollipop, the situation is similar, with the exception that a hexagonal vertex might not exist in $\ww(\bG)$ and thus the bident cannot readily be substituted by a $\sf Y$-cycle. Nevertheless, we can insert two consecutive hexagonal vertices with a candy twist – Move I in Figure <ref> – and then apply the same argument as above.

Let us continue with the example from Figure <ref> (upper left). In that figure, neither $\gamma_1$ or $\gamma_2$ are $\sf Y$-cycles. By implementing the argument in the proof of Proposition <ref>, we obtain a $\sf Y$-cycle $\wt\gamma_1$ homologous to $\gamma_1$, as depicted in Figure <ref> (upper left) in the same weave. Similarly, after performing a Move I to the initial weave $\ww(\bG)$ (for the first GP-graph $\bG$ in Figure <ref>), we are able to draw a $\sf Y$-cycle $\wt\gamma_2$ homologous to $\gamma_2$, as shown in Figure <ref> (upper right). The second row of Figure <ref> draws an absolute cycle $\gamma_3$ that is homologous to the sum $\gamma_1+\gamma_2=\wt\gamma_1+\wt\gamma_2$.

(First row) The $\sf Y$-cycles $\wt\gamma_1$, on the left, and $\wt\gamma_2$, on the right, which are homologous to $\gamma_1$, and $\gamma_2$, following (the proof of) Proposition <ref>. Note that the weave for $\wt\gamma_2$ has a Move I (candy twist) inserted so as to modify the bident in $\gamma_2$ to the $\sf Y$-cycle $\wt\gamma_2$. (Second row) An additional absolute $\sf I$-cycle $\gamma_3$ in this weave.

§.§ Naive Relative Cycles in $L(\ww(\bG))$

Let $L=L(\ww(\bG))$ be the initial filling of the GP-link $\Lambda=\Lambda(\bG)$. Subsection <ref> constructed an explicit set of generators for a basis of $H_1(L)$ in terms of $\sf Y$-cycles. Nevertheless, in order to construct cluster $\mathcal{A}$-variables, we also need access to the lattice given by the relative homology group $H_1(L, \Lambda)=H_1(L,\partial L)$. Recall that, by Poincaré duality, there exists a non-degenerate pairing between the absolute homology group $H_1(L)$ and the relative homology group $H_1(L,\Lambda)$:

$$\inprod{\cdot}{\cdot}:H_1(L)\otimes H_1(L,\Lambda)\lr\Z.$$

Let $\{\gamma_f\}$ be the basis of naive absolute cycles constructed in Subsection <ref>, where the index $f$ runs over all faces of $\bG$. Consider the Poincaré dual basis $\{\eta_f\}$ on $H_1(L,\Lambda)$.[Note that the dual of a given absolute cycle $\gamma_f$ is not constructed merely from $\gamma_f$ but rather from the entire basis of naive absolute cycles.] In order to perform computations in the moduli stack of sheaves, we also want to describe the relative cycles in $\{\eta_f\}$ combinatorially in terms of the weave $\ww=\ww(\bG)$. This is done according to the following discussion.

In general, given an $N$-weave $\ww\sse\bR^2$, we can consider an (unoriented) curve $\kappa\sse\bR^2$ that ends at unbounded regions in the complement of $\ww\sse\R^2$ and intersect weave lines of $\ww$ transversely and generically – in particular, away from the weave vertices. There are $N$ natural ways to lift $\kappa$ to the weave front $\Sigma(\ww)$, which in turn correspond to $N$ unoriented curves on $L$.
In consequence, any subset of these $N$ lifts, together with any orientation we choose for each of element of such a subset, defines a relative homology cycle $\eta\in H_1(L,\Lambda)$. Figure <ref> (left) depicts two possible oriented lifts of the (dashed) yellow curve $\kappa$ drawn to its right.

[yellow, decoration=markings,mark=at position 0.5 with stealth,postaction=decorate] (3.25,0.75) – (2.75,0.75) – (2,0) – (1.25,0) – (0.5,0.75) – (0,0.75);
[yellow, decoration=markings,mark=at position 0.4 with stealth,postaction=decorate] (3.25,0) – (2.75,0) – (1.25,1.5) – (0,1.5);
[fill=blue] (0.875,0.375) circle [radius=0.1];
[fill=blue] (2.375,0.375) circle [radius=0.1];
[fill=red] (1.625,1.125) circle [radius=0.1];
at (-0.5,0.75) [] bottom;
at (3.75,0.75) [] top;

[blue] (0,0) – (2,0);
[red] (0,0.75) – (2,0.75);
[blue] (0,1.5) – (2,1.5);
[dashed, yellow, thick] (1,-0.5) – (1,2);
at (1,0) [below right] $1$;
at (1,0.75) [below right] $1$;
at (1,1.5) [below right] $0$;

(Left) In yellow, two lifts of the $\kappa$ dashed curve depicted on the right. (Right) A 3-weave with a labeled dashed curve $\kappa$: the labels $0,1,1$ indicate the intersection number with each of the (pieces of cycles associated to) the weave lines.

Back to the case with $\ww=\ww(\bG)$, where we have the basis of naive absolute cycles available, we can compute the intersection numbers between such relative cycles $\eta$ and the naive absolute cycles. This leads to a tuple of integers $I(\eta):=(\inprod{\eta}{\gamma_f})_{\text{faces $f$}}$. By the non-degeneracy of the Poincaré pairing $\inprod{\cdot}{\cdot}$ and the fact that $\{\gamma_f\}_{\text{faces $f$}}$ is a basis, the tuple of intersection numbers $I(\eta)$ uniquely determines the relative homology class of $\eta$. In fact, given that all naive absolute cycles that $\eta$ intersects non-trivially must pass through weave lines, in order to describe the relative homology class of $\eta$ it suffices to draw the unoriented curve $\kappa\sse\R^2$ and record the collection of the intersection number of its lift $\eta$ with each of the weave lines. In order to distinguish such curves from weave lines, we will use dashed lines to depict such a curve $\kappa$.

A dashed curve $\kappa\sse\R^2$ as above, with the data of intersection numbers for each weave line it crosses, is called a labeled dashed curve.

Figure <ref> (right) depicts a labeled dashed curve $\kappa$. From a diagrammatic perspective, it is desirable to be manipulate labeled dashed curves in a weave diagram in the same manner that [14] explained how to combinatorially manipulate absolute cycles. For that, we have depicted in Figure <ref> the key moves on labeled dashed curves: these are all equivalences, in that these moves do not change the relative homology classes that the labeled dashed curves represent.

[blue] (0,-1) – (0,1);
[dashed, thick, yellow] (-1,0) – (0.5,0) – (0.5,-1);
at (0,0) [above left] any number;
at (0,-1) [below] exterior;
=   [baseline=0]
[blue] (0,-1) – (0,1);
[dashed, thick, yellow] (-1,0) – (-0.5,0) – (-0.5,-1);
at (0,-1) [below] exterior;
[blue] (0,-1) – (0,1);
[dashed, thick, yellow] (-1,0.7) – (0.5,0.7) – (0.5,-0.7) – (-1,-0.7);
at (0,0.7) [above right] $n$;
at (0,-0.7) [below right] $-n$;
=   [baseline=0]
[blue] (0,-1) – (0,1);
[dashed, thick, yellow] (-1,0.7) – (-0.5,0.7) – (-0.5,-0.7) – (-1,-0.7);

ıin 0,1,2

[blue] (0,0) – (90+ı*120:1.5);

[dashed, thick, yellow] (0,-0.5) – (-0.75,0.5) – (0.75,0.5);
at (-120:0.5) [left] $0$;
at (0,0.5) [above right] $0$;
=  [baseline=0]
ıin 0,1,2

[blue] (0,0) – (90+ı*120:1.5);

[dashed, thick, yellow] (0,-0.5) – (0.75,0.5);
at (-30:0.5) [below] $0$;
ıin 0,1,2

[red] (0,0) – (ı*120:1.5);
[blue] (0,0) – (60+ı*120:1.5);

[dashed, thick, yellow] (-0.5,-1.25) – (-0.5,1.25);
at (-0.5,1) [right] $i$;
at (-0.5,0) [above left] $j$;
at (-0.5,-1) [right] $k$;
=   [baseline=0]
ıin 0,1,2

[red] (0,0) – (ı*120:1.5);
[blue] (0,0) – (60+ı*120:1.5);

[dashed, thick, yellow] (0.5,-1.25) – (0.5,1.25);
at (0.5,1) [left] $k$;
at (0.5,0) [above right] $j$;
at (0.5,-1) [left] $i$;

Four equivalence moves for labeled dashed curves.

Finally, we can now diagrammatically describe a collection of labeled dashed curves that is a basis of the relative homology group $H_1(\Sigma, \Lambda)$, dual to the naive basis of $H_1(\Sigma)$ built in Subsection <ref>, as follows. First, for each face $f\sse\bG$, we select a Type 1 elementary column inside of $f$.
Second, consider the piece of the weave $\ww(\bG)$ associated to this column – weave lines arranged according to $\w_{0,n}$ – and draw a vertical dashed curve $\kappa_f$ transverse to this piece of the weave. It now suffices to specify the correct labels encoding the intersection between the cycle for $\kappa$ and the naive absolute cycle associated to this face $f\in\bG$. For that, consider the braid given by slicing the weave front $\Sigma$ along $\kappa$. Recall that there is a natural bijection between the appearances of the lowest Coxeter generator $s_i$ in $\w_{0,n}$ and the gaps in this Type 1 column (see Subsection <ref>). Locate the appearance of $s_i$ that corresponds to a gap belonging to $f$ and consider set of crossings in this braid which are contained within the rhomboid diamond whose unique lowest vertex is at this appearance of $s_i$. Figure <ref> (center) draws an example of such a diamond for the face $f\sse\bG$ depicted to its left. Then we label the curve $\kappa$, to a labeled curve $\kappa_f$, by assigning the intersection number $1$ for all the weave lines in $\ww(\bG)$ which are associated to crossings in the braid inside of the diamond, and by assigning the intersection number $0$ for all the remaining weave lines. Figure <ref> (right) depicts the corresponding curve $\kappa_f$ with its intersection labels for the face $f\sse\bG$.

ıin 0,...,4

(0,ı*0.6) – (2,ı*0.6);

at (1,0.9) [] $f$;

[yellow,  decoration=markings,mark=at position 0.4 with stealth,postaction=decorate] (0,0) – (0.3,0) – (1.5,2)– (2.7,2);
[yellow,  decoration=markings,mark=at position 0.5 with stealth,postaction=decorate] (0,0.5) – (0.3,0.5) – (0.6,0) – (0.9,0) – (1.8,1.5) – (2.7,1.5);

ıin 0,...,3

[fill=blue] (0.45+ı*0.6,0.25) circle [radius=0.1];

ıin 0,1,2

[fill=red] (0.75+ı*0.6,0.75) circle [radius=0.1];

ıin 0,1

[fill=green] (1.05+ı*0.6,1.25) circle [radius=0.1];

[fill=brown] (1.35,1.75) circle [radius=0.1];
[dotted] (1.05,-0.25) – (1.95,1.25) – (1.35,2.25) – (0.45,0.75) – cycle;
[blue] (0,0) – (4,0);
[red] (0,0.5) – (4,0.5);
[blue] (0,1) – (4,1);
[green] (0,1.5) – (4,1.5);
[red] (0,2) – (4,2);
[blue] (0,2.5) – (4,2.5);
[brown] (0,3) – (4,3);
[green] (0,3.5) – (4,3.5);
[red] (0,4) – (4,4);
[blue] (0,4.5) – (4,4.5);
[thick, dashed, yellow] (2,-0.5) – (2,5);
ıin 1,2,3,4,6,7

at (2,ı*0.5) [right] $1$;

ıin 0,5,8,9

at (2,ı*0.5) [right] $0$;

(Left) A face $f\in\bG$ chosen at an elementary Type 1 column. (Center) A slice of the weave $\ww(\bG)$ with the rhomboid diamond associated to the $s_1$-crossing, in blue, for the face $f\in\bG$. (Right) The labeled dashed curve $\kappa_f$, in yellow, in the weave $\ww(\bG)$, with $1$ in the weave lines for the crossings inside the diamond, and $0$ otherwise.

By construction, the intersection pairing between the labeled dashed curve $\kappa_f$ and the naive absolute cycles is given by $\langle \eta_f,\gamma_f\rangle=1$ and $\langle \eta_f,\gamma_g\rangle=0$ for $g\neq f$. Thus, the collection of relative cycles $\{\eta_f\}$, associated to these particular labeled dashed curves $\kappa_f$, are representatives of a dual naive basis of $H_1(L,\Lambda)$. We call this collection of relative cycles the naive basis of relative cycles of the relative homology group $H_1(L,\Lambda)$.

