Plabic links, quivers, and skein relations

We study relations between cluster algebra invariants and link invariants. First, we show that several constructions of positroid links (permutation links, Richardson links, grid diagram links, plabic graph links) give rise to isotopic links. For a subclass of permutations arising from concave curves, we also provide isotopies with the corresponding Coxeter links. Second, we associate a point count polynomial to an arbitrary locally acyclic quiver. We conjecture an equality between the top $a$-degree coefficient of the HOMFLY polynomial of a plabic graph link and the point count polynomial of its planar dual quiver. We prove this conjecture for leaf recurrent plabic graphs, which includes reduced plabic graphs and plabic fences as special cases.

The goal of this work is to compare knot invariants and cluster algebra invariants. The starting point is our earlier work [GL20] on open positroid varieties • Π π [KLS13], which are certain distinguished subvarieties of the Grassmannian equipped with a cluster structure [GL19] (see also [Sco06,MS16,Lec16,SSBW19]). In [GL20], we associated a positroid link L π to each open positroid variety • Π π , and constructed an isomorphism between the cohomology of • Π π and the top a-degree part of the Khovanov-Rozansky triply-graded link homology [KR08a,KR08b,Kho07] of L π . In subsequent work [GL21], we further developed this connection by defining positroid Catalan numbers as certain Euler characteristics of open positroid varieties. This invariant was studied for a subclass of permutations arising from concave curves, using a Dyck path recursion that did not require explicit mention of the cluster structure on The initial motivation of this work is to explain our positroid recursion in the broader setting of plabic (i.e., planar bicolored) graphs, making an explicit connection to recursions for quivers and for links. Plabic graphs were introduced by Postnikov [Pos06] who also speculated on the relation with cluster algebras. To an arbitrary plabic graph G one can associate a plabic graph link L plab G introduced in [FPST22,STWZ19]. On the other hand, the set of equivalence classes of reduced plabic graphs is in bijection with open positroid varieties, to which we associated positroid links in [GL20].
In the first part of this work, we establish (Theorem 2.5) that the various links associated to • Π π are isotopic, including the positroid link L π and the corresponding plabic graph link L plab G . In the second part of this work, we study the relation between the point count function of a quiver Q and the HOMFLY polynomial P (L; a, z) of a link. We introduce a class of simple plabic graphs. Our main conjecture (Conjecture 2.8) states that for a simple plabic graph G, the point count of the planar dual quiver Q G is equal to the top a-degree part of the HOMFLY polynomial of L plab G . We show (Theorem 2.9) that this conjecture holds for the class of leaf recurrent plabic graphs, that includes reduced plabic graphs and plabic fences. We interpret and generalize the locally acyclic recursion for positroids [MS16] in terms of the HOMFLY skein relation.
Acknowledgments. The second author thanks David Speyer for the previous collaborations [LS22,LS21], which have influenced this work. The authors thank Eugene Gorsky for explanations related to [CGGS21].
In [GL20], we described a way to associate a positroid link L π to such π. The number of components of L π is given by the number c(π) of cycles of π. Our running example will be (2.1) π = 1 2 3 4 5 6 7 8 4 6 8 5 1 3 2 7 , written in two-line notation, with different colors representing different cycles of π. We have N = 8 and c(π) = 2. The associated positroid link L π is shown in Figure 1(a); it appears under the name L4a1 in the Thistlethwaite Link Table [KAT1]. such that π(i) < π(j), then the segment [j, π(j)] is drawn above [i, π(i)]; see Figure 1(b). The union of these segments gives rise to a link diagram. We refer to the resulting link as the permutation link of π, denoted L perm π . Orienting each line segment from i to π(i) induces an orientation on L perm π . 2.1.2. Toric permutation link. Let T := R 2 /Z 2 be the torus. Consider its fundamental domain [0, 1] 2 which we view as a square subdivided into N 2 boxes of size 1 N × 1 N . We denote by b i,j , 1 ≤ i, j ≤ N , the box whose upper right corner has coordinates 1 N (i, j). For each i ∈ [N ], place a dot d i in box b i,N +1−i . Draw an arrow (in T) in the northeast direction from d i to d π(i) for each i ∈ [N ]. When two such arrows cross, the arrow with the higher slope is drawn above the arrow with the lower slope. We obtain a planar diagram of an oriented link L tor π drawn on the surface of T; see Figure 1(c). Embedding T inside R 3 in a standard way (Figure 1(d)), we may view L tor π as a link in R 3 . Remark 2.1. Viewing L tor π as drawn on the surface of T (resp., in a solid torus S 1 × [0, 1]) contains more information-it gives rise to a certain elliptic Hall algebra element [SV13,BS12] (resp., to a symmetric function [Tur88]) with deep connections to Khovanov-Rozansky homology. We aim to pursue this direction in an upcoming paper.
Remark 2.2. We may also view L tor π as coming from a grid diagram (see, e.g., [OSS15]). The grid diagram of π is obtained by placing an X in box b i,N +1−i and an O in box b i,N +1−π(i) for each i ∈ [N ]. We then connect the X's and the O's by horizontal and vertical segments, always drawing the vertical segments above the horizontal ones. More precisely, to a given grid diagram, we can associate two links. First, a toric grid linkL grid π , drawn on the surface of T, is obtained by drawing a horizontal arrow from O to X pointing right in each row, and a vertical arrow from X to O pointing up in each column; see Figure 2(b). (In Figure 2, for each i ∈ [N ], we place an i instead of an X in box b i,N +1−i .) Second, a planar grid link L grid π , whose planar diagram is drawn inside [0, 1] 2 , is obtained by drawing a horizontal arrow from O to X in each row, and a vertical arrow from X to O in each column, but in this case β(w) = σ 3 σ 2 σ 1 σ 4 σ 3 σ 2 σ 5 σ 4 σ 3 σ 7 σ 6 σ 5 σ 4 β(v) −1 = σ −1 2 σ −1 is also isotopic to L tor π .
Here, the colors represent the cycles of π = wv −1 . Let β(v), β(w) be the positive braid lifts of v, w to the braid group of S N . Consider the Richardson braid β v,w := β(w) · β(v) −1 shown in Figure 3(left). Taking the braid closure of β v,w , we obtain the Richardson link L Rich v,w = L Rich π shown in Figure 3(right). We orient the strands of β v,w from right to left; cf. Figure 14(left).
Remark 2.3. It is well known that k-Grassmannian permutations are in bijection with Young diagrams that fit inside a k × (N − k) rectangle. Thus, in Figure 3(left), we arrange the crossings in β(w) into a (rotated) Young diagram. The condition that v ≤ w in the Bruhat order implies that a reduced word for v is a subword of a reduced word for w, and thus we represent β(v) in Figure 3(left) by replacing some of the crossings in β(w) with "elbows." The above recipe was used in [GL20] more generally for pairs v ≤ w where w is not necessarily k-Grassmannian.

Plabic link.
A plabic graph is a non-empty planar graph G embedded in a disk with vertices colored black and white; see [Pos06]. A strand in G is a path that makes a sharp right turn at each black vertex and a sharp left turn at each white vertex. We assume that the graph G has N boundary vertices, all of which have degree 1 and are labeled 1, 2, . . . , N in clockwise order. An example of a plabic graph G is shown in Figure 4(a), and the strands in G are shown in Figure 4(b).
The following description of the plabic graph link L plab G , or plabic link for short, can be deduced from [A'C99, STWZ19,FPST22] by breaking the symmetry following [Hir02]. (See Section 3.4 for a more invariant description.) The planar diagram of L plab G will consist of the union of the strands of G. To specify the overcrossings, let p be an intersection point of two strands S 1 , S 2 in G. The tangent vector to S 1 (resp. S 2 ) at p can be considered a vector in the complex plane, and we let arg(S 1 , p) ∈ [0, 2π) (resp. arg(S 2 , p) ∈ [0, 2π)) denote the argument of this vector. We assume that 0 < arg(S 1 , p) = arg(S 2 , p) < 2π. Then S 1 is drawn above S 2 if arg(S 2 , p) > arg(S 1 , p), otherwise S 1 is drawn below S 2 .
A technical extra step is required to finalize the construction of L plab G ; see Figure 5. Consider a point p on a strand S such that arg(S, p) = 0, i.e., such that S is directed to the right at p. We add an extra segment S to S traveling from p to the boundary of the disk followed by a segment S traveling from the boundary back to p. In the neighborhood of p, S looks like a plot of a function. If this function is "concave up" at p (i.e., has positive second derivative), then S (resp., S ) is drawn below (resp., above) all other strands. If S is "concave down" at p then S (resp., S ) is drawn above (resp., below) all other strands.
Finally, each boundary vertex of G is an endpoint of exactly two strands (one outgoing and one incoming). We join these endpoints together and obtain a planar diagram of a link denoted L plab G . See Figure 4(c), where the points p such that arg(S, p) = 0 are marked by the symbol .  Figure 6. Converting a concave curve C into the link L tor π C for some π C ∈ S N ; see Example 2.4.
The strands in G that start and end at the boundary vertices give rise to the strand permutation π G of G. In this subsection, we will assume that G is reduced, i.e., that G has the minimal possible number of faces among all graphs with a given strand permutation. Importantly, in Section 2.2, we will drop this assumption. Postnikov [Pos06] showed that for any π ∈ S N , there exists a reduced plabic graph G with strand permutation π G = π. For example, for π given by (2.1), one can take G to be the graph in Figure 4(a). Up to isotopy, the resulting link L plab G depends only on π and is denoted L plab π .