§.§ Initial Absolute Cycles and Initial Relative Cycles in $L(\ww(\bG))$
Subsections <ref> and <ref> above explain the construction of the naive basis of absolute cycles and the corresponding naive basis of relative cycles. The generators of these basis are not geometrically appropriate: despite being $\sf Y$-cycles in the weave (or dual to them), they are often represented by immersed cycles and it is a priori unclear whether it is possible to mutate at them.[In any sense of the word mutation: geometrically, through a Lagrangian surgery, diagrammatically, via a weave mutation, or cluster-theoretically, mutating at the naive vertex representing them in the naive quiver.] A key idea in this manuscript is the consideration and study of sugar-free hulls, as introduced in Subsection <ref>. In this section, these two parts, sugar-free hulls and the study of homology cycles compatible with the weave $\ww(\bG)$, converge together: we show that it is possible to associate a $\sf Y$-tree absolute cycle on $\ww(\bG)$ to every sugar-free hull of $\bG$. In consequence, given that $\sf Y$-trees are embedded, it will be possible to perform a weave mutation at every sugar-free hull of $\bG$. This leads to the notion of initial basis, which eventually give rise to the initial seeds for our cluster structures.

In the study of sugar-free hulls, we must consider cycles which are associated to regions of $\bG$ – namely, the sugar-free hulls – and not just faces $f\sse\bG$. The simplest case is that of a region in an elementary column of Type 1, which is considered in the following simple lemma, where we use the weave $\n(\w_{0,n})$ introduced in Definition <ref>).

Let $\bG$ be a GP-graph and $C\sse\bG$ a Type 1 elementary column. Consider the region $R\sse C$ given by the union of $k$ consecutive gaps in $C$. Then the boundary of $\dd R$ is homologous to the lift of a unique weave line on the $k$th level.

Indeed, the boundary of a single gap has two connected components, and each of them is a deformation retract of a sheet of the spatial wavefront $\Sigma(C)$. For a single gap, the two sheets associated with its boundary intersect at a unique weave line at the bottom level. Now for a union $R$ of $k$ consecutive gaps, the two sheets associated with $\partial R$ intersect at the $k$th level and the conclusion follows.

Figure <ref> depicts four cases illustrating how to associate a cycle on a weave line for a region on a Type 1 column. The case of arbitrary strands can be readily imagined by examining these few cases.

Associating a (piece of a) cycle in the weave for regions on a Type 1 column. The first row depicts the case where the region is the entire column, for $n=3$ and $4$ strands. The second row depicts the remaining two cases for $n=4$ strands.

Proposition <ref> showed that it is possible, up to possibly performing a weave equivalence, to represent the naive absolute cycles with $\sf Y$-cycles. Nevertheless, these are typically immersed, e.g.  see both $\wt\gamma_1$ and $\wt\gamma_2$ in Figure <ref>. It is not possible, in general, to find embedded $\sf Y$-cycles representing these homology classes. Now, the following result, which we refer to as the $\sf Y$-representability Lemma, shows that it is possible to represent the boundary cycle $\dd R$ by a $\sf Y$-tree if the region $R$ is sugar-free.

Let $\bG$ be a GP-graph and $R\sse\bG$ a sugar-free region. Then the boundary cycle $\partial R$ is homologous to a $\sf Y$-tree on $\ww(\bG)$, up possibly performing a weave equivalence that adds $\n_k^\uparrow(\w_{n,0})\n_k^\uparrow(\w_{n,0})^{op}$ and $\n_k^\downarrow(\w_{n,0})\n_k^\downarrow(\w_{n,0})^{op}$ to $\ww(\bG)$.

Let $C$ be an elementary column of $\bG$. By Lemma <ref>, the intersection $R\cap C$ has at most one connected component. Therefore, if $R$ intersects $C$ non-trivially, $R$ must be a union of consecutive gaps in the column $C$. By Lemma <ref>, for a Type 1 or a Type 3 elementary column $C$, we can represent the boundary $\partial (R\cap C)$ by a single $\sf I$-cycle weave line going from left to right; note that the weave line color may change within a Type 3 column. It thus remains to treat the cases of elementary columns of Type 2.

By Lemma <ref> applied to a Type 2 column, we conclude that the following four cases – in correspondence with the four staircase patterns – are to be analyzed.

\[
\begin{tikzpicture}
\draw (0,0) -- (1,0) -- (1,1) -- (2,1);
\node at (0,1) [] {$R$};
\draw [fill=white] (1,1) circle [radius=0.1];
\draw [fill=black] (1,0) circle [radius=0.1];
\end{tikzpicture} \quad \quad
\begin{tikzpicture}
\draw (0,0) -- (-1,0) -- (-1,1) -- (-2,1);
\node at (0,1) [] {$R$};
\draw [fill=black] (-1,1) circle [radius=0.1];
\draw [fill=white] (-1,0) circle [radius=0.1];
\end{tikzpicture}\quad \quad
\begin{tikzpicture}
\draw (0,1) -- (1,1) -- (1,2) -- (2,2);
\node at (2,1) [] {$R$};
\draw [fill=black] (1,1) circle [radius=0.1];
\draw [fill=white] (1,2) circle [radius=0.1];
\end{tikzpicture}\quad \quad
\begin{tikzpicture}
\draw (0,2) -- (1,2) -- (1,1) -- (2,1);
\node at (0,1) [] {$R$};
\draw [fill=white] (1,1) circle [radius=0.1];
\draw [fill=black] (1,2) circle [radius=0.1];
\end{tikzpicture}
\]

First, let us consider the staircase pattern which is second from the left. The corresponding local weave pattern is $\c_k^\uparrow(\w_{0,n})$, as introduced in Definition <ref>. In the construction of this weave pattern, we first bring the $k$th strand in the bottom level upward, using the weave pattern $\n_k^\uparrow(\w_{0,n})$, subsequently insert a trivalent weave vertex at the top strand, and then insert the weave pattern $\n_k^\uparrow(\w_{0,n})^{op}$. Figure <ref> depicts a case with $4$ horizontal lines and $k=2$. (See also Figures <ref> and <ref>.) Since the $k$th horizontal line is the bottom boundary of $\partial R$, there must be a unique hexavalent weave vertex in the $\n_k^\uparrow(\w_{0,n})^{op}$ that connects to the weave line corresponding to the union of all gaps in $R$ at the right boundary. Therefore, we can create a $\sf Y$-tree in this region, with the required boundary conditions, by inserting a tripod leg that connects to the trivalent weave vertex (at the top) and continues to the left towards whichever weave line is required by the boundary condition of $R$. This resulting $\sf Y$-tree, in the shape of a tripod, then represents $\partial R$ locally, as desired. Figure <ref> (right) depicts this $\sf Y$-tree, highlighted in light green, for the region $R$ on the left, also drawn in the same color. This concludes the second case among the four staircase patterns; the third case, which also contains the region $R$ to the right of the crossing, can be resolved analogously.

[fill=lime] (0,1.4) – (1.5,1.4) – (1.5,0.7) – (3,0.7) – (3,2.1) – (0,2.1) – cycle;
ıin 0,...,3

(0,ı*0.7) – (3,ı*0.7);

(1.5,0.7) – (1.5,1.4);
[fill=black] (1.5,1.4) circle [radius=0.15];
[fill=white] (1.5,0.7) circle [radius=0.15];

[lime, line width = 5] (2,2) – (2.5,1.6) – (4,1.6);
[lime, line width=5] (0,2) – (1,2) – (2,1.2) – (2.5,1.6);
[blue] (0,0) – (4,0);
[red] (0,0.4) – (4,0.4);
[blue] (0,0.8) – (0.5,0.8) – (1.5,1.6) – (2.5,1.6) – (3.5,0.8) – (4,0.8);
[green] (0,1.2) – (0.5,1.2) – (1,0.8) – (3,0.8) – (3.5,1.2) – (4,1.2);
[red] (0,1.6) – (1.5,1.6) – (2,1.2) – (2.5,1.6) – (4,1.6);
[blue] (0,2) – (1,2) – (1.5,1.6);
[red] (1.5,1.6) – (2,2) – (2.5,1.6);
[blue] (2.5,1.6) – (3,2) – (4,2);
[red] (2,2) – (2,3);

(Left) Type 2 column with region $R$ highlighted in light green. (Right) The associated local weave and the $\sf Y$-tree, in the shape of a tripod.

Next let us study the first (leftmost) of the four staircase patterns. In this case, the chosen weave pattern $\c_k^\downarrow(\w_{0,n})$, as assigned in Subsection <ref>, does not already have a hexavalent weave vertex that meets our need. Nevertheless, we can create it by concatenating the local weave pieces $\n_k^\uparrow(\w_{0,n})$ and $\n_k^\uparrow(\w_{0,n})^{op}$ on the left first. Note that this can be achieved by inserting a series of Moves I and V, and thus the equivalence class of the weave remains the same. Now, inside of the weave piece $\n_k^\uparrow(\w_{0,n})^{op}$, there exists a unique hexavalent weave vertex that connects to both the weave line representative of $\partial R$, to the left, and the trivalent weave vertex in $\c_k^\downarrow(\w_{0,n})$, to the right. The $\sf Y$-tree, again in a tripod shape, represents $\partial R$ locally, as desired. Figure <ref> depicts an example of such tripod with $n=4$ and $k=2$. An analogous argument also resolves the case of the fourth (rightmost) staircase pattern.

[fill=lime] (0,0.7) – (1.5,0.7) – (1.5,1.4) – (3,1.4) – (3,2.1) – (0,2.1) – cycle;
ıin 0,...,3

(0,ı*0.7) – (3,ı*0.7);

(1.5,0.7) – (1.5,1.4);
[fill=white] (1.5,1.4) circle [radius=0.15];
[fill=black] (1.5,0.7) circle [radius=0.15];

[lime, line width = 5] (0,1.6) – (2.5,1.6) – (3,2) – (6.5,2);
[lime, line width=5] (2.5,1.6) – (3.5,0.8) – (4,0.8) – (5,0);
[blue] (0,0) – (4,0) – (4.5,0.4) – (5.5,0.4) – (6,0) – (6.5,0);
[red] (0,0.4) – (4.5,0.4) – (5,0) – (5.5,0.4) – (6.5,0.4);
[blue] (0,0.8) – (0.5,0.8) – (1.5,1.6) – (2.5,1.6) – (3.5,0.8) – (4,0.8) – (4.5,0.4);
[blue] (5.5,0.4) – (6,0.8) – (6.5,0.8);
[red] (5,0) – (5,-1);
[red] (4.5,0.4) – (5,0.8) – (5.5,0.4);
[green] (0,1.2) – (0.5,1.2) – (1,0.8) – (3,0.8) – (3.5,1.2) – (6.5,1.2);
[red] (0,1.6) – (1.5,1.6) – (2,1.2) – (2.5,1.6) – (6.5,1.6);
[blue] (0,2) – (1,2) – (1.5,1.6);
[red] (1.5,1.6) – (2,2) – (2.5,1.6);
[blue] (2.5,1.6) – (3,2) – (6.5,2);

Example of the first staircase.

Finally, by combining the local pictures for all three types of columns, we conclude that there exists a representative for $\partial R$ which is a $\sf Y$-tree.

The $\sf Y$-representability Lemma allows us to introduce the following definition.

Let $\bG$ be a GP-graph. The set of initial absolute cycles $\fS(\bG)$ is the set of all $\sf Y$-trees on $L(\ww(\bG))$ which are associated to the (non-empty) sugar-free hulls in $\bG$.

The inclusion relation between sugar-free hulls naturally puts a partial order on the set $\fS(\bG)$: by definition, $\partial R\leq \partial R'$ if $R\subset R'$ as sugar-free hulls. Also, given a face $f\sse\bG$, we note that a face $g\in \S_f$ in its sugar-free hall, must satisfy $\S_g\subset \S_f$ and hence $\partial \S_g\leq \partial \S_f$. Finally, note that multiple faces in $\bG$ can share the same sugar-free hull, and there may exist faces with an empty sugar-free hull. Thus, in general, the number of sugar-free hulls may be smaller than the number of faces in $\bG$. Nevertheless, we can prove that the set of initial absolute cycles $\fS(\bG)$ is always linearly independent.

Let $\bG$ be a GP-graph and let $\fS(\bG)$ be its set of initial absolute cycles. Then $\fS(\bG)$ is a linearly independent subset of $H_1(L(\ww(\bG)))$. In addition, it is possible to add naive absolute cycles to $\fS(\bG)$ to complete $\fS(\bG)$ into a basis of $H_1(L(\ww(\bG)))$.

Let $L=L(\ww(\bG))$ be the initial filling. Consider the naive basis $\{\gamma_f\}_{\text{faces $f$}}$ of $H_1(L)$. It suffices to show that we can replace $|\fS(\bG)|$ many naive basis elements with elements in $\fS(\bG)$ while maintain the spanning property of the set. This is can be done by using the partial order on $\fS(\bG)$ as follows.

Let us start with the minimal elements in $\fS(\bG)$ and work our way up, replacing the appropriate naive basis elements in $\{\gamma_f\}$ with elements in $\fS(\bG)$. At each turn, we select a face $f\sse\bG$ whose sugar-free hull $\S_f$ defines a $\sf Y$-tree $\partial \S_f\in \fS(\bG)$ and then replace the naive absolute cycle $\gamma_f$ with the $\sf Y$-tree $\partial \S_f$. Note that it is always possible to find such a face $f\sse\bG$ in this process: if no such face $f$ were available, any faces within the sugar-free hull would not have this particular sugar-free region as their sugar-free hull, which is tautologically absurd.