2.1.5. Coxeter link. In [GL21, Section 6], we studied concave permutations, defined as follows. Choose a generic concave curve C inside a k × (N − k) rectangle connecting (0, 0) to (k, N − k) as in Figure 6(a). We will shift it by the vector (−ε, ε) for some small ε > 0 as in Figure 6(b). Under the natural projection R 2 → T = R 2 /Z 2 , the curve C + (−ε, ε) projects to a curve C in the unit square [0, 1] 2 . For each self-intersection of C, we draw the segment of the higher slope above the segment of the lower slope; see Figure 6(c). The resulting link diagram is isotopic to L tor π for a permutation π = π C ∈ S N which can be explicitly read off from C as follows. Label the intersection points of C with the y = 1 − x diagonal of [0, 1] 2 by d 1 , d 2 , . . . , d N , proceeding in the northwest direction. Thus, d 1 = (1 − ε, ε) is the projection of the starting and ending points of C + (−ε, ε). In Figure 6(c), we represent each point d i by i. Since C was generic, we assume that the points d 1 , d 2 , . . . , d N are pairwise distinct. The permutation π C is defined so that each segment of C connects (in the northeast direction) d i to d π(i) for some i ∈ [N ]. See Figure 6(d). We refer to permutations that can be obtained in this way as concave permutations. They form a subclass of the class of repetition-free permutations introduced in [GL21].
Example 2.4. Let C be the curve shown in Figures 6 and 7. We have k = 5 and N = 9.
The following is our first main result.
Theorem 2.5. Let π ∈ S N be a permutation without fixed points. Then the links L perm π , L tor π , L Rich π , and L plab π are all isotopic. If π = π C for some concave curve C then each of these links is also isotopic to L Cox C . We refer to any one of the above links as the positroid link of π.
Remark 2.6. Recently, isotopies between the Richardson link L Rich π and some other closely related links (namely, closures of juggling braids, cyclic rank matrix braids, and Le-diagram braids) were independently constructed in [CGGS21]. Isotopies between cyclic rank matrix links and plabic graph links have also been observed in [STWZ19].
2.2. Quiver point count and the HOMFLY polynomial. In this subsection, we consider not necessarily reduced plabic graphs G. Throughout the paper, we assume that the interior faces of G are simply connected, i.e., that no interior face of G contains another connected component of G inside of it. For simplicity, we also assume that each plabic graph G is trivalent, i.e., that each interior vertex of G has degree 3. (Recall that the boundary vertices of G are always required to have degree 1.) Let G be a (trivalent) plabic graph. The planar dual of G is naturally a directed graph denoted Q G . Explicitly, place a vertex of Q G inside each interior face of G (i.e., a face not adjacent to the boundary of the disk). For every edge e of G whose endpoints are of different color and such that the two faces F, F adjacent to e are both interior, Q G contains an arrow between F and F . (These two faces F, F may or may not be equal.) The direction of the arrow is chosen so that the white endpoint of e is on the left as one moves along this arrow. See Figure 8.
The following definition is crucial for our analysis. Definition 2.7. A plabic graph G is called simple if the directed graph Q G is a quiver, i.e., contains no directed cycles of length 1 and 2.
For example, the plabic graph G in Figure 8(b) is not simple since Q G contains a pair of opposite arrows. The graph G in Figure 8(d) is not simple since Q G contains a loop arrow. The graphs in Figure 8(a,c) are simple.
To every locally acyclic 3 quiver Q (see Section 5.1), we associate a rational function R(Q; q) called the point count rational function. It can be computed via an explicit recurrence relation: see (5.2) and (5.11). The function R(Q; q) has the following interpretation in terms of cluster algebras [FZ02]; see Section 5.1 for the relevant definitions. Let Q be a quiver with n vertices. Consider an ice quiver Q with m frozen vertices and mutable part Q, and assume that the rows of the (n + m) × n exchange matrix B( Q) of Q span Z n over Z. Then the cluster algebra A( Q) gives rise to a cluster variety X ( Q), and its point count over a finite field F q with q elements (for q a prime power) is given by #X ( Q)(F q ) = (q − 1) m · R(Q; q); see Proposition 5.7.
A typical example of this phenomenon occurs when G is a reduced plabic graph: then, by [GL19], Here, denotes the unknot and L + , L − , L 0 are any three links whose planar diagrams locally differ as follows. Figure 9. Small plabic graph quivers Q G , links L plab G , point counts R(Q G ; q), and HOMFLY polynomials P (L plab ; a, z) illustrating Conjecture 2.8.
For a (not necessarily reduced) plabic graph G we let L plab G be the corresponding plabic graph link, defined as in Section 2.1. Following [GL20], we let P top (L; q) be obtained from the top a-degree term of P (L; a, z) by substituting a := q − 1 2 and z := q 1 2 −q − 1 2 . The following conjecture generalizes [GL20, Theorem 1.11] (see also [STZ17,STWZ19]).
Conjecture 2.8. Let G be a simple plabic graph with c(G) connected components. Then we have . For examples, see Figure 9, Example 5.11, and Section 9.
When the plabic graph G is reduced, Conjecture 2.8 becomes [GL20, Theorem 1.11]. We present a different proof in Section 7.2 by giving a plabic graph interpretation of the HOMFLY skein relation (2.3). More generally, in Section 7.1, we introduce a class of leaf recurrent plabic graphs and prove Conjecture 2.8 for them.
The fact that reduced plabic graphs are leaf recurrent was shown in [MS16,Remark 4.7]. Another natural subclass of leaf recurrent plabic graphs are the plabic fences of [FPST22, Section 12]. A plabic fence is a plabic graph obtained by drawing k horizontal strands and inserting an arbitrary number of black-white and white-black bridges between them; see Figure 10 for an example and Section 7.3 for a precise definition.
Proposition 2.10. Reduced plabic graphs and plabic fences are leaf recurrent. In particular, Conjecture 2.8 holds for these classes of plabic graphs.
This result gives a new proof of [GL20, Theorem 1.11].
A primary difficulty in showing Conjecture 2.8 in full generality comes from the examples described in Sections 9.1 and 9.2. Section 9.1 explains that the simple plabic graph G in Figure 8(a) is not leaf recurrent. The quiver Q G is locally acyclic (in fact, it is mutation equivalent to an acyclic quiver), and thus the point count polynomial R(Q G ; q) can be easily checked to coincide with P top (L plab G ; q). Section 9.2 describes a simple plabic graph G such that Q G is not locally acyclic, in which case the task of computing R(Q G ; q) becomes much harder.
Both sides of (2.4) are rational functions in q which are specializations of rational functions in (q, t) arising from the cohomology of cluster varieties on the one hand and Khovanov-Rozansky link homology on the other hand. We discuss these more general conjectures in Section 8.3.

Positroid combinatorics
In this section, we review some background on the combinatorics of plabic graphs and the associated objects.
The set of (k, N )-bounded affine permutations is denoted Bound(k, N ). Each f ∈ Bound(k, N ) gives rise to a unique permutationf ∈ S N defined byf (i) ≡ f (i) (mod N ) for each i ∈ [N ]. Given f ∈ Bound(k, N ), an integer i ∈ Z is called a loop (resp., a coloop) of f if f (i) = i (resp., f (i) = i + N ). Thus,f has no fixed points if and only if f has no loops and no coloops. In this case, f is uniquely determined byf , and the integer k is recovered fromf as follows: We define τ k,N ∈ Bound(k, N ) by and extend it to a function τ k,N : where ≤ denotes the Bruhat order on S N . Given a permutation u ∈ S N , we can extend it to an N -periodic mapũ : N ). Taking the inverse of this map, we obtain the following result, justifying the definition of a Richardson link in Section 2.1.3.
Corollary 3.1. If π ∈ S N has no fixed points then it can be uniquely factored as a product π = wv −1 , where (v, w) ∈ Q k,N and k = k(π).
The top-dimensional open positroid variety corresponds to the bounded affine permutation f k,N ∈ Bound(k, N ) sending i → i + k for all i ∈ Z. We let π k,N ∈ S N be the permutation 3.2. Plabic graphs. Let G be a plabic graph. Recall from Section 2.1.4 that G must be nonempty, that the boundary vertices of G are labeled by 1, 2, . . . , N in clockwise order, and that the strand permutation π G : [N ] → [N ] sends i → j whenever the strand starting at i terminates at j. We always assume that our plabic graphs G are such that π G has no fixed points, although most of our constructions can be easily extended to this case. Pos06]). We say that G is reduced if it has the minimal number of faces among all plabic graphs with strand permutation π G .
Alternatively [Pos06, Theorem 13.2], a plabic graph G is reduced if and only if G has no closed strands, no self-intersecting strands, and no pairs of strands forming a bad double crossing shown in Figure 17(right). Good double crossings shown in Figure 17(left) are allowed.
We consider several types of local moves on plabic graphs, shown in Figure 11. It was shown in [Pos06] that every permutation π ∈ S N is the strand permutation of some reduced plabic graph G, and that moreover, any two reduced plabic graphs with the same strand permutation are related by a sequence of the moves (a) and (b) in Figure 11. Below we review one construction of plabic graphs which will be particularly useful to us.
3.3. Le-diagrams. Let λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ k ≥ 0) be a partition, which we identify with its Young diagram. We assume that this Young diagram fits inside a k × (N − k) rectangle, i.e., satisfies λ 1 ≤ N − k. We draw Young diagrams using English (or matrix ) notation. We use matrix coordinates for the boxes of λ, thus, row i contains boxes with coordinates (i, j) for 1 ≤ j ≤ λ i , and box (1, 1) is the top left box. A Le-diagram Γ of shape λ is a way of placing a dot in some of the boxes of λ so that every box that is below a dot in the same column and to the right of a dot in the same row must also contain a dot. An example of a Le-diagram of shape λ = (3, 3, 3, 1) is shown in Figure 15(left). Le-diagrams whose shape fits inside a k × (N − k) rectangle are in bijection with the elements of Q k,N and Bound(k, N ). Given a Le-diagram Γ of shape λ, one can read off the corresponding pair (v, w) ∈ Q k,N as follows. First, w ∈ S (k) N is the k-Grassmannian permutation corresponding to λ. Specifically, let us label the unit steps in the southeast boundary of λ by 1, 2, . . . , N in increasing order from the northeast to the southwest corner. Then the labels of the vertical steps form a k-element subset of [N ], and this set is precisely {w(N − k + 1), . . . , w(N )}. This determines w uniquely. Alternatively, we can label each box (i, j) of λ with the simple transposition s k+j−i , and then a reduced word for w is obtained by reading these simple transpositions in the northwest direction. Thus, any reduced word for w ends with s k which labels the box (1, 1). To obtain a reduced word for v, we read these simple transpositions but ignore the ones labeled by dots in Γ. Conversely, given (v, w) ∈ Q k,N , the shape λ of Γ is reconstructed from w, and the empty boxes of Γ correspond to the rightmost subexpression for v inside the (unique up to commutation) reduced word for w. A Le-diagram Γ can be converted into a plabic graph G(Γ) using the local rules shown in the first two rows of Figure 12. This plabic graph is always reduced and has strand permutation π = wv −1 , where (v, w) ∈ Q k,N are recovered from Γ using the above procedure. 3.4. Properties of plabic graph links. Let G be a reduced plabic graph with strand permutation π = π G . Recall the description of the plabic graph link L plab π = L plab G from Section 2.1.4. It may appear from this description that L plab G depends on the precise way of drawing the strands in G as smooth curves, or that L plab G changes if one rotates the graph G (since the over/under-crossings information depends on the complex arguments of the strands). However, it turns out that the link L plab G is in fact independent of these choices, as explained in [A'C99, STWZ19,FPST22]. Let us give a more invariant description of L plab G . Recall that G is embedded in a disk D := {x ∈ C : |x| ≤ 1}. Consider the solid torus D × S 1 , and consider an equivalence relation ∼ on it defined as follows: x ∈ ∂D belong to the boundary of the disk and x = x . In other words, we collapse each fiber x × S 1 , where x ∈ ∂D, to a point. The resulting quotient An oriented divide is a smooth immersion of a union of oriented circles (called branches) into D. Divides are required to satisfy certain genericity conditions, e.g., that any intersections between the branches must be transversal and belong to the interior of D, and that there have to be no triple intersections; see [FPST22, Definitions 2.1 and 8.1] for a complete list.
Given a branch γ of an oriented divide D, we lift each point of γ to D × S 1 by letting the S 1 coordinate be the complex argument of the tangent vector of γ at this point. By the genericity assumptions, we obtain an embedding of a union of oriented circles into S 3 , i.e., a link, denoted L D .
The link L D is invariant under applying the local moves shown in Figure 13 to D; see [FPST22,Proposition 8.4].
Take the strands in our reduced plabic graph G. Each of the N boundary vertices of G has one incoming strand and one outgoing strand. Identifying their endpoints, we obtain an oriented divide D(G). It was shown in [Hir02] that the associated link L D(G) is isotopic to the link L plab Remark 3.4. In what follows, it will be convenient to us to use rotational invariance of L plab G . In our figures, we will fix some angle α ∈ [0, 2π), and assume that the link L plab G is obtained by first rotating G by the angle −α, then applying the description from Section 2.1.4, and then rotating the picture back by α. In other words, if two strands S 1 , S 2 in G intersect at some point p, we consider their complex arguments of tangent vectors in a different interval: arg(S 1 , p), arg(S 2 , p) ∈ [α, α + 2π). Then we draw S 1 above S 2 if and only if arg(S 2 , p) > arg(S 1 , p). And then for all points p where some strand S satisfies arg(S, p) = α, we insert line segments going to the boundary and back. In the figures, the direction α is indicated by the "over/under circle" o v e r u n d e r α and each point p on a strand S satisfying arg(S, p) = α is marked by as in Figure 5.

Positroid link isotopies
The goal of this section is to prove Theorem 2.5. We first discuss some properties of plabic graph links, and then we construct a sequence of isotopies, for any permutation π ∈ S N without fixed points, of the links π , in that order. When π = π C for some concave curve C, we will also show that Throughout this section, we fix a permutation π ∈ S N without fixed points, and let (v, w) ∈ Q k,N be the corresponding pair satisfying π = wv −1 , Γ the corresponding Lediagram of shape λ, and G the corresponding plabic graph, obtained from Γ using the recipe in the first two rows of Figure 12 Recall from Remark 2.3 that we are drawing the Richardson braid β v,w = β(w) · β(v) −1 in a particular way, so that the crossings of β(w) form a shape obtained by rotating λ clockwise by 135 • , and the crossings of β(v) −1 are drawn in some boxes of a shape obtained from the 135 • -rotation of λ by reflecting it along the vertical axis; see Figure 3(left). Moreover, the crossings of β(v) −1 are located precisely in the boxes of λ which do not contain a dot in Γ.
Our first goal is to draw the link L Rich v,w entirely within λ. To do that, we take β(v) −1 , reflect it along the vertical axis (flipping the over/under-crossings), and place the resulting braid on top of β(w). We rotate the result counterclockwise by 135 • . Every line segment in the boundary of λ contains an endpoint of a strand in β(v) −1 and an endpoint of a strand in β(w). We join these strand endpoints, obtaining a link diagram drawn inside of λ. This link L v,w , shown in Figure 14, is clearly isotopic to L v,w . Intuitively, the transformation L Rich v,w → L v,w can be described as follows: open a book, and place the left (resp., right) half of Figure 3(right) onto the left (resp., right) page of the book. Then close the book with your right hand, so that the front cover of the book is at the bottom. The page containing (the reflection of) β(v) −1 will be on top of the page containing β(w). Finally, rotate the closed book 135 • counterclockwise. An alternative description of the link L v,w can be obtained directly from the Le-diagram Γ by replacing each box containing a dot as shown in Figure 15(middle top) and each box not containing a dot as shown in Figure 15(middle bottom). For example, the Le-diagram in Figure 15(left) gets converted into the link L v,w in Figure 15(right); this is the same link as in Figure 14(right). Since π has no fixed points, each row and each column of Γ contains at least one dot. Consider some column j of Γ and let b j be the highest box in it which contains a dot. The part of L v,w above b j contains a strand going vertically to the top boundary and then back, passing between all the horizontal strands that it encounters along the way. We may therefore contract this piece of the strand to be fully contained inside b j , as shown in Figure 16(far left). Repeating this procedure for each column j = 1, 2, . . . , λ 1 , we obtain a link L v,w isotopic to L v,w ; see Figure 16(middle).
Finally, we claim that the link L v,w is isotopic to the plabic graph link L plab G . To see this, we choose the angle α from Remark 3.4 to be 80 • . Then combining the rules from Figure 12 for converting the Le-diagram Γ into the plabic graph G with the description of L plab G given in Section 3.4, we see that a planar diagram of L plab G is obtained from Γ via the rules shown in the last two rows of Figure 12. Replacing each square (where arg(S, p) = α) by a horizontal segment going to the left boundary above all other strands and back below all other strands, we see that L plab G = L v,w ; see Figure 16(far right). To summarize, we have constructed explicit isotopies good double crossing bad double crossing  Figure 13 give rise to link isotopies, so L D ∼ = L D . Let us move the boundary vertices 1, 2, . . . , N smoothly to the points 1 , 2 , . . . , N which are located clockwise on the left semicircle of ∂D, i.e., on the subset of ∂D with negative real part. Let D be the oriented divide obtained by drawing a straight arrow i → π(i) for each i ∈ [N ]. It follows that L D ∼ = L D . Let i, j ∈ [N ] be such that the corresponding arrows S i := (i → π(i) ) and S j := (j → π(j) ) of D cross at some point p. Consider the arguments of the tangent vectors 0 ≤ arg(S i , p) = arg(S j , p) < 2π. Then it is easy to check that arg(S i , p) > arg(S j , p) if and only if π(i) < π(j). Comparing to the description in Section 2.1.1, we get that L D ∼ = L perm π . 4.3. From permutation links to toric permutation links. Our goal is to find an isotopy L perm π ∼ = L tor π . Recall from Remark 2.2 that L tor π is isotopic to the planar grid link denoted L grid π ; see Figure 2(a-c). InL grid π and L grid π , the vertical arrows are drawn above the horizontal arrows.
The main diagonal y = 1 − x of the square [0, 1] 2 divides L grid π into the lower-triangular and the upper-triangular parts. In the lower-triangular part, all vertical arrows point down and all horizontal arrows point right, while in the upper-triangular part, all vertical arrows point up and all horizontal arrows point left. Let us take the lower-triangular part and reflect it around the main diagonal, placing it below the upper-triangular part; see Figure 2(c-d).
We obtain a planar diagram of a link which we denote L π . Clearly, L π ∼ = L grid π . The diagram of L π contains arrows pointing, up, left, down, and right. It follows that the over/undercrossings rule in L π is given by the following total order on the arrow directions: up > left > down > right. In particular, L π is a divide link for the angle α = 45 • from Remark 3.4  (where the main diagonal is treated as part of the boundary of the disk), and we get that 4.4. From toric permutation links to Coxeter links. First, we introduce a more general class of links, associated to arbitrary curves inside the k × (N − k) rectangle.
We say that C g is a monotone curve if it is the plot of a strictly monotone increasing ∈ Z for all j = 1, 2, . . . , N − k − 1. Thus, the curve C g passes through the points (0, 0) and (N − k, k), but through no other lattice points. To any monotone curve C g we associate a link L g drawn on the surface of T. The planar diagram of L g is obtained from the projection C g of C g to T via the following rule: whenever two points (x, g(x)), (x , g(x )) on C g (with x < x ) project to the same point of C g , we draw the projection of (x, g(x)) above the projection of (x , g(x )).