In the process of implementing each of these replacements, we must argue that the resulting set is still spanning the homology group $H_1(L)$. For that, we observe that if a replacement of $\gamma_g$ by $\partial \S_g$ is done before the replacement of $\gamma_f$ by $\S_f$, the sugar-free hull $\S_g$ must not contain the face $f$ inside. This follows from the fact that $f\in \S_g$ implies $\S_f\subset \S_g$. Therefore, we can use the set $\{\partial \S_g \mid g\in \S_f\}$, which is already in the basis set due to the partial order, and $\partial \S_f$ to recover $\gamma_f=\partial f$. This shows that the resulting set after the replacement of $\gamma_f$ by $\partial \S_f$ as above, is still a basis for $H_1(L)$.

Example <ref> below partially illustrates this replacement procedure in a particular Hasse diagram for the initial absolute cycles $\fS(\bG)$. The choice of completion of $\fS(\bG)$ to a basis in $H_1(L(\ww(\bG))$ is a neat instance of the natural appearance of quasi-cluster structures in (symplectic) geometry. There is no particular canonical manner by which we can typically choose a basis for this complement but, as we shall explain, the different choices all lead to the same cluster structure up to monomials in the frozen variables, i.e. a quasi-cluster structure, see Appendix I.

The dualization process, via the Poincaré pairing, requires a choice of basis. Therefore, there is no canonical choice of initial relative cycles associated to the set $\fS(\bG)$ unless the later spans $H_1(L(\ww(\bG))$. In a general situation, we can at least consider the following concept.

Let $\bG$ be a GP-graph, $L=L(\ww(\bG))$, $\Lambda=\Lambda(\bG)$ and $\fS(\bG)$ its set of initial absolute cycles on $L$. Let $\fB$ be a basis of $H_1(L)$ which is obtained by adding naive absolute cycles to the set $\fS(\bG)$. Then the initial relative cycles associated to $\fB$ is the collection $\fB^\vee$ of linear combinations of naive relative cycles whose relative homology classes form a basis of $H_1(L,\Lambda)$ dual to the basis $\fB$ of $H_1(L)$.

We emphasize that the basis $\fB^\vee$ depends not only on $\fS(\bG)$, but also on the chosen basis completion $\fB$. Note that a few simple choices of $\fB$ are available as a result of the replacement construction in the proof of Proposition <ref>. Namely, we start with the naive absolute basis $\{\gamma_f\}$, and then swaps some of the basis elements $\gamma_f$ with their corresponding initial absolute cycles $\partial \S_f$. In case that there are multiple faces sharing the same sugar-free hull, only one of the naive absolute cycles gets replaced, and the rest remain in the basis, which will become frozen basis elements.

Furthermore, if the basis $\fB$ is chosen via such a basis replacement process, then the corresponding dual relative cycle basis $\fB^\vee$ can be described with respect to the partial order on sugar-free hulls as well. Indeed, suppose that we have an equality $\S_f=\S_g=\cdots$ of sugar-free hulls, for some faces $f,g\sse\bG$, among others, and $\gamma_f$ was chosen to be replaced by $\partial \S_f$ in the replacement process. Then, the naive relative cycle $\eta_g$ needs to be replaced by $\eta_g-\eta_f$ for each $g$ with $\S_g=\S_f$. Similarly, $\eta_f$ would need to be replaced by $\eta_f+N$, where the $N$ summand is a linear combination of the naive relative cycles $\eta_h$ associated to the chosen faces $h$ with $\S_f\subsetneq \S_h$, such that the pairing of $(\eta_f+N)$ with $\partial \S_h$ vanishes for all such $h$.

Let us suppose that a set of initial absolute cycles $\fS(\bG)$ conforms to the Hasse diagram below – i.e.  this is the inclusion partial order on sugar-free hulls for some GP-graph $\bG$ with $16$ faces $\{f_1,\ldots,f_{16}\}$, where we have denoted $\S_i:=\S_{f_i}$, $i\in[1,16]$.

(1) [red] at (0,0) [] $\S_1$;
(10) at (3,0) [] ${\color{red} \S_{10}}=\S_{15}$;
(2) at (-1,1) [] ${\color{red}\S_2}=\S_3$;
(4) [red] at (1,1) [] $\S_4$;
(6) at (3,1) [] $\S_5={\color{red}\S_6}$;
(11) [red] at (5,1) [] $\S_{11}$;
(8) at (0,2) [] $\S_7={\color{red} \S_8}=\S_9$;
(13) [red] at (3,2) [] $\S_{13}$;
(14) at (5,2) [] ${\color{red}\S_{14}}$;
(16) [red] at (2.5,3) [] $\S_{16}$;
(1) – (2);
(1) – (4);
(1) – (6);
(1) – (11);
(10) – (6);
(10) – (11);
(2) – (8);
(4) – (8);
(6) – (13);
(11) – (14);
(11) – (16);
(8) – (16);
(13) – (16);

The Hasse diagram in Example <ref>, where the sets are the sugar-free hulls and the partial order relation is given by inclusion.

We have marked in red a possible choice of naive absolute cycles for replacement, e.g. the cycles $\gamma_1,\gamma_2,\gamma_4,\gamma_6,\gamma_8\gamma_{10},\gamma_{11},\gamma_{13},\gamma_{14},\gamma_{16}$ will be modified accordingly, whereas $\gamma_3,\gamma_5,\gamma_7,\gamma_9\gamma_{15}$ will remain part of the basis. Note that, as drawn, there is an additional face $f_{12}$ whose sugar-free hull is empty, and thus $\gamma_{12}$ also remains in the basis. Given this choice, the replacement construction applied to the basis of naive absolute cycles $\{\gamma_i\}_{i=1}^{16}$, leads to the following basis $\fB$ for $H_1(L)$:

\[
\{{\color{red} \gamma_1},\ \gamma_1+{\color{red}\gamma_2}+\gamma_3,\ {\color{blue}\gamma_3},\ \gamma_1+{\color{red} \gamma_4},\ {\color{blue}\gamma_5},\ \gamma_1+\gamma_5+{\color{red}\gamma_6}+\gamma_{10}+\gamma_{15},\ {\color{blue}\gamma_7},\ \gamma_1+\gamma_2+\gamma_3+\gamma_4+\gamma_7+{\color{red}\gamma_8}+\gamma_9,
\]

\[
{\color{blue}\gamma_9}, \ {\color{red}\gamma_{10}}+\gamma_{15}, \ \gamma_1+\gamma_{10}+{\color{red}\gamma_{11}}+\gamma_{15},\ {\color{blue}\gamma_{12}},\ \gamma_1+\gamma_5+\gamma_6+\gamma_{10}+{\color{red}\gamma_{13}}+\gamma_{15},\ \gamma_1+\gamma_{10}+\gamma_{11}+{\color{red}\gamma_{14}}+\gamma_{15},
\]

\[
{\color{blue}\gamma_{15}}, \ \gamma_1+\gamma_2+\gamma_3+\gamma_4+\gamma_5+\gamma_6+\gamma_7+\gamma_8+\gamma_9+\gamma_{10}+\gamma_{11}+\gamma_{13}+\gamma_{15}+{\color{red}\gamma_{16}}\},
\]

We have highlighted in red the naive absolute cycles in $\{\gamma_i\}$ that correspond to the red sugar-free hulls in the Hasse diagram. For completeness, we have also highlighted in blue the basis elements that will become frozen variables. Similar, the basis $\fB^\vee$ of $H_1(L,\Lambda)$ dual to $\fB$ is

\[
\{{\color{red}\eta_1}-\eta_2-\eta_4-\eta_6+\eta_8-\eta_{11}+2\eta_{16},\ {\color{red}\eta_2}-\eta_8,\ {\color{blue}\eta_3-\eta_2},\ {\color{red}\eta_4}-\eta_8,\ {\color{blue}\eta_5-\eta_6},\ {\color{red}\eta_6}-\eta_{13},\ {\color{blue}\eta_7-\eta_8},
\]

\[
{\color{red} \eta_8}-\eta_{16},\ {\color{blue}\eta_9-\eta_8},\ {\color{red}\eta_{10}}-\eta_6-\eta_{11}+\eta_{16},\ {\color{red} \eta_{11}}-\eta_{14}-\eta_{16},\ {\color{blue} \eta_{12}}, \ {\color{red}\eta_{13}}-\eta_{16},\ {\color{red}\eta_{14}},\ {\color{blue}\eta_{15}-\eta_{10}},\ {\color{red}\eta_{16}}\}.
\]

This illustrates that there is a rather simple algorithmic procedure to complete $\fS(\bG)$ to a basis $\fB$ of $H_1(L)$ and to find the dual basis $\fB^\vee$ of $H_1(L,\Lambda)$.

§.§ The naive quiver of a GP-graph
Let us start by emphasizing that the quiver of the initial seed for the cluster structure we construct is not always the dual quiver of the GP-graph $\bG$. Nevertheless, that dual quiver is useful in order to construct the actual quiver of the initial seed because it can be used to compute the intersection form on the initial filling, which is needed to define the initial quiver. Let us provide the details.

Following the description of M. Gross, P. Hacking, S. Keel, and M. Kontsevich [29], in order to define a cluster structure, we first fix an integer lattice with a skew-symmetric form on it. In the context of a GP-graph $\bG$, the integer lattice is $H_1(L(\ww(\bG)))$, and a natural skew-symmetric form on it is given by the intersection pairing between absolute homology classes. By using the naive basis $\{\gamma_f\}$ of naive absolute cycles and the GP-graph $\bG$, we can describe the intersection pairing form $\{\cdot,\cdot\}$ combinatorially using a quiver.

Let $\bG\sse\R^2$ be a GP-graph. The naive quiver $Q_0(\bG)$, or dual quiver, associated to $\bG$ is the quiver constructed as follows:

* A quiver vertex is associated to each face $f\sse\bG$.

* For every bipartite edge in $\bG$, we draw an arrow according to $\begin{tikzpicture}[baseline=0]
\draw (0,-0.5) -- (0,0.5);
\draw [fill=black] (0,0.5) circle[radius=0.1];
\draw [fill=white] (0,-0.5) circle[radius=0.1];
\node [blue] (l) at (-1,0) [] {$\bullet$};
\node[ blue](r) at (1,0) [] {$\bullet$};
\draw [->,blue] (l) -- (r);
\end{tikzpicture}$.

* For each pair of quiver vertices, sum up the arrows between them.

Note that in the third step (3) there might be cancellations.

The following lemma is proved by A. Goncharov and R. Kenyon:

Let $\epsilon_{fg}$ be the exchange matrix of the quiver $Q_0(\bG)$. Then the intersection pairing between $\gamma_f$ and $\gamma_g$ is given by $\{\gamma_f,\gamma_g\}=\epsilon_{fg}$.

Since $\{\gamma_f\}_{\text{faces $f$}}$ is a basis of $H_1(L(\ww(\bG)))$, Lemma <ref> uniquely determines the intersection skew-symmetric form. This intersection form, i.e. the quiver $Q_0(\bG)$, is then used to compute the correct quiver $Q(\bG)$ for the initial seed. The unfrozen vertices of the correct initial quiver $Q(\bG)$ will be indexed by $\fS(\bG)$, the set of initial absolute cycles, and the remaining frozen vertices are determined by the choice of completion of $\fS(\bG)$ to a basis of $H_1(L)$. Since we will elaborate more on this in Section <ref>, we conclude this discussion for now and revisit $Q(\bG)$ then.

In the case of a plabic fence $\bG$, all naive absolute cycles are $\sf I$-cycles and thus they also are initial absolute cycles. Thus, for a plabic fence, $Q(\bG)$ coincides with $Q_0(\bG)$.

§.§ A few examples
For the purpose of clarity, let us explicitly apply the constructions in this section to three examples of GP-graphs, drawing the associated fronts, weaves and associated bases. In the first and second examples below show, not all naive absolute cycles are (homologous to) initial absolute cycles.[Only in particular cases, such as for plabic fences, all naive absolute cycles are initial absolute cycles. These two sets do not coincide for general examples of GP-graphs $\bG$.] In these two examples, the set of initial absolute cycles will form a basis of $H_1(L(\ww(\bG)))$.  The third example shows that the set of initial absolute cycles might not span $H_1(L(\ww(\bG)))$ (even if the set of naive cycles does), and that there are different natural choices for extending the set of initial absolute cycles to a basis.

The GP-graph $\bG_1$, on the left, and its associated Legendrian front, on the right. The Legendrian knot $\La(\bG_1)$ is a max-tb Legendrian representative of the $(3,4)$-torus knot.

Example 1. Let us consider the GP-graph $\bG_1$ in Figure <ref> (left). The faces $f_2,f_3,f_4,f_5$ and $f_6$ are their own sugar-free hulls, whereas the sugar-free hull of $f_1$ is $f_1\cup f_2$. The associated Legendrian front is depicted to in Figure <ref> (right). This is a front for the unique max-tb Legendrian $(3,4)$-torus knot, which determines the Legendrian isotopy type of $\La(\bG_1)$. Note that the Legendrian knot $\La(\bG_1)$ is isotopic to the Legendrian knot associated to the alternating strand diagram of the plabic fence in Figure <ref> (right), but that the GP-graph from Figure <ref> (left) is itself not a plabic fence.

The Legendrian weave $\ww(\bG_1)$ associated to the GP-graph $\bG_1$. Supports of relative cycles $\eta_j$, $j\in[1,6]$, are also drawn.

The weave $\ww(\bG_1)$ associated to this GP-graph $\bG_1$ is drawn in Figure <ref>. It represents a genus 3 embedded exact Lagrangian filling $L=L(\ww(\bG_1))$ for $\La=\La(\bG_1)$. A basis of $H_1(L)$ is given by the absolute cycles $\{\gamma_1+\gamma_2\}\cup \{\gamma_i\}_{i\in[2,6]}$ depicted in Figure <ref>, where $\gamma_i$ is the naive cycle associated to $f_i$, $i\in[1,6]$. In this example, each naive absolute cycles $\gamma_2,\gamma_3,\gamma_4,\gamma_5,\gamma_6$ is also an initial absolute cycle. The naive absolute cycle $\gamma_1$ is immersed, and its associated initial absolute cycle is $\gamma_1+\gamma_2$, as drawn in Figure <ref>; this is coherent with the fact that the sugar-hull of $f_1$ is $f_1\cup f_2$, as noted above.

A basis of absolute cycles $\{\gamma_i\}_{i\in[1,6]}$ for the lattice $H_1(L(\ww(\bG_1)))$.

A basis of relative cycles $\eta_j$, $j\in[1,6]$, through its support, is also drawn in Figure <ref>. The intersection matrix between these two basis is

$$\langle \wt\gamma_i,\eta_j\rangle=
\left(
\begin{array}{cccccc}
1 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & 0 & 1
\end{array}
\right),\quad \wt\gamma_1:=\gamma_1+\gamma_2,\quad \wt\gamma_i:=\gamma_i, i\in[2,6],

and thus the dual basis of $\{\gamma_1+\gamma_2\}\cup\{\gamma_i\}_{i\in[2,6]}$ in terms of $\{\eta_j\}_{j\in[1,6]}$ is

and the associated quiver $Q(\bG_1)$ is depicted in Figure <ref>. All vertices are in fact mutable, due to the fact that they are represented by initial cycles, as we show in Section <ref>. It is readily verified that the mutation class of this quiver is that of the simple type $E_6$, though it is not the standard $E_6$-Dynkin diagram (which could have obtained from a plabic fence).

The quiver associated to the intersection form and basis for $\bG_1$.

Example 2. Consider the GP-graph $\bG_2$ in Figure <ref> (left). We note that the sugar-free hull of $f_1$ is $f_1\cup f_3\cup f_4 \cup f_5$, that of $f_5$ is $f_3\cup f_5$, and $f_2,f_3,f_4,f_6$ are their own sugar-free hulls. The corresponding front is drawn in <ref> (right). The associated Legendrian knot $\La(\bG_2)$ is a max-tb representative of the $(2,7)$-torus knot, and we thus would expect the quiver to be in the mutation class of the $A_6$-Dynkin diagram (with any orientation), with all variables being mutable.

A GP-graph $\bG_2$ (left) and its associated Legendrian front (right).

The first step is to draw the weave $\ww(\bG_2)$, which is done in Figure <ref>. This weave defines a Lagrangian filling $L=L(\ww(\bG_2))$ of $\La(\bG_2)$, in this case a once-punctured genus 3 surface, and the second step is finding the initial basis of $H_1(L)\cong \Z^6$. Figure <ref> draws the set of naive absolute cycles $\{\gamma_i\}_{i\in[1,6]}$, which indeed span this lattice.

The weave $\ww(\bG_2)$ associated to the GP-graph of Figure <ref>, which represents a genus 3 embedded exact Lagrangian filling of $\La(\bG_2)$.

With respect to the naive absolute basis, the intersection pairing has matrix

$$\langle \gamma_i,\gamma_j\rangle=
\left(
\begin{array}{cccccc}
0 & 1 & -1 & 1 & 0 & 0 \\
-1 & 0 & 1 & 0 & -1 & 0\\
1 & -1 & 0 & 0 & 0 & 0\\
-1 & 0 & 0 & 0 & 1 & -1\\
0 & 1 & 0 & -1 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0
\end{array}
\right)

In this example, the naive cycles $\gamma_2,\gamma_3,\gamma_4$ are initial absolute cycles, as they are short $\sf I$-cycles. Nevertheless, the naive cycles $\gamma_1,\gamma_5,\gamma_6$ are not initial absolute cycles. In order to obtain an initial absolute cycle from $\gamma_1$, resp. $\gamma_5$, we can add $\gamma_3+\gamma_4+\gamma_5$ to it, resp. $\gamma_3$. Again, this is coherent with the sugar-free hulls of $f_1$ and $f_5$. These two initial cycles $\gamma_1+\gamma_3+\gamma_4+\gamma_5$ and $\gamma_3+\gamma_5$ are depicted in the weave $\ww(\bG_2)$ drawn in Figure <ref>. The cycle $\gamma_6$ is in fact homologous to a $\sf Y$-cycle (using the hexagonal vertex exactly at the left of its bident), and thus we can take that representative, also denoted $\gamma_6$, as an initial absolute cycle.

The set of naive absolute cycles $\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5,\gamma_6$ for $\ww(\bG_2)$.

The two initial absolute cycles $\gamma_1+\gamma_3+\gamma_4+\gamma_5$ and $\gamma_3+\gamma_5$ in the weave $\ww(\bG_2)$. Note that both homology classes are represented by $\sf Y$-cycles.

In summary, the set of initial absolute cycles $\fS(\bG_2)$ is

which is a basis of $H_1(L(\ww(\bG_2)))$. Its dual basis, in terms of the relative cycles associated to the dashed curves $\{\eta_i\}_{i\in[1,6]}$ in Figure <ref>, then reads

as $\langle \gamma_i,\eta_j\rangle=\delta_{ij}$ for $i,j\in[1,6]$. The quiver $Q(\bG_2)$ is drawn in Figure <ref>, and all vertices will be mutable (because they are initial cycles). This quiver is indeed in the mutation equivalence of the simple $A_6$-type.

The quiver $Q(\bG_2)$ associated to the GP-graph $\bG_2$.

Example 3. Consider the GP-graph $\bG_3$ in Figure <ref> (right) (equivalently the upper left of Figure <ref>), a front for which is drawn in Figure <ref> (left). Denote the two faces of $\bG_3$ by $f_1,f_2$, and note that $f_1\cup f_2$ is the sugar-free hull of both of them. The associated initial weave $\ww(\bG_3)$ is depicted in Figure <ref> (right), where the Lagrangian filling $L=L(\ww(\bG_3))$ is a once-punctured 2-torus. There are two naive cycles $\gamma_1$ and $\gamma_2$, drawn in Figure <ref> (top row), respectively associated to the two faces $f_1$ (left) and $f_2$ (right). The naive quiver $Q(\bG)$ is

\[
\begin{tikzpicture}
\node (l) at (-1,0) [] {$\gamma_1$};
\node (r) at (1,0) [] {$\gamma_2$,};
\draw [->] (l) -- (r);
\end{tikzpicture}
\]

and it defines the intersection pairing on the lattice $H_1(L)\cong \mathbb{Z}\gamma_1\oplus \mathbb{Z}\gamma_2$. However, this is not the quiver needed to define the cluster structure associated with the Legendrian knot $\La(\bG_3)$. In order to obtain the correct quiver, we note that there is a unique (non-empty) sugar-free hull $f_1\cup f_2$ available. In consequence, there is only one initial cycle $\delta:=\gamma_1+\gamma_2$, which does not span $H_1(L)$. Now, to complete the set $\{\delta\}$ into a basis of this lattice, we can throw in either $\gamma_1$ or $\gamma_2$ (as a frozen vertex). In each such ways, we can produce a quiver using the intersection pairing form. The resulting two possible quivers are

\[
\begin{tikzpicture}
\node (l) at (-1,0) [] {$\delta$};
\node [blue] (r) at (1,0) [] {$\gamma_2$,};
\draw [->] (l) -- (r);
\end{tikzpicture} \quad \quad \quad \quad
\begin{tikzpicture}
\node (l) at (-1,0) [] {$\delta$};
\node [blue] (r) at (1,0) [] {$\gamma_1$.};
\draw [->] (r) -- (l);
\end{tikzpicture}
\]

As shown in Section <ref>, either of these two quiver will define a cluster structure for this max-tb Legendrian representative $\La(\bG_3)$ of the smooth knot $m(5_2)$, and the resulting cluster structures are quasi-equivalent. We remark that the existence of freedom in completing the set of initial cycles to a basis, is what leads naturally to a quasi-cluster structure, rather than a cluster structure.

As we will see in the next section, in general we need to decorate GP-links $\Lambda=\Lambda(\bG)$ with a set $T$ of marked points subject to the requirement of at least one marked point per link component, and the general lattice we need is $H_1(L,T)$, where $L=L(\ww(\bG))$ is the initial filling. However, since all three examples given in this subsection are knots, we can take $T$ to be a singleton set, and hence it is enough to just consider $H_1(L,T)=H_1(L)$.

§.§ Bases and homology lattices in the presence of marked points
The construction of the cluster structures in Theorem <ref>, and the definition of the moduli space $\fM(\Lambda, T)$, in general require an additional piece of data:
a set $T$ of marked points on $\La(\bG)$.

Let $\bG$ be a GP-graph and let $\Lambda=\Lambda(\bG)$ be its GP-link. A set of marked points $T\subset \Lambda$ is a subset of distinct points in $\La$, where we require that there is at least one marked point on each link component of $\Lambda$ and, without loss of generality, the set $T$ is disjoint from all crossings and all cusps in the front $\mathfrak{f}(\bG)$ of $\Lambda$.

All prior statements in Section <ref> remain unchanged by the addition of marked points, as they do not affect the associated weaves or the Hamiltonian isotopy class of exact Lagrangian fillings. Therefore, we can still consider the initial embedded exact filling $L=L(\ww(\bG))$ of the GP-link $\Lambda$. As before, we select the collection of initial absolute cycles $\fS(\bG)$ associated with sugar-free hulls, and they form a linearly independent subset of $H_1(L)$. The addition of marked points affects only the cluster-theoretic constructions: we need to replace the lattice of absolute homology $H_1(L)$ by the lattice of relative homology $H_1(L,T)$. The natural inclusion $H_1(L)\subset H_1(L,T)$ induced by the inclusions $T\sse\La=\dd L\sse L$, allows us to include the initial absolute cycles $\fS(\bG)$ as a linearly independent subset of $H_1(L,T)$. The only difference is that, in order to fix a cluster structure, we must expand $\fS(\bG)$ further to a basis $\fB$ of $H_1(L,T)$. This expansion can be done in two steps: we first expand $\fS(\bG)$ to a basis of $H_1(L)$, as done via the replacement process in Subsection <ref>, and then expand this basis of $H_1(L)$ to a basis of $H_1(L,T)$.

As was the case for $H_1(L)$ and its dual $H_1(L,\La)$, we shall need a dual space of $H_1(L,T)$ together with a basis dual to a chosen basis $\fB$ of $H_1(L,T)$. In fact, there is a natural intersection pairing

$$\inprod{\cdot}{\cdot}:H_1(L,T)\otimes H_1(L\setminus T,\Lambda\setminus T)\lr\Z$$

obtained by algebraically counting geometric intersections of relative cycles in generic position. In the same manner that Poincaré duality was used in Subsection <ref>, a duality also exists in the setting with marked points. We record the precise statement in the following.

Let $L$ be a connected smooth surface with boundary $\La=\dd L$, and $i:T\lr\La$ an inclusion of a set of marked points with $\pi_0(i)$ being surjective. Then
$$\rank (H_1(L,T))=\rank (H_1(L\setminus T,\Lambda\setminus T)),$$

and the intersection pairing $\inprod{\cdot}{\cdot}$ is non-degenerate.

There are cases in which marked points are not needed, such as the case where $\La$ is a knot. Nevertheless, marked points are always needed if $\La$ is a link. Let us illustrate a simple example where marked points will be used, using the the simplest link.

Consider the Legendrian representative $\La\sse(\R^3,\xi_\st)$ of the Hopf link with both max-tb components, that admits an embedded exact Lagrangian annulus filling $L$. Let us consider the set $T=\{t_1,t_2\}\sse\La$ of marked points, with $t_1$ in one component of $\La$ and $t_2$ in the other. Then the relative homology group is $H_1(L,T)\cong \mathbb{Z}^2$, and Figure <ref> depicts two blue paths which are a basis of $H_1(L,T)$. Note that exactly one of the generators of the basis naturally lives in $H_1(L)\sse H_1(L,T)$. With respect to the intersection pairing $\inprod{\cdot}{\cdot}$, the blue paths give rise to a dual basis consisting of the red paths. By applying the duality from Proposition <ref>, we obtain a corresponding basis for $H_1(L\setminus T,\Lambda\setminus T)$, which we have depicted in red in the same figure.