The link diagram of L g is drawn on the torus T, and therefore it is natural to think of L g itself as a link inside the thickened torus T × [0, 1]. Given a point p = (x, g(x)) ∈ C g , we denote by p ∈ T the projection of p to T, and we let p, 1 − x N −k be the corresponding point in T × [0, 1]. This yields an embedding of C g into T × [0, 1]. Under this embedding, the points (0, 0) and (N − k, k) map to ((0, 0), 1) and ((0, 0), 0), respectively. Adding a line segment connecting ((0, 0), 1) to ((0, 0), 0), we obtain a representation of the link L g inside T × [0, 1]. An obvious consequence of this construction is that the link L g only depends on the set of lattice points below the plot of g.
Corollary 4.1. Let C g , C h be two monotone curves, and assume that for each j = 1, 2, . . . , N − k − 1, we have g(j) = h(j) . Then the links L g and L h are isotopic.
Suppose now that π = π C for a concave curve C. Our goal is to find an isotopy L tor π ∼ = L Cox C . Comparing the above description to the one in Section 2.1.5, we see that when g is the concave down function satisfying C = C g , we have L tor π ∼ = L g . Our goal is to choose a monotone curve C h such that L g ∼ = L h by Corollary 4.1, and such that L h ∼ = L Cox C . Let λ = λ C = (λ 1 ≥ λ 2 ≥ · · · ≥ λ k = 0) be the Young diagram above C. For r = 0, 1, . . . , k − 1, let p r := λ k−r + r k , r . Thus, p 0 = (0, 0). Let p k := (N − k, k). Let h be the piecewise-linear function whose plot passes through the points p 0 , p 1 , . . . , p k , and let C h be the corresponding monotone curve; see Figure 18(b). Indeed, by Corollary 4.1, we have L g ∼ = L h . It remains to establish the isotopy L h ∼ = L Cox C .
Let β = β(a(C)) be the Coxeter braid associated to C, defined in (2.2). It is a braid on k strands. For i = 1, 2, . . . , k, let S i (β) be the strand of β whose left endpoint is labeled by i. Thus, the right endpoint of S i (β) is labeled by i + 1 for i < k and by 1 for i = k. On the other hand, the link diagram of L h in T intersects the line y = 0 in exactly k points with coordinates r k for r = 0, 1, . . . , k − 1. We would like to view L h as the closure of a k-strand braid β h in the solid torus; see Figure 18(c-d). The braid β h is obtained from L h via the following procedure: whenever a strand of L h passes through some point (1, y) ∈ [0, 1] 2 and continues from the point (0, y) ∈ [0, 1] 2 , we connect the points (0, y) and (1, y) by a line segment that is drawn below all strands of L h . The result is a drawing of a braid connecting k points at the bottom of [0, 1] 2 to k points at the top of [0, 1] 2 .
We claim that the braids β h and β are isotopic. We compare them strand-by-strand, identifying S i (β) with S k+1−i (β h ) for i = 1, 2, . . . , k, in that order. (This correspondence is indicated via colors in Figure 18(d-e).) For convenience, we rotate the drawing of β 90 • counterclockwise. The strand S 1 (β) moves straight up during the a 2 2 · · · a k k part, and then it moves left under all other strands. We apply an isotopy to make S k (β h ) move straight up towards the top boundary of [0, 1] 2 , and then move left under all other strands. Recall that we have a sequence a(C) = (a 2 , . . . , a k ) given by a i = λ i−1 − λ i for i = 2, . . . , k. Thus, the strand S 2 (β) wraps around S 1 (β) exactly a 2 times. On the other hand, S k−1 (β h ) is the projection to T of the line segment of C h connecting p k−1 to p k . This projection intersects the vertical boundary of T exactly a 2 times. Therefore S k−1 (β h ) wraps around S k (β h ) exactly a 2 times. Continuing in this fashion, we see that for i = 3, . . . , k, the strand S i (β) wraps around the union S 1 (β) ∪ S 2 (β) ∪ · · · ∪ S i−1 (β) exactly a i times. On the other hand, the strand S k+1−i (β h ) wraps around the union S k (β h ) ∪ S k−1 (β) ∪ · · · ∪ S k+2−i exactly a i times. This shows that the braids β and β h are isotopic, and therefore we get an isotopy L Cox C ∼ = L h between their closures.

Quivers
In this section, we introduce the point count polynomial R(Q; q) of a locally acyclic quiver, and study its basic properties. We use this polynomial to define the Q-Catalan number of Q, an integer that we conjecture to be nonnegative in Conjecture 5.10. 5.1. Quiver mutation. Recall that a quiver is a directed graph without directed cycles of length 1 and 2.
Given a quiver Q and a vertex j ∈ V (Q), one can define another quiver Q = µ j (Q) called the mutation of Q at j. This operation preserves the set of vertices: V (Q ) = V (Q), and changes the set of arrows as follows: • for every length 2 directed path i → j → k in Q, add an arrow i → k to Q ; • reverse all arrows in Q incident to j; • remove all directed 2-cycles in the resulting directed graph, one by one.
An ice quiver is a quiver Q whose vertex set V = V ( Q) is partitioned into frozen and mutable vertices: V = V fro V mut . We automatically omit all arrows in Q both of whose endpoints are frozen. For an ice quiver Q, its mutable part is the induced subquiver of Q with vertex set V mut . For a set S ⊂ V mut , let Q[S] denote the ice quiver obtained from Q by further declaring all vertices in S to be frozen. We write Q − S for the ice quiver obtained from Q by removing the vertices in S. For a simply-laced Dynkin diagram D, we say that Q or Q has type D, or is a D-quiver, if the underlying graph of Q is isomorphic to D.
Let Q be an ice quiver with V = [n + m], V mut = [n], and V fro = [n + 1, n + m] := {n + 1, n + 2, . . . , n + m}. We represent it by an (n + m) × n exchange matrix The top n × n submatrix B(Q) of B( Q) is skew-symmetric, and is called the principal part of B( Q). Let B be an (n + m) × n matrix. We denote corank( B) := n − rank( B). Following [LS22], we say that an (n+m)×n matrix B is really full rank if the rows of B span Z n over Z. We say that Q is really full rank if B( Q) is really full rank, and write corank( Q) := corank( B( Q)). We say that Q is torsion-free if the abelian group Z n+m / B( Q)Z n is torsion-free, equivalently, if Z n / B( Q) T Z n+m is torsion-free. Thus, Q is really full rank if and only if it is both full rank and torsion-free.
Definition 5.1. Given a quiver Q, we say that an ice quiver Q with mutable part Q is a minimal extension of Q if Q has m = corank(Q) frozen vertices and is really full rank.
Lemma 5.2. If Q is torsion-free then it admits a minimal extension.
Proof. If Q is torsion-free, we can find vectors d 1 , . . . , d m ∈ Z n that form a basis of Z n /B(Q) T Z n . Define Q with m frozen vertices so that the bottom m rows of the exchange matrix B are equal to d 1 , . . . , d m .
Example 5.3. The exchange matrices of the two quivers Q, Q in Figure 19(c) are given by Thus, corank(Q) = 1, corank(Q ) = 0, and neither quiver is torsion-free.
We say that an ice quiver Q is isolated if it has no arrows. For example, the quiver in Figure 8(c) is isolated.
We call an ice quiver Q acyclic if it has no directed cycles, and mutation acyclic if it is mutation equivalent to an acyclic ice quiver. For example, any quiver of Dynkin type is acyclic, as is the quiver in Figure 19(a). The quiver in Figure 8(a) and the left quiver in Figure 19(b) are not acyclic, but they are both mutation acyclic.
An edge u → v in a quiver Q is called a separating edge [Mul13] if it does not belong to a bi-infinite walk in Q. Here, a bi-infinite walk is a sequence (w j ) j∈Z of vertices in Q such that for each j ∈ Z, Q contains an arrow w j → w j+1 .
• Any isolated quiver Q is locally acyclic.
• Any quiver that is mutation equivalent to a locally acyclic quiver is locally acyclic.
• Suppose that a quiver Q has a separating edge u → v, and that all three quivers Q − {u}, Q − {v}, Q − {u, v} are locally acyclic. Then Q is locally acyclic.  Figure 19. Acyclic, locally acyclic and non-locally acyclic quivers.
We say that an ice quiver Q is locally acyclic if its mutable part Q is locally acyclic. See Figure 19.
By repeatedly mutating, we generate (possibly infinitely) many seeds and cluster variables. We denote by A( Q) = A( B( Q)) the cluster algebra associated to Q. This is the Z-subalgebra of Q(x 1 , x 2 , . . . , x n+m ) generated by all cluster variables and the inverses of frozen variables. The cluster variety X ( Q) = X ( B( Q)) is defined to be the scheme X ( Q) := Spec(A( Q)).
We say that A( Q) (resp., X ( Q)) is a cluster algebra (resp., cluster variety) of type Q, where Q is the mutable part of Q. We say that A( Q) is isolated, acyclic, (really) full rank, if Q is isolated, mutation acyclic, (really) full rank, respectively. If Q is a locally acyclic quiver, then A( Q) is locally acyclic in the sense of Muller [Mul13]. . Suppose that Q is locally acyclic and really full rank. Then X ( Q) C is a smooth complex algebraic variety and X ( Q)F q is smooth for any prime power q.  (1) The function R(Q; q) defined in (5.1) does not depend on the choice of Q.
(2) Suppose Q is locally acyclic. Then R(Q; q) is a rational function in q of degree n.
If u → v is a separating edge in Q and Q − {u}, Q − {v}, Q − {u, v} are locally acyclic, then we have the recurrence which follows from Propositions 5.5 and 5.6.
Proposition 5.8 ([LS21, Proposition 3.9]). Let Q be an acyclic quiver with n vertices. Then where a k is the number of independent sets of size k in the underlying undirected graph of Q.
When Q itself is really full rank, R(Q; q) is a polynomial in q. However, R(Q; q) is in general a genuine rational function whose denominator is a power of (q − 1). E.g., for Q a single isolated vertex (we denote this quiver by A 1 ; it has corank 1), we have For an ice quiver Q with mutable part Q and m frozen vertices, we set R( Q; q) := (q − 1) m R(Q; q).
Thus, when Q is really full rank and locally acyclic, R( Q; q) is a polynomial in q of degree n + m satisfying #X ( Q)(F q ) = R( Q; q) for all prime powers q.
Definition 5.9. Let Q be a torsion-free quiver, and Q be a minimal extension as in Lemma 5.2. If R( Q; q) is a polynomial in q, define χ Q := R( Q; 1).
By Proposition 5.7, χ Q does not depend on the choice of Q.
We call χ Q the Catalan number of the quiver Q, or simply the Q-Catalan number.
Conjecture 5.10. Suppose that Q is a locally acyclic quiver. Then χ Q ≥ 0.
Example 5.11. In Table 1, which can be computed using Proposition 5.8 or Corollary 5.15, we take Q to be any orientation of a Dynkin diagram and Q to be any minimal extension of Q. Note that R(Q; q) = R( Q; q)/(q − 1) m , where m = corank(Q) is given in Table 1.
The corresponding HOMFLY polynomials are computed via the following observation, which amounts to a single application of (2.3). Let G be a connected simple plabic graph such that Q := Q G is an orientation of a tree, and let Q (resp., Q ) be obtained from Q by adjoining a path of length 1 (resp., of length 2) to some vertex of Q. Each of Q and Q is the planar dual of some simple plabic graph, and by Lemma 7.6, the HOMFLY polynomials of the associated links are related by (5.4) P (L ) = z a P (L ) + 1 a 2 P (L). For example, we can take (Q , Q , Q) to be one of (A n , A n−1 , A n−2 ), (D n , D n−1 , D n−2 ), (E 6 , D 5 , A 4 ), (E 7 , E 6 , D 5 ), or (E 8 , E 7 , E 6 ). Writing P (Q) in place of P (L) where L is the link corresponding to Q, we find: z a P (A n−1 ) + 1 a 2 P (A n−2 ) for n ≥ 2. Here A 0 denotes the empty quiver whose associated link is the unknot; the links corresponding to A 1 and A 2 are the Hopf link and the trefoil knot, respectively; see Figure 9. The above recurrence can be solved explicitly: for each n ≥ 1, we have Similarly, we find P (D n ) = z a P (D n−1 ) + 1 a 2 P (D n−2 ) for n ≥ 4.
Substituting a := q − 1 2 and z := q 1 2 − q − 1 2 into the top a-degree term in the above formulas, one recovers the point count formulas in Table 1, in agreement with Theorem 2.9.
5.4. Leaf recurrence. Let i ∈ V (Q) be a leaf vertex of a quiver Q, connected to some other vertex j ∈ V (Q). We call i a recurrent leaf if both quivers Q − {i} and Q − {i, j} are locally acyclic. This automatically implies that Q itself is locally acyclic.
Proposition 5.13. Fix a field F, and consider cluster varieties over F. Let Q be a really full rank ice quiver with m frozen vertices and mutable part Q. Let i ∈ V (Q) be a recurrent leaf in Q connected to j ∈ V (Q). Then the subvariety of X ( Q) given by x i = 0 is isomorphic to F × X ( Q ), where Q is a really full rank quiver with m frozen vertices and mutable part Q := Q − {i, j}.
Proof. Our goal is to construct a quiver Q satisfying the above assumptions and an isomorphism By Proposition 5.5, the subvariety of X ( Q) given by x i = 0 is contained inside the subvariety of X ( Q) given by x j = 0. The quiver Q − {j} is the disjoint union of Q and a single isolated vertex. By assumption, Q is locally acyclic and thus so is Q − {j}. By Proposition 5.6, we have where M, M are monomials in the frozen variables of Q[j] − {i}. We therefore see that the right-hand side of (5.6) is given by We see that the variable x i is free, and thus the right-hand side of (5.8) is isomorphic the direct product of F and (5.9) x ∈ X x j M M = −1 .
It remains to show that the locus (5.9) is isomorphic to X ( Q ) for some quiver Q satisfying the conditions in Proposition 5.13. By assumption, B := B( Q) is really full rank. Thus, there exist integers (α k ) k∈ [n+m] such that k α kbk = e i , whereb k denotes the k-th row of B and e i = (0, . . . , 0, 1, 0, . . . , 0) is the i-th standard basis vector in Z n . Since i is a leaf, the rowb i has a single nonzero entry in the j-th column. Let B (1) be obtained from B by removing the i-th row and the j-th column. 5 Then we have whereē i is the i-th standard basis vector in Z n−1 . Moreover, the matrix B (1) is still really full rank.
Let V fro := [n + 1, n + m] be the set of frozen vertices of Q. Then the set of frozen vertices of Q[j] − {i} is V fro {j}. We have gcd(α k | k ∈ V fro {j}) = 1 since all the rows with a nonzero entry in column i of B belong to V fro {j}. By [LS22, Section 5], replacing a frozen row by its negative, or adding a frozen row to another frozen row produces an isomorphic cluster variety. After applying a series of such transformations, we obtain a really full rank (n + m − 1) × (n − 1) exchange matrix B (2) and a family of integers (α such that moreover for some ∈ V fro {j} we have α (2) = 1.  k ) k =i describe a one-parameter subgroup F * of the cluster automorphism torus T = Aut(A( B (2) )) (see [LS22, Section 5]), with z ∈ F * acting by x k → z α (2) k x k . Equation (5.10) implies that z acts on the monomial x j M M by z ±1 . Thus, every F * -orbit contains a unique point in the locus (5.9). In other words, the locus (5.9) is isomorphic to the quotient X ( B (2) )/F * . Since we assumed that α (2) = 1, the quotient X ( B (2) )/F * can be obtained by setting x = 1. Let B be the (n + m − 2) × (n − 2) obtained from B (3) by removing the -th row. Let Q be the associated quiver, with m frozen vertices corresponding to the rows (V fro {j}) \ { } and mutable part Q − {i, j}. Clearly, Q is still really full rank. We find that indeed the locus (5.9) is isomorphic to X ( Q ).
Remark 5.14. Proposition 5.13 generalizes to the case of i being either a source or a sink in Q.
The following result will be later compared to the HOMFLY skein relation (2.3). Proof. Let Q be really full rank with mutable quiver Q, and X := X ( Q). Let U := {x i = 0} and Z := {x i = 0} be two subvarieties of X . Then U is a really full rank cluster variety of type Q and Z is described in Proposition 5.13. The result follows since #X (F q ) = #U(F q ) + #Z(F q ).

Quivers, links, and plabic graphs
In this section, we study some basic relations between the properties of plabic graphs and the associated quivers and links.
6.1. Plabic graph quivers. Recall the background on plabic graphs from Sections 2.1.4 and 3.2. We continue to assume that each plabic graph G is trivalent, and that each interior face of G is simply connected.
We say that two plabic graphs are move equivalent if they are connected by a sequence of moves in Figure 11. For the square move in Figure 11 Proposition 6.1. Let G be a connected plabic graph. Then G can be transformed using only tail removal (Figure 11(c)) into: • the graph in Figure 20(a) if G has no interior faces; • the graph in Figure 20(b) or Figure 20(c) if G has one interior face; • a plabic graph with no boundary vertices, if G has two or more interior faces. We call this graph the tail reduction of G.
Proof. Let G be a connected plabic graph. By repeatedly applying tail removal to G (without changing connectedness or the number of interior faces), we may assume no more tail removals are possible. If G has no boundary vertices then G must have two or more interior faces. If G has a boundary vertex b connected to an interior vertex v, then v must be incident to a loop edge, for otherwise we may apply a tail removal to b. In this case, G has one interior face. Finally, if G has no interior vertices, then G consists of a single edge connecting two boundary vertices, and has no interior faces. Proposition 6.2.
• Suppose that two simple plabic graphs G and G are move equivalent. Then Q G and Q G are mutation equivalent. • Any plabic graph G move equivalent to a simple plabic graph G is also simple.
Proof. Direct check.
Recall from Section 2.2 that each plabic graph gives rise to a link L plab G and that our main conjecture (Conjecture 2.8) yields a relation between the polynomials R(Q G ; q) and P (L plab G ; a, z). For a rational function R(q) = P (q)/Q(q), the leading coefficient of R(q) is defined as the ratio of the coefficient of q deg P in P (q) and the coefficient of q deg Q in Q(q). While Conjecture 2.8 applies only to simple plabic graphs (cf. Section 9.3), the following more basic statement appears to hold for arbitrary plabic graphs. In particular, we allow plabic graphs with interior leaves, though we still insist that interior faces are simply connected. The link L plab G is defined as before, and the behavior of the strands at an interior leaf is shown in Figure 21.
Conjecture 6.3. Let G be a plabic graph with c(G) connected components and n interior faces. Then the top a-degree of P (L plab G ; a, z) equals (6.1) deg top a (P (L plab G )) = c(G) − n − 1. The degree of P top (L; q) (as a rational function in q) is given by (6.2) deg(P top (L; q)) = n, and the leading coefficient of P top (L; q) is equal to 1.
Cluster varieties X ( Q) are irreducible, with dimension equal to the number of vertices in Q. Thus, when R(Q; q) is a rational function, it has leading coefficient 1 and degree equal to the number of vertices of Q. So Conjecture 2.8 implies (6.2) for simple plabic graphs.