(0,0) arc (-180:0:1 and 0.25);
[dashed] (2,0) arc (0:180:1 and 0.25);
(0,0) – (0,2);
(2,0) – (2,2);
(1,2) ellipse (1 and 0.25);
[blue, decoration=markings,
mark=at position 0.4 with >, postaction=decorate] (0,1) arc (-180:0:1 and 0.25);
[blue, dashed] (2,1) arc (0:180:1 and 0.25);
[red, decoration=markings,
mark=at position 0.4 with >, postaction=decorate] (0,0.75) arc (-180:0:1 and 0.25);
[red, dashed] (2,0.75) arc (0:180:1 and 0.25);
[blue, decoration=markings,
mark=at position 0.4 with >, postaction=decorate] (1,1.75) – (1,-0.25);
at (1,-0.25) [] $\ast$;
at (1,-0.5) [] $t_2$;
at (1,1.75) [] $\ast$;
at (1,2) [] $t_1$;
[red, decoration=markings,
mark=at position 0.4 with >, postaction=decorate] (1.25,1.76) – (1.25,-0.24);

The two blue paths are cycles that represent a basis of $H_1(L,T)$. The two red paths are relative cycles in $H_1(L\setminus T,\Lambda\setminus T)$ that represent the dual bases with respect to the intersection pairing.

We emphasize that the lattice $H_1(L\setminus T,\La\setminus T)$, or from a cluster theoretical viewpoint, the character lattice for the cluster $\mathcal{A}$-structure, does not carry a natural skew-symmetric form. It is the lattice $H_1(L,T)$ (or more generally, the character lattice for a cluster $\mathcal{X}$-structure) that carries a natural skew-symmetric form, which in turn defines the quiver for the (quasi-)cluster structure. This is why every cluster $\mathcal{X}$-variety is equipped with a natural log-canonical Poisson structure, which further admits a canonical quantization [21].

Finally, we conclude this section by noting that it is possible, and in many cases useful, to consider intermediate lattices $M$ and $N$ in between the lattices discussed above. Namely, we can consider sublattices $N$ of $H_1(L,T)$ which include $H_1(L)$, and dually quotients $M$ of $H_1(L\setminus T,\La\setminus T)$:

\begin{tikzcd}
H_1(L,T) \arrow[r,<->] & H_1(L\setminus T,\La\setminus T) \arrow[d, two heads] \\
N \arrow[u,hook] \arrow[r,<->] & M \arrow[d, two heads] \\
H_1(L)  \arrow[u,hook]\arrow[r,<->] & H_1(L,\La)
\end{tikzcd}$$

In the diagram above all horizontal arrows are dual lattices. The intermediate lattices are only interesting if $T$ is large enough; e.g.  if $\La$ is a knot and $T$ consists of a unique marked point, these lattices are all isomorphic. To wit, already in the study of max-tb Legendrian $(2,n)$-torus links $\La\sse(\R^3,\xi_\st)$, i.e. a plabic fence $\bG$ with two $\bG$-strands and $n$ vertical edges, it can be desirable to consider the intermediate lattices $M,N$. Indeed, there are natural cluster structures arising from the study of triangulations of $(n+3)$-gons <cit.>, with a varying number of frozen vertices. In the literature on cluster algebras, a rather standard choice of frozens is the set of boundary frozens, one per boundary edge of the $(n+3)$-gon. This choice is not as intuitive from the perspective of contact topology, where assigning one marked point per component appears to be more natural; thus having a unique frozen vertex if $n$ is even, and none if $n$ is odd. Nevertheless, the addition of such frozen variables often leads to more tractable combinatorics and novel results for cluster variables, e.g.  the saturation property of Newton polytopes [51, 58]. It can thus be of interest to be able to discuss such cluster structures (with the boundary frozens) from a contact and symplectic topology perspective. With the flexibility in choosing intermediate lattices, this can be achieved. We illustrate this in the following example.

Adding marked points $T$ to a front of the trefoil in order to incorporate boundary frozen variables. The weave represents an embedded exact Lagrangian filling of this Legendrian. In this case, the lattices $M,N$ are intermediate lattices.

Consider the weave depicted in Figure <ref>, which represents an embedded exact Lagrangian once-punctured 2-torus filling of the max-tb Legendrian (right-handed) trefoil $\La(3_1)$. It is associated to a plabic fence, and dual to the triangulation drawn in the figure. If we choose the set $T=\{t_0\}$ to be a unique marked point, the $\mathcal{X}$–lattice is $H_1(L,T)\cong H_1(L)\cong \Z^2\langle\gamma_1,\gamma_2\rangle$, and the $\mathcal{A}$–lattice is generated by their duals. In particular, there are no frozen variables, and the intersection form yields the $A_2$-quiver with two mutable vertices. In order to incorporate the boundary frozen variables, we enlarge the set of marked point to $T=\{t_0,t_1,t_2,\ldots,t_{10}\}$, adding 10 marked points in the front, as depicted in Figure <ref>: near each of the five crossings of the front associated to the weave lines we incorporate two marked points, one on the left and one on the right, always in the upper strand. This enlargement is logical: we are incorporating the relative cycles $\tau_1,\ldots,\tau_5$, which now belong to $H_1(L,T)$, and appear naturally in the weave.

The lattice $H_1(L,T)$ has rank 11, and it contains the sublattices $H_1(L)=\Z^2\langle\gamma_1,\gamma_2\rangle$ and $N_f:=\Z^5\langle\tau_1,\tau_2,\tau_3,\tau_4,\tau_5\rangle$. Notice that the marked point $t_0$ is kept so as to avoid acquiring the relation $\tau_1+\tau_2+\tau_3+\tau_4+\tau_5=0$, which would make $N_f$ of rank 4 instead. At this point we can then consider the sublattice $N:=H_1(L)\oplus N_f$ of rank 7, which precisely captures the two mutable variables (dual to the diagonal edges of the triangulation) and the five boundary frozen variables. By construction, the lattice $N$ intermediately (and strictly) lies as $H_1(L)\sse N\sse H_1(L,T)$, and similarly for its dual lattice $M$.

§ CONSTRUCTION OF QUASI-CLUSTER STRUCTURES IN SHEAF MODULI

In this section we develop the necessary results to study the geometry of the moduli stack $\FM(\La,T)$ associated to $\La=\La(\bG)$ and prove Theorem <ref>. In particular, we introduce microlocal merodromies in Subsection <ref> which, as we will prove, become the cluster $\mathcal{A}$-variables. which. The construction of the cluster structures is obtained purely by symplectic geometric means, using the results for Legendrian weaves from Section <ref> above and [14] and the microlocal theory of sheaves [33, 54, 74]. The proof of its existence also requires a couple of results from algebraic geometry, which we present in detail as well.

Let us review what we have developed in Sections <ref> and <ref> thus far. Given a GP-graph $\bG$, we constructed the following list of objects:

(i) A Legendrian link $\La=\La(\bG)$, which is a $(-1)$ closure of a positive braid $\beta(\bG)$.

(ii) An exact Lagrangian filling $L=L(\ww)$ of $\La$ called the initial filling. This exact Lagrangian filling $L$ is obtained as the Lagrangian projection of the Legendrian lift associated with the spatial front defined by the initial weave $\ww=\ww(\bG)$.

(iii) A collection of initial absolute cycles $\fS(\bG)$, which form an $\mathbb{L}$-compressing system for $L$ and can be described by $\sf Y$-trees on $\ww$.

(iv) A skew-symmetric intersection pairing on the lattice $H_1(L)$. This intersection pairing can be computed directly from the GP-graph $\bG$.

By specifying an additional generic set of marked points $T\subset \La$ with at least one marked point per component, we also obtain the lattice $H_1(L,T)$, which contains $H_1(L)$ and hence the linearly independent subset $\fS(\bG)$. The skew-symmetric pairing on $H_1(L)$ extends naturally to a skew-symmetric pairing on $H_1(L,T)$. By Poincaré duality, we can identify the dual lattice of $H_1(L,T)$ with the relative homology $H_1(L\setminus T, \Lambda\setminus T)$. Any completion of $\fS(\bG)$ to a basis $\fB$ of $H_1(L,T)$ gives rise to a unique dual basis $\fB^\vee$ of $H_1(L\setminus T,\Lambda\setminus T)$.

The outline for this section is as follows. First, we give working definitions of the moduli space $\fM(\La,T)$, which allows us to draw connections to Lie-theortical moduli spaces and also deduce the factoriality of $\mathcal{O}(\fM(\La,T))$. Next, on the moduli space $\fM(\La,T)$, we construct a new family of $\mathbb{C}^\times$-valued rational functions called microlocal merodromies, which are associated with relative cycles in $H_1(L\setminus T,\Lambda\setminus T)$. Although the definition of microlocal merodromies depends on the initial filling $L$, we show that for elements in the dual basis $\fB^\vee$, their microlocal merodromies actually extend to $\mathbb{C}$-valued regular functions on the entire moduli space $\fM(\La,T)$. Moreover, we also prove that within these special microlocal merodromies, those dual to $\fS(\bG)$ can be mutated according to the cluster $\mathcal{A}$-mutation formula as the initial weave $\ww$ undergoes weave mutation, corresponding to a Lagrangian disk surgery on $L(\ww)$. Then, we show that the codimension-2 argument in cluster varieties can be applied by studying immersed Lagrangian fillings represented by non-free weaves. Finally, these results together with a theorem of Berenstein-Fomin-Zelevinsky allow us to conclude existence of a cluster $\mathcal{A}$-structure on $\fM(\La,T)$, where the initial and adjacent seeds are constructed via the Lagrangian filling $L(\bG)$, its Lagrangian surgeries and the associated microlocal merodromies.

§.§ Descriptions of sheaves with singular support on the Legendrian $\La(\bG)$
Let $\La\sse(\R^3\xi_\st)$ be a Legendrian link, $T\sse\La$ a set of marked points, and consider the moduli stacks $\M_1(\Lambda)$ and $\FM(\Lambda, T)$ discussed in Section <ref>. These stacks classify (complexes of) constructible sheaves on $\R^2$ with a singular support condition. In this section, we provide Lie-theoretical descriptions for $\M_1(\Lambda)$ and $\FM(\Lambda, T)$ which are suited for our computations, using [54] and closely following <cit.> and Section 5 in ibid.[An expert in the results of [54, 74] might be able to quickly move forward to Subsection <ref>.] These are more combinatorial presentations of these stacks, as the constructible and microlocal aspects of the original definition are translated into explicit quiver representations satisfying certain conditions.

First, it suffices to focus on sheaves, rather than complexes of sheaves, as explained in Section <ref>, since $\La(\bG)$ admits a binary Maslov index <cit.>. By <cit.>, the sheaves parametrized by $\M_1(\Lambda)$ and $\FM(\Lambda, T)$ are constructible. A rather combinatorial presentation of (the category of) constructible sheaves is then provided by work of R. MacPherson: for a stratified topological space $(M,S)$, the category
of $S$-constructible sheaves of sets on $M$ is equivalent to the category of set-valued functors on the exit-path category. See <cit.> and <cit.>. In our case, the space being stratified is $M=\R^2$ and the strafication $S$ is provided by a front $\pi_F(\La)\sse\R^2$ of $\La\sse(\R^3\xi_\st)$, where $\pi_F:\R^3\lr\R^2$ denotes a front (Legendrian) projection. Since we can ensure that this particular fronts are stratifying $\R^2$ as a regular cell-complex, and that the strate are simply-connected, the exit-path category is equivalent to the poset category of $(M,S)$, whose objects are cells and morphisms are inclusions. Thus, by virtue of being constructible, sheaves being parametrized can be described as quiver representations (with certain conditions), where the quiver is the one associated to the (cell) poset category of $M$. Second, the condition of having singular support contained in $\La$ translates into constraints, which further simplifies the description, as beautifully explained in <cit.>. Indeed, it follows from that result that it suffices to consider only maps between the top-dimensional 2-cells $\R^2\setminus \pi_F(\La)$, only in the direction opposite to co-orientation, and certain condition must be imposed for the maps near cusps and crossings. Following <cit.>, the combination of these results of R. MacPerson, [54] and [74] lead to the following description for $\M_1(\Lambda)$ and $\FM(\Lambda, T)$.

Given a co-oriented front projection $\pi_F(\La)$, consider the following quiver $Q_F(\Lambda)$:

- A vertex of $Q_F(\La)$ is placed at each connected component of the $\mathbb{R}^2\setminus \pi_F(\Lambda)$,

- For each (1-dimensional) connected component of $\pi_F(\Lambda)\setminus S_0$, where $S_0$ denotes the set of crossings and cusps in $\pi_F(\Lambda)$, draw an arrow connecting the two vertices associated to the two adjacent 2-dimensional cell (that contain that stratum in their closure). The direction of the arrow is opposite to the co-orientation of the front $\pi_F(\La)$.

The following is then proven in <cit.>:

Let $\La\sse(\R^3,\xi)$ be a Legendrian and $\pi_F(\La)\sse\R^2$ a front, with a binary Maslov potential, such that $(\R^2,\pi_F(\La))$ is a regular stratification. Consider the stack $\mathcal{M}(Q_F(\La))$ classifying linear representations of the quiver $Q_F(\Lambda)$ that satisfy the following conditions:

* The vector space associated with the unbounded region in $\mathbb{R}^2$ is $0$.

* Any two vector spaces associated with neighboring vertices differ in dimension by $1$.

* At each cusp, the composition depicted in Figure <ref> (left) is the identity map.

* At each crossing, the four linear maps involved form a commuting square which is exact, as precised in Figure <ref> (right).

[lightgray] (0.75,0.75) to [out=180,in=30] (-0.5,0) to [out=-30,in=180] (0.75,-0.75);
(v) at (-1,0) [] $V$;
(w) at (1,0) [] $W$;
[->] (v) to [out=-45,in=-135] node [below] $f$ (w);
[->] (w) to [out=135,in=45] node [above] $g$ (v);
at (0,-1.75) [] $g\circ f=\mathrm{id}_V$;

[lightgray] (-0.75,-0.75) – (0.75,0.75);
[lightgray] (-0.75,0.75) – (0.75,-0.75);
(e) at (1,0) [] $V_e$;
(s) at (0,-1) [] $V_s$;
(w) at (-1,0) [] $V_w$;
(n) at (0,1) [] $V_n$;
[->] (s) – node[below right] $f_{se}$ (e);
[->] (s) – node [below left] $f_{sw}$ (w);
[->] (e) – node [above right] $f_{en}$ (n);
[->] (w) – node [above left] $f_{wn}$ (n);
at (0,-1.75) [] $0\longrightarrow V_s\xrightarrow{(f_{sw},f_{se})} V_w\oplus V_e\xrightarrow{f_{wn}-f_{en}} V_n\longrightarrow 0$;

Identity condition at cusps and exactness condition at crossings.

Then the stack $\M_1(\Lambda)$ is isomorphic to $\mathcal{M}(Q_F(\La))$.

The dimension vector of such quiver representations is uniquely determined by $\pi_F(\Lambda)$. Moreover, the identity condition at cusps and the exactness condition at crossings determine that each homomorphism is one of the following:

- injective with a 1-dimensional cokernel if it crosses a strand with Maslov potential $0$;

- surjective with a 1-dimensional kernel if it crosses a strand with Maslov potential $1$.

The left picture in Figure <ref> draws the front from Figure <ref> (left) for a Legendrian $m(5_2)$, and the vector spaces assigned to each top-dimensional cell of the complement of the front, according to the quiver $Q(\pi_F(\La))$ and Proposition <ref>.

[xscale=0.5, yscale=0.7,baseline=0]
(-0.5,0) – (-1,0) – (-2,-1) – (-4,-1) – (-4.5,-0.5) – (-5,0) – (-7,0) – (-8,-1) – (-9,-1) – (-8,-2) – (0.5,-2) – (-0.5,-1) – (-1,-1) – (-2,0) – (-4.5,0) – (-5,-0.5) – (-6,-0.5) – (-4,-2.5) – (-1.5,-2.5) – (-3.5,-0.5) – (-4,-0.5) – (-4.5,-1) – (-7,-1) – (-8,0) – (-11,0) – node [left] $0$ (-8,-3) – (2.5,-3) – cycle;
[red] (red) at (-1,-2.5) [] $\mathbb{C}$;
[cyan] (cyan) at (-1.5,-1.5) [] $\mathbb{C}^2$;
[teal] (teal) at (-7.5,-1.5) [] $\mathbb{C}^2$;
[purple] (purple) at (-3,-2.25) [] $\mathbb{C}^2$;
[orange] (orange) at (-3.75,-0.75) [] $\mathbb{C}^2$;
[blue] (blue) at (-3,-0.25) [] $\mathbb{C}$;
[brown] (brown) at (-6.5,-0.5) [] $\mathbb{C}$;
[pink] (pink) at (-5,-0.75) [] $\mathbb{C}^2$;
(black) at (-4,-1.5) [] $\mathbb{C}^3$;
[blue] (blue) at (0.7,0) [] $\mathbb{C}$;
[brown] (brown) at (-3.5,0) [] $\mathbb{C}$;
[cyan] (cyan) at (1,-1.5) [] $\mathbb{C}^2$;
[orange] (orange) at (-0.3,-0.75) [] $\mathbb{C}^2$;
[pink] (pink) at (-2.7,-0.75) [] $\mathbb{C}^2$;
[teal] (teal) at (-4,-1.5) [] $\mathbb{C}^2$;
(black) at (-1.5,-1.5) [] $\mathbb{C}^3$;
(0) at (-1.5,0.5) [] $0$;
[purple] (purple) at (-1.5,-2.5) [] $\mathbb{C}^2$;
[red] (red) at (-1.5,-3.5) [] $\mathbb{C}$;
[->>] (cyan)– (blue);
[->>] (orange) – (blue);
[->>] (pink) – (blue) ;
[->>] (pink) to [out=90,in=0] (brown) ;
[right hook->] (brown) to [out=-90,in=180] (pink);
[->>] (teal)– (brown) ;
[->>] (black)–(cyan);
[right hook->]  (teal)–(black);
[->>] (black)–(orange);
[->>] (black)–(pink);
[left hook->]  (purple)–(black);
[left hook->]  (red)to [out=120,in=-120] (purple);
[right hook->] (red)–(cyan);
[left hook->] (red)–(teal);
[->>](cyan) to [out=0,in=90] (2,-2) to [out=-90,in=20] (red) ;
[->>] (teal) to [out=180,in=90] (-5,-2) to [out=-90,in=160] (red) ;
[->>] (purple) to [out=-60,in=60] (red);
[->>] (blue) – (0);
[->>] (brown)– (0) ;
[->>] (red)to[out=180,in=-90] (-6,-1.5) to [out=90,in=180] (0) ;
[->>] (red) to [out=0,in=-90] (3,-1.5) to [out=90,in=0] (0);
[left hook->] (0) to [out=90,in=90] (-6.5,-1.5) to [out=-90,in=-90] (red);

(Left) Defining the vertices of the quiver $Q(\pi_F(\La))$ from the front, by assigning a vector space to each top-dimensional strata. In this example $\La=\La(\bG)$ from Figure <ref>, a Legendrian representative of $m(5_2)$. (Right) The (representations of the) quiver associated to the front. The representations consist of injective and surjective homomorphisms between vector spaces going across strands, as indicated.

Figure <ref> (right) depicts the associated quiver $Q(\pi_F(\La))$. Each quadrilateral piece in the quiver forms a short exact sequence; each bigon consists of one injective map and a surjective map, and the composition satifies $($surjective map$)\circ($injective map$)=$ identity.

Proposition <ref> describes $\mathcal{M}_1(\La)$. Now we gear towards the decorated moduli $\mathfrak{M}(\La,T)$. First, we need a description of microlocal monodromy in terms of these quiver representations, which is provided in <cit.> and we briefly summarize as follows. Let $S$ be the set of singular points (crossings and cusps) in $\pi_F(\Lambda)$. By Remark <ref>, each connected components of $\pi_F(\Lambda)\setminus S$ is associated with a 1-dimensional kernel or a 1-dimensional cokernel. These kernels and cokernels can be glued together along strands of $\Lambda$ using the identity condition at cusps and the exactness condition at crossings, as follows:

- At a cusp, as in Figure <ref> (left), the condition $g\circ f=\id_V$ forces the composition $\ker g\hookrightarrow W\twoheadrightarrow \coker f$ to be an isomorphism. By definition, we glue $\ker g$ and $\coker f$ using this isomorphism.

- At a crossing, as in Figure <ref> (right), there are three cases depending, on the injectivity or surjectivity of the four maps: the four maps can be all injective, all surjective, or two injective with two surjective. In each of the three cases, we have the following isomorphisms from the exactness condition:

\begin{align}
\begin{tikzpicture}[baseline=0]
\draw [lightgray] (-0.75,-0.75) -- (0.75,0.75);
\draw [lightgray] (-0.75,0.75) -- (0.75,-0.75);
\node (e) at (1,0) [] {$V_e$};
\node (s) at (0,-1) [] {$V_s$};
\node (w) at (-1,0) [] {$V_w$};
\node (n) at (0,1) [] {$V_n$};
\draw [right hook->] (s) -- node[below right] {$f_{se}$} (e);
\draw [left hook->] (s) -- node [below left] {$f_{sw}$} (w);
\draw [left hook->] (e) -- node [above right] {$f_{en}$} (n);
\draw [right hook->] (w) -- node [above left] {$f_{wn}$} (n);
\end{tikzpicture} \quad & \quad \quad \begin{array}{l}
\coker f_{sw} \hookrightarrow\dfrac{V_n}{V_s} \twoheadrightarrow \coker f_{en}       \\ \\
\coker f_{wn} \twoheadleftarrow\dfrac{V_n}{V_s} \hookleftarrow \coker f_{se}
\end{array} \label{eq:gluing with inclusion maps}
\\
\begin{tikzpicture}[baseline=0]
\draw [lightgray] (-0.75,-0.75) -- (0.75,0.75);
\draw [lightgray] (-0.75,0.75) -- (0.75,-0.75);
\node (e) at (1,0) [] {$V_e$};
\node (s) at (0,-1) [] {$V_s$};
\node (w) at (-1,0) [] {$V_w$};
\node (n) at (0,1) [] {$V_n$};
\draw [->>] (s) -- node[below right] {$f_{se}$} (e);
\draw [->>] (s) -- node [below left] {$f_{sw}$} (w);
\draw [->>] (e) -- node [above right] {$f_{en}$} (n);
\draw [->>] (w) -- node [above left] {$f_{wn}$} (n);
\end{tikzpicture} \quad & \quad \quad \begin{array}{l}
\ker f_{sw} \hookrightarrow \ker(f_{wn}\circ f_{sw})=\ker(f_{en}\circ f_{se}) \twoheadrightarrow \ker f_{en}       \\ \\
\ker f_{wn} \twoheadleftarrow\ker(f_{wn}\circ f_{sw})=\ker(f_{en}\circ f_{se}) \hookleftarrow \ker f_{se}
\end{array}
\\
\begin{tikzpicture}[baseline=0]
\draw [lightgray] (-0.75,-0.75) -- (0.75,0.75);
\draw [lightgray] (-0.75,0.75) -- (0.75,-0.75);
\node (e) at (1,0) [] {$V_e$};
\node (s) at (0,-1) [] {$V_s$};
\node (w) at (-1,0) [] {$V_w$};
\node (n) at (0,1) [] {$V_n$};
\draw [->>] (s) -- node[below right] {$f_{se}$} (e);
\draw [left hook->] (s) -- node [below left] {$f_{sw}$} (w);
\draw [left hook->] (e) -- node [above right] {$f_{en}$} (n);
\draw [->>] (w) -- node [above left] {$f_{wn}$} (n);
\end{tikzpicture} \quad & \quad \quad \begin{array}{l}
\coker f_{sw} \xrightarrow{f_{wn}} \coker f_{en}       \\ \\
\ker f_{wn} \xleftarrow{f_{sw}} \ker f_{se}
\end{array}
\end{align}

The result of gluing these 1-dimensional vector spaces is a rank-1 local system $\Phi$ on $\Lambda$. In fact, it coincides with the microlocal monodromy functor, see <cit.> for more details. Given the set $T$ of marked points on $\Lambda$, with at least one marked point per link component, $\Lambda\setminus T$ is a collection of open intervals. Thus, along each such open interval $I$, we can trivialize the rank-1 local system $\Phi$ by specifying an isomorphism $\phi_I:I\times\mathbb{C}\xrightarrow{\cong}\Phi|_I$. By definition, a collection of such maps $\{\phi_I\}$ are said to be a framing for the local system $\Phi$.

In conclusion, a point in the decorated moduli space $\FM(\Lambda,T)$, as defined in Subsection <ref>, is a point in $\M_1(\Lambda)$, which is combinatorialized via Proposition <ref>, together with a framing for the local system $\Phi$, i.e.  a trivialization of the (trivial) local system $\Phi|_{\Lambda\setminus T}$. In this description, wwo framings are considered equivalent if they differ by a global scaling $\mathbb{C}^\times$-factor. It follows that $\dim \FM(\Lambda,T)=\dim \M_1(\Lambda)+|T|-1$.

§.§.§ Descriptions for GP-graph $\bG$.
In the case that $\La=\La(\bG)$ comes from a GP-graph $\bG$, Section <ref> provides a specific front $\mathfrak{f}(\bG)\sse\R^2$. For this front, the description from Proposition <ref> can be translated in terms of configurations of flags, as follows. The front projection $\pi_F(\Lambda)=\mathfrak{f}(\bG)$ can be sliced into the three types of elementary columns.