Empty quivers.
Proposition 6.4. Let G be a connected plabic graph. The following are equivalent: (1) Q G is empty.
(2) G is a tree.
(3) The tail reduction of G is a graph with no interior vertices.
Proof. The mutable quiver Q G is empty if and only if G has no interior faces, which is equivalent to G being a tree, since G is assumed to be connected. By Proposition 6.1, this is also equivalent to (3).  Figure 22. A loop edge in G whose sole vertex is adjacent to a boundary face results in having the Hopf link as a connected summand of L plab G ; see the proof of Proposition 6.7. 6.3. Disconnected graphs. For two plabic graphs G , G , we let G G denote the disconnected plabic graph obtained by taking the disjoint union of G and G . We denote by the same symbol the disjoint union of two quivers and of two links.
Proof. Follows directly from the definitions.
6.4. Isolated plabic graph quivers. Recall that the connected sum of two oriented knots K, K is defined as follows. Find planar projections of the two knots that are disjoint. Then find a (topological) rectangle R in the plane, with oriented boundary consisting of sides S 1 , S 2 , S 3 , S 4 in order, such that S 1 (resp. S 3 ) is an oriented arc along K (resp. K ), and S 2 , S 4 are disjoint from K and K . The oriented connected sum knot K#K is obtained by deleting S 1 , S 3 and adding S 2 , S 4 . To construct the connected sum of two links L, L , we choose components K ⊂ L and K ⊂ L and perform the above surgery. We call any link produced in this way the connected sum of L and L , denoted L#L . The following result is well known; see e.g. [Lic97, Proposition 16.2].
Proposition 6.6. Let L and L be two oriented links. Then P (L L ) = a−a −1 z P (L)P (L ) and P (L#L ) = P (L)P (L ).
In particular, the r-component unlink has HOMFLY polynomial a−a −1 z r−1 . Let L Hopf := denote the positively oriented Hopf link. Then we have Recall that a quiver Q is isolated if it has no arrows.
Proposition 6.7. Let G be a simple plabic graph with n interior faces and c(G) connected components. Suppose that Q G is isolated. Then L plab G is a disjoint union of c(G) links, each of which is a connected sum of some number (possibly zero) of positively oriented Hopf links.
Proof. By Proposition 6.5, we may assume that G is connected. Replace G by its tail reduction (Proposition 6.1). Then the result holds when n = 0: the link L plab G is an unknot. The result also holds when n = 1: the link L plab G is the Hopf link. So suppose that n > 1. In particular, we are assuming that G has no boundary vertices. For the remainder of the proof it is convenient to replace G by its bipartite reduction, obtained by contracting all interior edges whose endpoints have the same color. In particular, interior vertices are allowed to have degree higher than three. Suppose first that G contains an interior face F which is bounded by a loop e whose sole endpoint is v ∈ V (G). We show that v must be adjacent to a boundary face of G. Without loss of generality, we may assume that the edge e is not fully contained inside any other loop edge incident to v. Let F be the face on the other side of e to F . If F is a boundary face then we are done. Otherwise, we assume that F is an interior face. Let e be any non-loop edge incident to v that is on the boundary of F . By our bipartite assumption, e connects v to a vertex of opposite color. The assumptions that G is simple and Q G is isolated then imply that e separates F from a boundary face. It follows that v is adjacent to a boundary face of G. Letting L := L plab G−e , we see from Figure 22 that G is a connected sum of L and L Hopf . Since G − e has fewer interior faces, the result follows by induction.
Suppose now that G, still assumed to be bipartite, has no interior faces that are bounded by loop edges. Let F be an interior face of G. Since G is simple, the perimeter of F does not contain both sides of any edge. Moreover, since Q G is isolated, every edge adjacent to F is adjacent to a boundary face on the other side. Thus, G consists of a number of interior faces connected in a tree-like manner as in Figure 23. The interior faces at the leaves of this tree must be bounded by loop edges, contradicting our assumption. 6.5. Torsion. By definition, for G a (possibly not simple) plabic graph, the quiver Q − G is obtained from the directed graph Q G (Section 2.2) by removing all 1-cycles, and canceling out directed 2-cycles one by one. This agrees with the procedure in [FPST22], and the mutation class of Q − G is again invariant under the moves in Figure 11. Note that G is simple if and only if Q G = Q − G . Proposition 6.8. Let G be a plabic graph. Then Q − G is torsion-free. Proof. We prove the result by induction on the number of vertices of Q − G . In the following, we index rows of the B(Q − G ) by interior faces F of G. We may suppose that G is connected. If G has fewer than two interior faces, then checking the graphs in Figure 20, we see that the result holds. So suppose that G has at least two interior faces. Applying tail removal, we may suppose that G is not reduced. Then by [Pos06], one of the following holds: (a) G has an interior leaf, or (b) the bipartite reduction (see the proof of Proposition 6.7) has a loop, or (c) G is move equivalent to a plabic graph with a two parallel edges between vertices of opposite color.
In case (a), removing the leaf vertex does not change Q − G . We may thus assume that all interior leaves have been removed. In case (b), deleting the loop edge removes a zero row and a zero column from B(Q − G ), so the result holds by induction. We may thus assume that G is leafless and loopless and we are in situation (c), that is, G has a double edge: a pair e , e of edges between vertices u, v of opposite colors. Let F be the interior face between e and e , and let F , F be the faces on the other side of e and e to F ; see Figure 24. Let G be obtained from G by removing both edges e , e . Let B := B(Q − G ), C := B(Q − G ), and n := |V (Q − G )|. If F , F are both boundary faces, then Q − G is obtained from Q − G by removing an isolated vertex, so the result holds by induction.
Suppose that one of the faces F , F is boundary and one, say F , is interior. Then Q − G is obtained from Q − G by removing the vertices F and F . Let Z V ∼ = Z n denote the free Z-module with basis vectors {e H } H∈V indexed by the vertex set V = V (Q − G ) = {interior faces H of G} and Z n−2 ∼ = Z V ⊂ Z V the submodule spanned by basis elements other than e F , e F . The rows of B belong to Z V and the rows of C belong to Z V ⊂ Z V . By induction, C is torsionfree, so Z V /C T Z V has a basis given by some vectors d 1 , d 2 , . . . , d r ∈ Z V . Furthermore, we where H is any interior face other than F, F , and b H and c H denote the rows of B and C, respectively. It is clear that Z V /B T Z V has a basis given by {d 1 , . . . , d r }. So B is torsion-free. Now suppose that F , F are both interior faces. If F = F , then one of u, v, say v, must be a cut vertex of G. Thus, G contains a non-trivial induced subgraph H containing v and connected to G − H only via e , e . The subgraph H is contained "inside" F = F ; we may consider H to be a plabic graph, necessarily not reduced, with a single boundary vertex incident to v. Replacing G by H and repeating the argument, we are reduced the situation where F = F .
With F = F both interior, Q − G is obtained from Q − G by removing the vertex F and identifying the vertices F and F to give a new vertexF of Q − G . The rows of B belong to Z V ∼ = Z n and the rows of C belong to Z V ∼ = Z n−2 , where V = V (Q − G ). Define the inclusion ι : Z V → Z V by sending eF to e F , and mapping the other basis vectors in the obvious way. By induction, C is torsion-free, so Z V /C T Z V has a basis given by some vectors d 1 , d 2 , . . . , d r ∈ Z V . We have that where H is an interior face other than F, F , Combining Proposition 6.8 with Lemma 5.2, we obtain the following result.
Corollary 6.9. Let G be a simple plabic graph. Then Q G admits a minimal extension.

Skein relations
In this section, we introduce leaf recurrent plabic graphs. This class of plabic graphs includes the well-studied reduced plabic graphs and plabic fences. We show that the leaf recurrence for plabic graphs corresponds to the skein relation for the associated links, thus establishing Conjecture 2.8 for leaf recurrent plabic graphs.  Figure 25. Leaf recurrence for plabic graphs corresponds to the HOMFLY skein relation; see Lemma 7.6. 7.1. Leaf recurrent plabic graphs. In this section, we assume that all plabic graphs satisfy the assumptions of Section 6.1.
Definition 7.1. Let G be a plabic graph. An interior face F of G is a boundary leaf face if, in the tail reduction of G, the face F has the form shown in Figure 24, i.e., F is bounded by two edges e, e with endpoints u, v of distinct color, with e separating F from a boundary face and e separating F from an interior face F . Proof. The directed graph Q G is obtained from Q G by removing the vertex corresponding to F . Similarly, Q G is obtained from Q G by removing the vertices corresponding to F and F . Both directed graphs are clearly quivers, and thus G , G are simple.
Definition 7.3. The class of leaf recurrent plabic graphs is defined as follows.
(a) Every leaf recurrent plabic graph is simple.
If G is move equivalent to a leaf recurrent plabic graph then G is leaf recurrent.
(d) Suppose G has a boundary leaf face F as in Lemma 7.2. If the plabic graphs G and G are leaf recurrent then G is leaf recurrent.
Remark 7.4. It is immediate that if G is leaf recurrent then Q G is locally acyclic.
Lemma 7.6. Suppose that G is a plabic graph with a boundary leaf face F . Let L + := L plab G , L 0 := L plab G , and L − := L plab G , where G, G , G are the three graphs in Lemma 7.2. Then the HOMFLY polynomials of (L + , L 0 , L − ) are related by (2.3).
Theorem 7.7. Suppose G is a leaf recurrent plabic graph. Then Conjectures 2.8 and 6.3 hold for G.