For a Type 1 column, there are $n$ strands in the bottom region, with Maslov potential 0, and $n$ strands in the top region, with Maslov potential 1. By Proposition <ref>, the vector space associated with the central region must be $\mathbb{C}^n$. From the quiver representation data, we can construct the following pair of flags in $\mathbb{C}^n$. Since all of the linear maps in the bottom region are injective, their images in the middle $\mathbb{C}^n$ naturally form a first flag. Similarly, since all of the linear maps in the top region are surjective, their kernels in the middle vector space $\mathbb{C}^n$ form a second flag. See Figure <ref> for a depiction of the front in Type 1 and its associated pair of flags. We adopt the convention of indexing flags from the bottom region with a subscript, and indexing flags from the top region with a superscript, so as to distinguish between these two types of two flags.

ıin 0,1,2,3

[lightgray] (0,ı*0.75) – (2,ı*0.75);
[lightgray] (0,ı*0.75+3.5) – (2,ı*0.75+3.5);

ıin 2

(bı) at (1,ı*0.75-0.375) [] $\mathbb{C}^\i$;
(tı) at (1,3*0.75+3.5+0.375-ı*0.75) [] $\mathbb{C}^\i$;

(b1) at (1,0.375) [] $\mathbb{C}$;
(t1) at (1,3*0.75+3.5+0.375-0.75) [] $\mathbb{C}$;
(m)  at (1,2.875) [] $\mathbb{C}^n$;
(b3) at (1.2,1.925) [] $\vdots$;
(t3) at (1.2,3.875) [] $\vdots$;
(b0) at (1,-0.375) [] $0$;
(t0) at (1,3.5+3*0.75+0.375) [] $0$;
[blue, ->] (b0) to [out=30,in=-30] node [right] $f_1$ (b1);
[blue, ->] (b1) to [out=30,in=-30] node [right] $f_2$ (b2);
[blue, ->] (b2) to [out=30,in=-30] (b3);
[blue, ->] (b3) to [out=30,in=-30] node [right] $f_n$ (m);
[blue, ->] (m) to [out=30,in=-30] node [right] $g_n$ (t3);
[blue, ->] (t3) to [out=30,in=-30] (t2);
[blue, ->] (t2) to [out=30,in=-30] node [right] $g_2$ (t1);
[blue, ->] (t1) to [out=30,in=-30] node [right] $g_1$ (t0);
at (8,1.25) [] $\SF_0=\left(\begin{array}{l} 0\subset \img (f_2\circ \cdots \circ f_n) \subset \img(f_3\circ  \cdots \circ f_n) \\ \subset \cdots\subset \img f_n\subset \mathbb{C}^n\end{array}\right)$;
at (7,4.5) [] $\SF^0=\left(\begin{array}{l} 0\subset \ker g_n\subset \ker (g_{n-1}\circ g_n)  \subset \\ \cdots \subset \ker(g_2\circ \cdots \circ g_n)\subset \mathbb{C}^n\end{array}\right)$;

The pair of flags associated to a Type 1 column.

Before discussing Type 2 and 3 columns, we recall that the relative positions relations between two flags in $\mathbb{C}^n$ are classified by elements of the symmetric group $S_n$, which is a Coxeter group with Coxeter generators $\{s_i\}_{i=1}^{n-1}$. For two flags $\SF=(0\subset \SF_1\subset \cdots \subset \mathbb{C}^n)$ and $\SF'=(0\subset \SF'_1\subset \cdots\subset \mathbb{C}^n)$, we denote

* $\SF\stackrel{s_i}{\sim}\SF'$ if $F_i\neq F'_i$ but $F_j=F'_j$ for all $j\neq i$;

* $\SF\stackrel{w}{\sim}\SF'$ if there exists a sequence of flags $\SG_0, \SG_1,\dots, \SG_l$ such that

\[
\SF=\SG_0\stackrel{s_{i_1}}{\sim} \SG_1\stackrel{s_{i_2}}{\sim}\SG_2\stackrel{s_{i_3}}{\sim}\cdots \stackrel{s_{i_l}}{\sim}\SG_l=\SF'
\]

and $s_{i_1}s_{i_2}\cdots s_{i_l}$ is a reduced word of $w$.

This classification can be identified with the Tits distance obtained from a Bruhat decomposition of $\GL_n$. In particular, being in $w$ relative position does not depend on the choice of reduced word of $w$. If $\SF\stackrel{w}{\sim}\SF'$, then for each choice of reduced word $(i_1,\dots, i_l)$ for $w$, there exists a unique sequence of flags $(\SG_k)_{k=0}^l$ that relate the two flags $\SF$ and $\SF'$.

We can now translate the local quiver representation data associated with a Type 2 column into relative position relations between flags. Suppose the pair of flags to the left of a Type 2 column is $(\SL_0,\SL^0)$, and the pair of flags to the right of a Type 2 column is $(\SR_0,\SR^0)$. If there is a crossing in the bottom region at the $i$th gap, counting from the bottom in both the front and in the GP-graph, then from the exactness condition at the crossing obtain the constraints

\begin{equation}\label{eq: type 2 bottom relative position}
\SL_0\overset{s_i}{\sim}\SR_0 \quad \quad \text{and} \quad \quad \SL^0=\SR^0.
\end{equation}

Similarly, if there is a crossing in the top region at the $i$th gap, counting from the top in the front projection or counting from the bottom in the GP-graph, then the exactness condition at the crossing yields

\begin{equation}\label{eq: type 2 top relative position}
\SL_0=\SR_0 \quad \quad \text{and} \quad \quad \SL^0\overset{s_{n-i}}{\sim}\SR^0.
\end{equation}

Since crossings in Type 2 columns correspond to vertical edges in the GP-graph, we can infer the relative position relation between pairs of flags from the GP-graph as well.

The following four Type 2 columns of a GP-graph give rise to the following relative position relations between pairs of flags:

\[
\begin{tikzpicture}[scale=0.7]
\foreach \i in {0,1,2}
\draw (0,\i) -- (2,\i);
\draw (1,0) -- (1,1);
\draw [fill=black] (1,0) circle [radius=0.2];
\draw [fill=white] (1,1) circle [radius=0.2];
\node at (1,-1) [] {$\begin{array}{l} \SL^0=\SR^0 \\  \SL_0\overset{s_1}{\sim}\SR_0\end{array}$};
\end{tikzpicture} \quad \quad \quad
\begin{tikzpicture}[scale=0.7]
\foreach \i in {0,1,2}
\draw (0,\i) -- (2,\i);
\draw (1,2) -- (1,1);
\draw [fill=black] (1,1) circle [radius=0.2];
\draw [fill=white] (1,2) circle [radius=0.2];
\node at (1,-1) [] {$\begin{array}{l} \SL^0=\SR^0 \\  \SL_0\overset{s_2}{\sim}\SR_0\end{array}$};
\end{tikzpicture}
\quad \quad \quad
\begin{tikzpicture}[scale=0.7]
\foreach \i in {0,1,2}
\draw (0,\i) -- (2,\i);
\draw (1,0) -- (1,1);
\draw [fill=white] (1,0) circle [radius=0.2];
\draw [fill=black] (1,1) circle [radius=0.2];
\node at (1,-1) [] {$\begin{array}{l} \SL^0\overset{s_2}{\sim}\SR^0 \\  \SL_0=\SR_0\end{array}$};
\end{tikzpicture}\quad \quad \quad
\begin{tikzpicture}[scale=0.7]
\foreach \i in {0,1,2}
\draw (0,\i) -- (2,\i);
\draw (1,2) -- (1,1);
\draw [fill=black] (1,2) circle [radius=0.2];
\draw [fill=white] (1,1) circle [radius=0.2];
\node at (1,-1) [] {$\begin{array}{l} \SL^0\overset{s_1}{\sim}\SR^0 \\  \SL_0=\SR_0\end{array}$};
\end{tikzpicture}
\]

For a Type 3 column, the pairs of flags on the (Type 1 column on the) left and on the (Type 1 column on the) right are not in the same ambient vector space, as the dimensions of the two vector spaces differ by one. Instead, there is a linear map from the ambient vector space for the pair of flags on the left to the ambient vector space for the pair of flags on the right. This linear map is injective if the lollipop is white and it is surjective if the lollipop is black. Let us investigate how the two pairs of flags are related.

Suppose first that the lollipop is white, so that the linear map $h:\mathbb{C}^{n-1}\rightarrow \mathbb{C}^{n}$ between the two (middle) adjacent ambient vector spaces is injective. Given any flag $\SF=(0\subset \SF_1\subset \cdots\subset \SF_{n-1}=\mathbb{C}^{n-1})$ in $\mathbb{C}^{n-1}$, we can use $h$ to naturally extend it to a flag $h(\SF)$ in $\mathbb{C}^{n}$ by defining

\[
h(\SF):=(0\subset h(\SF_1)\subset h(\SF_2)\subset \cdots\subset h(\SF_{n-1})\subset \mathbb{C}^{n}).
\]

This extension from $(\SL_0,\SL^0)$ to $(h(\SL_0),h(\SL^0))$ can be achieved geometrically by a sequence of RII moves that pulls the left cusp upward in the front projection. Indeed, consider the following local example:

ıin 0,1,3

[lightgray] (0,ı*0.7) – (3,ı*0.7);
[lightgray] (0,6-ı*0.7) – (3,6-ı*0.7);

[lightgray] (3,4.6) – (1,4.6) – (2,1.4) – (3,1.4);
(bl0) at (0.5,-0.35) [] $0$;
(bl1) at (0.5,0.35) [] $\mathbb{C}$;
(bl2) at (0.5,1.4) [] $\mathbb{C}^2$;
(l3) at (0.5,3) [] $\mathbb{C}^3$;
(tl2) at (0.5,4.6) [] $\mathbb{C}^2$;
(tl1) at (0.5,5.65) [] $\mathbb{C}$;
(tl0) at (0.5,6.35) [] $0$;
(br0) at (2.5,-0.35) [] $0$;
(br1) at (2.5,0.35) [] $\mathbb{C}$;
(br2) at (2.5,1.05) [] $\mathbb{C}^2$;
(br3) at (2.5,1.8) [] $\mathbb{C}^3$;
(r4) at (2.5,3) [] $\mathbb{C}^4$;
(tr3) at (2.5,4.25) [] $\mathbb{C}^3$;
(tr2) at (2.5,4.95) [] $\mathbb{C}^2$;
(tr1) at (2.5,5.65) [] $\mathbb{C}$;
(tr0) at (2.5,6.35) [] $0$;
[->] (bl0) to [out=150,in=-150] (bl1);
[->] (bl1) to [out=150,in=-150] (bl2);
[->] (bl2) to [out=150,in=-150] (l3);
[->] (l3) to [out=150,in=-150] (tl2);
[->] (tl2) to [out=150,in=-150] (tl1);
[->] (tl1) to [out=150,in=-150] (tl0);
[->] (l3) – node [above] $h$ (r4);
[->] (br0) to [out=30,in=-30] (br1);
[->] (br1) to [out=30,in=-30] (br2);
[->] (br2) to [out=30,in=-30] (br3);
[->] (br3) to [out=30,in=-30] (r4);
[->] (r4) to [out=30,in=-30] (tr3);
[->] (tr3) to [out=30,in=-30] (tr2);
[->] (tr2) to [out=30,in=-30] (tr1);
[->] (tr1) to [out=30,in=-30] (tr0);
at (-1,1) [] $\SL_0$;
at (-1,5) [] $\SL^0$;
at (4,1) [] $\SR_0$;
at (4,5) [] $\SR^0$;
ıin 0,1,3

[lightgray] (-1,ı*0.7) – (3,ı*0.7);
[lightgray] (-1,6-ı*0.7) – (3,6-ı*0.7);

[lightgray] (3,4.6) – (2,4.6) – (0,6.7) – (2,1.4) – (3,1.4);
[blue] (0,6.7) – (2,1.4);
(bl0) at (-0.5,-0.35) [] $0$;
(bl1) at (-0.5,0.35) [] $\mathbb{C}$;
(bl2) at (-0.5,1.4) [] $\mathbb{C}^2$;
(l3) at (-0.5,3) [] $\mathbb{C}^3$;
(tl2) at (-0.5,4.6) [] $\mathbb{C}^2$;
(tl1) at (-0.5,5.65) [] $\mathbb{C}$;
(tl0) at (-0.5,6.35) [] $0$;
(br0) at (2.5,-0.35) [] $0$;
(br1) at (2.5,0.35) [] $\mathbb{C}$;
(br2) at (2.5,1.05) [] $\mathbb{C}^2$;
(br3) at (2.5,1.8) [] $\mathbb{C}^3$;
(r4) at (2.5,3) [] $\mathbb{C}^4$;
(tr3) at (2.5,4.25) [] $\mathbb{C}^3$;
(tr2) at (2.5,4.95) [] $\mathbb{C}^2$;
(tr1) at (2.5,5.65) [] $\mathbb{C}$;
(tr0) at (2.5,6.35) [] $0$;
(m3) at (1.5,4.6) [] $\mathbb{C}^3$;
(m2) at (0.8,5.65) [] $\mathbb{C}^2$;
(m1) at (0.25,6.35) [] $\mathbb{C}$;
(m0) at (1,6.5) [] $0$;
[teal,->] (bl0) to [out=150,in=-150] (bl1);
[teal,->] (bl1) to [out=150,in=-150] (bl2);
[teal,->] (bl2) to [out=150,in=-150] (l3);
[->] (l3) to [out=150,in=-150] (tl2);
[->] (tl2) to [out=150,in=-150] (tl1);
[->] (tl1) to [out=150,in=-150] (tl0);
[teal,->] (l3) – node [above] $h$ (r4);
[->] (br0) to [out=30,in=-30] (br1);
[->] (br1) to [out=30,in=-30] (br2);
[->] (br2) to [out=30,in=-30] (br3);
[->] (br3) to [out=30,in=-30] (r4);
[->] (r4) to [out=30,in=-30] (tr3);
[->] (tr3) to [out=30,in=-30] (tr2);
[->] (tr2) to [out=30,in=-30] (tr1);
[->] (tr1) to [out=30,in=-30] (tr0);
[red, ->] (r4) – (m3);
[red, ->] (m3) – (m2);
[red, ->] (m2) – (m1);
[red, ->] (m1) – (m0);
[->] (tl2) – (m3);
[->] (tl1) – (m2);
[->] (tl0) – (m1);
at (-2,1) [] $\SL_0$;
at (-2,5) [] $\SL^0$;
at (4,1) [] $\SR_0$;
at (4,5) [] $\SR^0$;

Pulling up a left cusp

The green maps in the bottom region define the extension $h(\SL_0)$. By the exactness of the quadrilaterals in the top region, the red maps define the extension $h(\SL^0)$. In particular, the extensions $h(\SL_0)$ and $h(\SL^0)$ are completely determined by the original data of the quiver representations.

Now, with these extensions defined, it follows that if there is a white lollipop emerging in the $i$th gap with $0\leq i\leq n-1$, counting from below in the GP-graph[The case $i=0$ is a lollipop at the bottom, and $i=n-1$ is a lollipop at the top.], then the corresponding relative position conditions are

\begin{equation}\label{eq: type 3 white lollipop relative position}
h(\SL_0)\overset{s_{n-1}\cdots s_{i+1}}{\sim}\SR_0 \quad \text{and} \quad h(\SL^0) \overset{s_{n-1}\cdots s_{n-i}}{\sim} \SR^0.
\end{equation}

Suppose that there is a black lollipop, and thus the linear map between the two ambient vector spaces $h:\mathbb{C}^{n}\rightarrow \mathbb{C}^{n-1}$ is surjective. Then, given any flag $\SF=(0\subset \SF_1\subset \cdots\subset \SF_{n-1}=\mathbb{C}^{n-1})$ in $\mathbb{C}^{n-1}$, we consider $h^{-1}(\SF_i)$ and insert $\ker(h)$ in front of it so as to form a flag in $\mathbb{C}^n$:

\[
h^{-1}(\SF) := (0\subset \ker(h)\subset h^{-1}(\SF_1)\subset \cdots \subset h^{-1}(\SF_{n-1})=\mathbb{C}^{n}).
\]

ıin 0,1,3

[lightgray] (0,ı*0.7) – (3,ı*0.7);
[lightgray] (0,6-ı*0.7) – (3,6-ı*0.7);

[lightgray] (0,4.6) – (1,4.6) – (2,1.4) – (0,1.4);
(bl0) at (2.5,-0.35) [] $0$;
(bl1) at (2.5,0.35) [] $\mathbb{C}$;
(bl2) at (2.5,1.4) [] $\mathbb{C}^2$;
(l3) at (2.5,3) [] $\mathbb{C}^3$;
(tl2) at (2.5,4.6) [] $\mathbb{C}^2$;
(tl1) at (2.5,5.65) [] $\mathbb{C}$;
(tl0) at (2.5,6.35) [] $0$;
(br0) at (0.5,-0.35) [] $0$;
(br1) at (0.5,0.35) [] $\mathbb{C}$;
(br2) at (0.5,1.05) [] $\mathbb{C}^2$;
(br3) at (0.5,1.8) [] $\mathbb{C}^3$;
(r4) at (0.5,3) [] $\mathbb{C}^4$;
(tr3) at (0.5,4.25) [] $\mathbb{C}^3$;
(tr2) at (0.5,4.95) [] $\mathbb{C}^2$;
(tr1) at (0.5,5.65) [] $\mathbb{C}$;
(tr0) at (0.5,6.35) [] $0$;
[->] (bl0) to [out=30,in=-30] (bl1);
[->] (bl1) to [out=30,in=-30] (bl2);
[->] (bl2) to [out=30,in=-30] (l3);
[->] (l3) to [out=30,in=-30] (tl2);
[->] (tl2) to [out=30,in=-30] (tl1);
[->] (tl1) to [out=30,in=-30] (tl0);
[->] (r4) – node [above] $h$ (l3);
[->] (br0) to [out=150,in=-150] (br1);
[->] (br1) to [out=150,in=-150] (br2);
[->] (br2) to [out=150,in=-150] (br3);
[->] (br3) to [out=150,in=-150] (r4);
[->] (r4) to [out=150,in=-150] (tr3);
[->] (tr3) to [out=150,in=-150] (tr2);
[->] (tr2) to [out=150,in=-150] (tr1);
[->] (tr1) to [out=150,in=-150] (tr0);
at (-1,1) [] $\SL_0$;
at (-1,5) [] $\SL^0$;
at (4,1) [] $\SR_0$;
at (4,5) [] $\SR^0$;
ıin 0,1,3

[lightgray] (0,ı*0.7) – (4,ı*0.7);
[lightgray] (0,6-ı*0.7) – (4,6-ı*0.7);

[lightgray] (0,4.6) – (1,4.6) – (3,-0.7) – (1,1.4) – (0,1.4);
[blue] (3,-0.7) – (1,4.6);
(bl0) at (3.5,-0.35) [] $0$;
(bl1) at (3.5,0.35) [] $\mathbb{C}$;
(bl2) at (3.5,1.4) [] $\mathbb{C}^2$;
(l3) at (3.5,3) [] $\mathbb{C}^3$;
(tl2) at (3.5,4.6) [] $\mathbb{C}^2$;
(tl1) at (3.5,5.65) [] $\mathbb{C}$;
(tl0) at (3.5,6.35) [] $0$;
(br0) at (0.5,-0.35) [] $0$;
(br1) at (0.5,0.35) [] $\mathbb{C}$;
(br2) at (0.5,1.05) [] $\mathbb{C}^2$;
(br3) at (0.5,1.8) [] $\mathbb{C}^3$;
(r4) at (0.5,3) [] $\mathbb{C}^4$;
(tr3) at (0.5,4.25) [] $\mathbb{C}^3$;
(tr2) at (0.5,4.95) [] $\mathbb{C}^2$;
(tr1) at (0.5,5.65) [] $\mathbb{C}$;
(tr0) at (0.5,6.35) [] $0$;
(m3) at (1.5,1.4) [] $\mathbb{C}^3$;
(m2) at (2.2,0.35) [] $\mathbb{C}^2$;
(m1) at (2.75,-0.35) [] $\mathbb{C}$;
(m0) at (2,-0.5)[] $0$;
[->] (bl0) to [out=30,in=-30] (bl1);
[->] (bl1) to [out=30,in=-30] (bl2);
[->] (bl2) to [out=30,in=-30] (l3);
[->, red] (l3) to [out=30,in=-30] (tl2);
[->, red] (tl2) to [out=30,in=-30] (tl1);
[->, red] (tl1) to [out=30,in=-30] (tl0);
[->, red] (r4) – node [above] $h$ (l3);
[->] (br0) to [out=150,in=-150] (br1);
[->] (br1) to [out=150,in=-150] (br2);
[->] (br2) to [out=150,in=-150] (br3);
[->] (br3) to [out=150,in=-150] (r4);
[->] (r4) to [out=150,in=-150] (tr3);
[->] (tr3) to [out=150,in=-150] (tr2);
[->] (tr2) to [out=150,in=-150] (tr1);
[->] (tr1) to [out=150,in=-150] (tr0);
at (-1,1) [] $\SL_0$;
at (-1,5) [] $\SL^0$;
at (5,1) [] $\SR_0$;
at (5,5) [] $\SR^0$;
[teal, <-] (r4) – (m3);
[teal, <-] (m3) – (m2);
[teal, <-] (m2) – (m1);
[teal, <-] (m1) – (m0);
[<-] (bl2) – (m3);
[<-] (bl1) – (m2);
[<-] (bl0) – (m1);

Pulling down a right cusp

Similar to the white lollipop case, the extension of $(\SR_0,\SR^0)$ to $(h^{-1}(\SR_0),h^{-1}(\SR^0))$ can be achieved geometrically by a sequence of RII moves that pulls the right cusp downward in the front projection, as depicted in Figure <ref>. Note that the green maps in bottom region define the extension $h^{-1}(\SR_0)$, whereas the red maps in the top region define the extension $h^{-1}(\SR^0)$.

It follows that if there is a black lollipop occurring in the $i$th gap with $0\leq i\leq n$, counting from below in the GP-graph, then the corresponding relative position conditions are

\begin{equation}\label{eq: type 3 black lollipop relative position}
\SL_0\overset{s_{i-1}\cdots s_1}{\sim}h^{-1}(\SR_0) \quad \text{and} \quad  \SL^0\overset{s_{n-i+1}\cdots s_1}{\sim}h^{-1}(\SR^0).
\end{equation}

In summary, given a GP-graph $\bG$, we can divide $\bG$ into columns of three types such that every consecutive pair of non-Type 1 columns is separated by a Type 1 column and every consecutive pair of Type 1 columns is separated by a non-Type 1 column. In between Type 3 columns there is a unique ambient vector space $V_i= \mathbb{C}^n$ for some $n$, and they are linked by linear maps $h_i:V_{i-1}\rightarrow V_i$ that are either injective with a 1-dimensional cokernel or surjective with a 1-dimensional kernel. The above discussion proves the following lemma:

For a GP-graph $\bG$ with a decomposition into columns as above, the moduli space $\M_1(\Lambda)$ can be described by the following data:

* a pair of flags in $V_i$ for each Type 1 column contained in the $V_i$ part of $\bG$;

* for each Type 2 column, the neighboring flags satisfy the relative position condition according to (<ref>) and (<ref>);

* for each Type 3 column, the neighboring flags satisfy the relative position condition according to (<ref>) and (<ref>);

where we quotient this data by the equivalence relation $(\SF,h)\sim (\SF',h')$ for a collection of elements $g_i\in \GL(V_i)$ such that $h_i\circ g_{i-1}=g_i\circ h'_i$.

In the flag description of Lemma <ref>, the rank-1 local system $\Phi$ on $\Lambda$ can be constructed by taking quotients of consecutive vector subspaces in each flag and then gluing them along strands of $\Lambda$ at crossings and cusps in the same manner before. Note that in this context, only (<ref>) is used when gluing these rank-1 local systems at crossings because all linear maps near a crossing are now inclusions of vector subspaces. In particular, the surjective maps in the top region of the front projection are now turned into inclusions of kernels.

Note that the trivialization of the rank-1 local system $\Phi$ on $\Lambda$ can be translated into the flags as well, e.g. by considering affine flags and allowing for a $\C^\times$-rescaling of the (volume form associated to the) flag at the $i$th position, top or bottom, if a marked point is at the $i$th strand, top or bottom.

§.§.§ Descriptions for (-1)-closures.
Finally, there is another description of $\M_1(\Lambda)$ and $\FM(\La,\T)$ as moduli space of configurations of flags, which aligns better when comparing with the flag moduli of the weaves $\ww(\bG)$. In that latter case, there will be only one ambient vector space. The description in Lemma <ref>, which is associated to the specific front $\mathfrak{f}(\bG)$, after using RII and RIII moves, can be shown to be equivalent to a description with a unique ambient (top-dimensional) vector space. Indeed, rather than using flags from different ambient spaces with varying dimensions, we can perform additional RII and RIII moves to push strands as the blue one in Figure <ref> all the way to the left and push strands as the blue one in Figure <ref> all the way to the right (see also Lemma <ref>). This will extend all flags from all Type 1 columns to flags in $\mathbb{C}^h$ where $h$ is the total number of horizontal lines in the GP-graph $\bG$. Moreover, these flags will satisfy the relative position conditions imposed by the external weave lines of the initial weave $\ww=\ww(\bG)$, or equivalently, the cyclic positive braid word $\beta=\beta(\bG)$ for which $\Lambda$ is its $(-1)$ closure. In this context, <cit.>, or Lemma <ref>, reads:

Let $\beta=(i_1,i_2,\dots,i_l)\in\mbox{Br}_h^+$ be a positive braid word on $h$  strands and $\Lambda$ be the Legendrian link associated to the front given by the $(-1)$-closure of $\beta$. Then

\[
\M_1(\Lambda)\cong\left.\left\{(\SF_0,\SF_1,\SF_2,\dots, \SF_l)\ \middle| \ \begin{array}{l}\text{$\SF_i$ is a flag in $\mathbb{C}^h$ for all $i$} \\
\SF_0\overset{s_{i_1}}{\sim}\SF_1\overset{s_{i_2}}{\sim}\cdots \overset{s_{i_l}}{\sim}\SF_l=\SF_0\end{array}\right\}\right/\PGL_h.
\]