Proof. Suppose G has a boundary leaf face F such that the graphs G , G from Lemma 7.2 are both leaf recurrent. Denote Q := Q G , Q := Q G , Q := Q G , L + := L plab G , L 0 := L plab G , and L − := L plab G , and note that G, G , G all have the same number of connected components. By Lemma 7.6, the HOMFLY polynomials of (L + , L 0 , L − ) are related by (2.3). By Remark 7.4, the quivers Q, Q , Q are locally acyclic. By Corollary 5.15, their point counts are related by (5.11). Comparing (2.3) and (5.11), we see that (2.4) holds for Q if it holds for both Q and Q . Now suppose that Q G is isolated with n vertices. If G is the disjoint union of G 1 and G 2 then by Proposition 6.6, we have P (L plab , the result follows. We may now assume that G is connected. By Proposition 6.7, L plab G must be the connected sum of n Hopf links. By Proposition 6.6, (6.3) and (5.3) we have Thus, the result holds when Q G is isolated.
Since the validity of (2.4) is unchanged under moves on G, it follows that (2.4) holds for all leaf recurrent plabic graphs G. The equality (6.1) also follows from the same argument. We denote the set of double braid words of length m by (±I) m . To a double braid word w ∈ (±I) m we associate a plabic graph G(w) with 2N boundary vertices. First, draw N horizontal strands, with both endpoints of each strand on the boundary of the disk. Then for each j = 1, 2, . . . , m, let h := |i j | and consider the strands at height h and h + 1. If i j > 0, add a black at h, white at h + 1 bridge between these two strands, and if i j < 0, add a white at h, black at h+1 bridge between these two strands. The bridges are added one by one proceeding from left to right. See Figure 26 for an example. Applying moves (B3)-(B4), we may assume that w has no negative indices. We say that a word w = (i 1 , i 2 , . . . , i m ) ∈ I m is reduced if the associated permutation π(w) := s i 1 s i 2 · · · s im has precisely m inversions. If w is not reduced then we may apply moves (B1)-(B2) to transform w into a word of the form (w 1 , i, i, w 2 ) for some i ∈ I and some words w 1 ∈ I m 1 , w 2 ∈ I m 2 . Let − rev(w 1 ) ∈ (−I) m 1 denote the word obtained from w 1 by negating all indices and reversing their order. One can transform w 1 into − rev(w 1 ) by applying moves (B3)- (B4). Thus, applying moves (B3)-(B4), we transform The double bridge in G(i, i, w 2 , w 1 ) corresponding to (i, i, . . . ) gives rise to a boundary leaf face. Applying the induction hypothesis to the words w := (i, w 2 , w 1 ) and w := (w 2 , w 1 ), we see that the plabic graphs G(w ) and G(w ) are leaf recurrent. Thus, by Definition 7.3, G(w) is also a leaf recurrent plabic graph.
Finally, suppose that the word w = (i 1 , i 2 , . . . , i m ) ∈ I m is reduced. Then G(w) is a reduced plabic graph and therefore we are done by Theorem 7.5. Alternatively, applying moves (B3)-(B4), we may transform w as follows: We refer to this transformation as a conjugation move. Applying conjugation and braid moves repeatedly, we either arrive at a non-reduced word and proceed by induction, or we find that π(w) ∈ S N is a minimal length representative in its conjugacy class; see [GP93, Theorem 1.1]. Recall that conjugacy classes in the symmetric group S N correspond to cycle types, and π(w) has minimal length if and only if it is a product of cycles of the form (a, a + 1, . . . , b) on disjoint intervals [a, b] ⊂ [N ]. The resulting plabic fence G(w) has no interior faces and therefore is leaf recurrent.

Invariants of quivers and links
In this section, we investigate the relation between some other invariants of the quiver Q G and invariants of the link L plab G . 8.1. Corank. Recall that we set corank(Q) := corank(B(Q)) = n−rank(B(Q)) for a quiver Q with n vertices.
Proposition 8.1. Let G be a plabic graph. Then c(G) + corank(Q G ) = c(L plab G ). Proof. We prove the result using the same induction as in the proof of Proposition 6.8. The result can be checked directly for the graphs in Figure 20. We now suppose that G has one of the following: (a) an interior leaf, or (b) a loop edge, or (c) two parallel edges between vertices of opposite color.
In case (a), removing the leaf vertex preserves c(G), c(L plab G ), and corank(Q G ). In case (b), removing the loop edge reduces c(L plab G ) by 1 and corank(Q G ) by 1. In case (c), we use the same notation as in the proof of Proposition 6.8; see Figure 24. Let G be obtained from G by removing both edges e , e . This operation preserves c(L plab G ). If F , F are boundary faces, then c(G ) = c(G) + 1 and corank(Q G ) = corank(Q G ) − 1. If one of the faces F , F is boundary and one is interior, then c(G ) = c(G) and corank(Q G ) = corank(Q G ), as shown in the proof of Proposition 6.8. If F , F are both interior, we may assume they are distinct. Thus, c(G ) = c(G) and as shown in the proof of Proposition 6.8, we have corank(Q G ) = corank(Q G ). Thus, c(Q G ) + corank(Q G ) = c(L plab G ) follows from the same equation for G . See Table 2 for some examples.
Proposition 8.4. Suppose that Conjecture 2.8 holds for a simple plabic graph G. Then Proof. Let Q be a minimal extension of Q G (cf. Corollary 6.9), and set m = corank(Q G ). Then Problem 8.5.
• For which links is χ L defined?
• For which links is χ L nonnegative, or positive?
8.3. Cluster cohomology versus link homology. Let Q be a locally acyclic and torsionfree quiver. Let Q be a minimal extension of Q and set m = corank(B(Q)). Let T = T ( Q) be the cluster automorphism torus of Q defined in [LS22, Section 5]. The torus T acts on A( Q) sending every cluster variable to a nonzero multiple of itself, and the character lattice of T is naturally identified with Z n+m / B( Q)Z n . We may associate to Q the cohomology and the compactly-supported T -equivariant cohomology of the cluster variety X ( Q). Both cohomologies are equipped with mixed Hodge structures and Proposition 8.7. The cohomology H * (Q) := H * (X ( Q)) is of mixed Tate type, that is, Moreover, it does not depend on the choice of Q.
Proof. The first statement is part of [LS22, Theorem 8.3]. Let Q and Q be two really full rank ice quivers with mutable part Q and the same number m of frozen variables. In [LS22, Proposition 5.11], it is shown that we have an isomorphism of mixed Hodge structures H * (X ( Q)) = H * (X ( Q )).
The results of [LS22] are only established in the case of ordinary cohomology. In the following, we assume that the results there extend to equivariant cohomology. That is, we have and it does not depend on the choice of Q. We expect to return to equivariant cohomology of cluster varieties in future work. Let L be an oriented link. Let HHH(L) denote the Khovanov-Rozansky triply-graded homology [KR08a,KR08b,Kho07] of L, and HHH 0 (L) the Hochschild degree 0 part of HHH(L). Similarly, let HHH C denoted the variant of Khovanov-Rozansky homology defined in [GL20, Section 3], and let HHH 0 C denote the Hochschild degree 0 part. We assume these homology groups have been normalized so they become link invariants; cf. [GL20, Equations (3.11) and (3.15)].
Conjecture 8.8. Let G be a connected simple plabic graph. Then we have bigraded isomorphisms Figure 27. Two connected simple plabic graphs G, G which are move equivalent. The associated quivers Q G , Q G are not locally acyclic.
On the other hand, the plabic graph link L plab G is listed as L9n19 in [KAT1]. Its HOMFLY polynomial is given by This agrees with Conjectures 2.8 and 6.3. 9.2. Example: a simple plabic graph quiver that is not locally acyclic. Consider the plabic graph G and the associated quiver Q G in Figure 27(b). This quiver comes from a cluster algebra associated with a 4-punctured sphere; see [Mul13, Section 11.5].
Muller defines a cluster variety to be locally acyclic if it can be covered by cluster localizations that are acyclic.
Proposition 9.1 ([Mul13, Section 11.5]). The quiver Q G is not locally acyclic. The associated cluster algebra A(Q G ) is not locally acyclic.
Proof. If Q G is locally acyclic then A(Q G ) would be locally acyclic, so it is enough to establish the second statement. The proof in [Mul13] that A(Q G ) is not locally acyclic has the following gap: in [Mul13, Theorem 8.3], it is assumed that it is enough to check the cluster exchange relations to produce a point of a cluster variety. However, the ideal of relations between cluster variables is in general larger than the ideal generated by the exchange relations. G. Muller has supplied us with the following alternative argument.
The cluster algebra A(Q G ) includes into the tagged skein algebra S q=1 at q = 1 of the 4-punctured sphere. This is the algebra generated by immersed curves in the surface with taggings at each puncture, modulo certain relations [Mul16]. The two algebras A(Q G ) and S q=1 may or may not be equal. One can define a point p 0 in Spec(S q=1 ) by giving every arc the value 0, and giving every loop the value 2(−1) i , where i is the number of punctures in the interior of the loop. The point p 0 defines a point in Spec(A(Q G )), and since every arc takes value 0 on p 0 , so does every cluster variable. Since every cluster variable vanishes on p 0 , it cannot be contained in any proper cluster localization (see [Mul13,Definition 3.3]). Therefore, p 0 cannot be contained in any locally acyclic cover of Spec(A(Q G )).
We do not know the point count function R(Q; q) of this quiver, or even whether it is a polynomial. On the other hand, the HOMFLY polynomial of the link L plab G (see Figure 27(c)) Figure 28. Two plabic graphs G (left) and G (right) with the same quiver is given by Thus, according to Conjecture 2.8, the prediction for the point count R(Q; q) is 9.3. Example: conjectures fail when the plabic graph is not simple. The plabic graph G in Figure 8(d) is not simple as the directed graph Q G has a loop arrow. If we remove this loop, we obtain the quiver Q := Q − G = A 1 A 1 consisting of two isolated vertices. The quiver point count is On the other hand, the link L plab G is listed as L10n94 in [KAT1]. Its HOMFLY polynomial is Thus, Conjecture 2.8 predicts that the point count R(Q; q) should be equal to q 2 + 1, which does not match (9.1). The plabic graph G in Figure 8(b) is not simple as the directed graph Q G has a directed 2-cycle. If we cancel out the arrows in this 2-cycle, we obtain a quiver Q := Q − G that is a directed 4-cycle, which is mutation equivalent to the quiver of type D 4 . Using Proposition 5.8, we obtain that the quiver point count is R(D 4 ; q) = (q 6 −2q 5 +2q 4 −q 3 +2q 2 −2q +1)/(q −1) 2 ; see Table 1. On the other hand, the HOMFLY polynomial of L plab G is P (L plab G ; a, z) = z 4 + 4z 2 + 2 a 4 − z 2 + 3 + 2z −2 a 10 + z −2 a 8 + 1 + z −2 a 12 .
Thus, Conjecture 2.8 predicts that the point count R(Q; q) should be given by q 4 + 1, which does not agree with R(D 4 ; q).
9.4. Connected sum and disjoint union. Consider the graphs G and G in Figure 28.
However, the links L plab G and L plab G are not isotopic: L plab G is a disjoint union of two Hopf links while L plab G is a connected sum of two Hopf links. The HOMFLY polynomials are given by We see that the HOMFLY polynomials of these two links are different, however, their top a-degree terms satisfy (q − 1)P top (L plab G ; q) = P top (L plab G ; q) = R(Q; q), in agreement with Conjecture 2.8. We conjecture that when one restricts to the class of connected plabic graphs, the full HOMFLY polynomial becomes an invariant of the quiver.
Conjecture 9.2. Let G, G be two connected simple plabic graphs. Assume that the quivers Q G and Q G are mutation equivalent. Then P (L plab G ; a, z) = P (L plab G ; a, z). Remark 9.3. A stronger cohomological version of Conjecture 9.2 is that HHH C (L plab G ) ∼ = HHH C (L plab G ) and HHH(L plab G ) ∼ = HHH(L plab G ); see Section 8.3. Remark 9.4. Let G be a reduced plabic graph with strand permutation π = π G . The top a-degree coefficient P top (L plab G ; q) of P (L plab G ; a, z) can be extracted from the open positroid variety Π • π by computing the point count. However, observe that the first two rows in Figure 9 correspond to isomorphic open positroid varieties. Thus, neither the link L plab G nor the full HOMFLY polynomial P (L plab G ; a, z) are determined by Π • π . Conjecture 9.2 implies that when G is connected, P (L plab G ; a, z) is fully determined by Π • π . It would be interesting to find a geometric interpretation of the lower a-degree coefficients in this case.
Remark 9.5. According to M. Shapiro's conjecture [FPST22,Conjecture 6.17], if the quivers Q G and Q G are mutation equivalent then the graphs G and G are move-and-switch equivalent. When two plabic graphs G, G differ by a switch (see [FPST22, Figure 25]), the corresponding links L plab G , L plab G need not be isotopic. Nevertheless, by [FPST22, Proposition 6.16], Conjecture 9.2 implies P (L plab G ; a, z) = P (L plab G ; a, z). The switch operation can be defined more generally for oriented links, and we expect that this operation still preserves the HOMFLY polynomial.
Conjecture 9.6 (Switch preserves HOMFLY). Suppose that the planar diagram of a link L intersects the vertical line x = 0 at four points (0, −b), (0, −a), (0, a), (0, b) for 0 < a < b. Suppose that L is directed to the right at (0, ±b) and to the left at (0, ±a). The switch of L is a new link L obtained by taking the x > 0 part of L and reflecting it along the x axis, flipping all crossings. Then P (L; a, z) = P (L ; a, z). Figure 29. The switch operation for links; see Conjecture 9.6.
See Figure 29 for an example of a switch. 9.5. Small knots. It is natural to ask which knots and links can potentially be associated to a quiver. Assuming Conjecture 2.8, a necessary condition for a link L to come from a locally acyclic quiver would be that up to a power of (q − 1), P top (L; q) is a polynomial in q. First, let K be the trefoil knot (Figure 9), with P (K; q) = z 2 +2 a 2 − 1 a 4 . Let K * be the mirror image of K. Taking the mirror image results in replacing a with −a −1 ; cf. (2.3). Thus, P (K * ; q) = −a 4 + (z 2 + 2)a 2 and P top (K * ; q) = −q −2 , which is not a polynomial. Next, let E be the figure-eight knot. It coincides with its mirror image: E ∼ = E * . We have P (E; q) = a 2 − (z 2 + 1) + 1 a 2 and P top (E; q) = −q −1 , which is again not a polynomial. We therefore do not expect K * and E to be associated to locally acyclic quivers. (By [GL20, Theorem 1.11], it follows that neither K * nor E is a Richardson knot.) In Table 2, we give a list of all knots K with up to 9 crossings such that P top (K; q) is a polynomial in q. The knot numbering is taken from [Rol90,KAT2], and the knots are considered up to taking the mirror image. We see that for K = 8 21 , the leading coefficient of P top (K; q) is 2, which violates Conjecture 6.3; thus, we do not expect 8 21 to be associated to a quiver.  Figure 30. A reduced plabic graph G(π) and two affine plabic fences.
9.6. Affine plabic fences. While reduced plabic graphs were classified by Postnikov [Pos06], the problem of classifying simple plabic graphs appears much harder. We consider an affine version of plabic fences, which contains a large subclass of simple plabic graphs that can be parametrized; see Proposition 9.7. Recall from [Pos06] that for any (k, N )-bounded affine permutation f ∈ Bound(k, N ), we have a reduced plabic graph G(f ), satisfying the conditions of Section 2.2. For each fixed point i ∈ [N ] of π :=f ∈ S N , the boundary vertex i of G(f ) is adjacent to an interior leaf, called a lollipop. This lollipop is white if f (i) = i + N and black if f (i) = i. In particular, if π has no fixed points then G(f ) has no interior leaves. Recall from Section 3.1 that in this case, f can be recovered from π, and we denote G(π) := G(f ).
Given a double affine braid word w = (i 1 , i 2 , . . . , i m ), and f ∈ Bound(k, N ), we define the affine plabic fence G(w, f ) as follows. Begin with the plabic graph G(f ). For each j = m, m − 1, . . . , 1, let h := |i j | and consider the two boundary vertices h and h + 1, considered modulo N . If i j > 0, add a black at h, white at h + 1 bridge between these two boundary vertices, and if i j < 0, add a white at h, black at h + 1 bridge between these two boundary vertices. When π :=f has no fixed points, we denote G(w, π) := G(w, f ). See Figure 30 for an example.
Proposition 9.7. For any w ∈ (±Ĩ) m and any permutation π ∈ S N without fixed points, the plabic graph G(w, π) is simple.
Proof. It is clear from the construction that every time we add a bridge to an affine plabic fence, the faces F and F on the two sides of the bridge are distinct. Thus, the quiver of G(w, π) cannot have 1-cycles. Let F, F be two adjacent (interior or boundary) faces of G(w, π). If F and F are both adjacent to the bridge e, then e must be the only edge separating F and F since this is the case if e is the bridge added to G(f ).
In G(f ), any two faces have at most one edge in common. So if F and F do not have a bridge edge in common in G(w, π), then their common boundary must belong to a single edge e of G(f ). The edge e is divided into a number of segments by the vertices of G(w, π) that lie on e . By considering what happens when a single bridge is added, we see that two distinct such segments cannot belong to the boundary of the same two faces of G(w, π). It follows that the quiver of G(w, π) has no directed 2-cycles.
Remark 9.10. A strengthening of Conjecture 9.9 can be given when f = f k,N is the "top cell" bounded affine permutation (Section 3.1). Consider the affine braid group with generators β ±1 i for i ∈ I, lifting the Coxeter generators s i , together with the shift braids that lift the affine permutations i → i + k. For a word w ∈ (±I) m , consider an affine braid w 1 f k,N w 2 , where w 1 (resp., w 2 ) is the positive braid lift of the subword of w consisting of the letters fromĨ (resp., positive braid lift of the reverse of the subword of w consisting of letters from −Ĩ). We conjecture that for w, w ∈ (±I) m , the affine plabic fences G(w, f k,N ) and G(w , f k,N ) are restricted move equivalent if and only if the affine braids w 1 f k,N w 2 and w 1 f k,N w 2 are braid equivalent, or equivalently, the affine braids w 1 (w 2 + k) and w 1 (w 2 + k) are braid equivalent, where (w 2 +k) is the positive braid word obtained by adding k (modulo N ) to each letter in w 2 . 9.7. Trees and acyclic quivers.
Definition 9.11. A link L is called acyclic (resp. tree-like) if there is a simple plabic graph G such that L = L plab G , and Q G is mutation acyclic (resp. mutation equivalent to a tree). Problem 9.12.
• Which acyclic quivers Q can occur as Q G for a simple plabic graph G?
• Can we characterize acyclic, or tree-like links?
It is clear that any tree T can occur as Q G for some simple plabic graph G. In fact, Lam and Speyer (unpublished) have shown that one can choose such a plabic graph to be reduced.
We expect that acyclic links have particularly simple link invariants, including link homology groups. Proposition 5.8 gives a closed formula for the top a-degree part of the HOMFLY polynomial of an acyclic link. It would be interesting to obtain a closed formula for the entire HOMFLY polynomial.
In [LS21, Theorem 1.3], it is shown that the mixed Hodge numbers of an acyclic really full rank cluster variety satisfy a strong vanishing condition. It would be interesting to interpret this vanishing via the link homology groups of acyclic links.