Combinatorial cluster expansion formulas from triangulated surfaces

We give a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces in terms of perfect matchings of angles. Our formula simplifies the cluster expansion formula given by Musiker-Schiffler-Williams in terms of perfect matchings of snake graphs. A key point of our proof is to give a bijection between perfect matchings of angles in some triangulated polygon and perfect matchings of the corresponding snake graph. Moreover, they also correspond bijectively with perfect matchings of the corresponding bipartite graph and minimal cuts of the corresponding quiver with potential.


Introduction
Cluster algebras, introduced by Fomin and Zelevinsky in 2002 [FZ1], are commutative algebras with a distinguished set of generators, which are called cluster variables. Their original motivation was coming from an algebraic framework for total positivity and canonical bases in Lie Theory. In recent years, it has interacted with various subjects in mathematics, for example, quiver representations, Calabi-Yau categories, Poisson geometry, Teichmüller spaces, exact WKB analysis, etc.
We study cluster algebras defined from triangulated surfaces that are developed in [FoG1,FoG2,FST,FT,GSV]. In this case, Musiker, Schiffler and Williams gave a cluster expansion formula in terms of perfect matchings of snake graphs. Using it, they proved the positivity conjecture [MSW1] and constructed two bases [MSW2] for these cluster algebras. The first aim of this paper is to give a cluster expansion formula for these cluster algebras in terms of perfect matchings of angles (Theorem 1.5). This simplifies their formula as we will discuss later. The second aim of this paper is to give bijections between several different combinatorial objects containing perfect matchings of snake graphs (Theorem 1.4).
This paper is organized as follows. In the rest of this section, we give our results and some examples. For simplicity, we first specialize Theorem 1.5 to the coefficient-free case, that is y i = 1 for all i (Theorem 1.3). Using Theorem 1.5, we also study f -vectors of cluster variables. In Section 2, we recall basic definitions and facts on cluster algebras, triangulated surfaces and the cluster expansion formula of Musiker-Schiffler-Williams. We prove Theorem 1.5 and a part of Theorem 1.4 simultaneously in Section 3. We prove our results for the corresponding bipartite graphs in Section 4 and study minimal cuts of the corresponding quivers with potential in Sections 5. Finally, some elements in A(T ) correspond to essential loops in T (see Section 6 for details). In the case of a marked surface without punctures, it is known that these elements and cluster variables form a base of A(T ) (see Theorem 6.3). We give the formula for these elements in terms of good perfect matchings of angles in Theorem 6.5.
1.1. Our results in the coefficient-free case. Let (S, M ) be a marked surface and T a tagged triangulation of (S, M ). Let A(T ) be the cluster algebra with principal coefficients defined from T (see Subsection 2.3). Then there is a bijection between cluster variables in A(T ) and tagged arcs of (S, M ), which are obtained from ordinary arcs by tagging their endpoints plain or notched (see Theorem 2.10). We represent tagged arcs as follows: plain notched ⊲⊳ For simplicity, in this paper, we assume that if (S, M ) is a closed surface with exactly one puncture, all tagged arcs are plain arcs. For a tagged arc δ, we denote by x δ the corresponding cluster variable in A(T ). We index the tagged arcs of T by [1, N ] := {1, . . . , N }. In particular, x i (resp., y i ) is the corresponding initial cluster variable (resp., coefficient) in A(T ) for i ∈ [1, N ].
Definition 1.1. We call a tagged arc δ • a plain arc if its both ends are tagged plain, • a 1-notched arc if an end of δ is tagged plain and the other end is tagged notched, • a 2-notched arc if its both ends are tagged notched.
To give cluster expansion formulas, by changing tags, we can make the following assumption (see Proposition 2.11).
Assumption 1. The initial tagged triangulation T consists of plain arcs and 1-notched arcs, with at most one 1-notched arc incident to each puncture.
In this case, for each 1-notched arc of T , the corresponding plain arc is also in T . Then there is a unique ideal triangulation T 0 obtained from T by replacing every 1-notched arc with the corresponding loop cutting out a once-punctured monogon and by forgetting tags.
For a tagged arc δ of (S, M ), we denote by δ the plain arc corresponding to δ. Now, we only consider the case of γ := δ / ∈ T . Let p and q be the endpoints of γ. Let γ (p) be the 1-notched arc obtained from γ by tagging its end p notched. Similarly, we define the 2-notched arc γ (pq) with both ends tagged notched:
Our cluster expansion formula for x γ (resp., x γ (p) , x γ (pq) ) comes down to type A (resp., D, D) corresponding to polygons with no punctures (resp., one puncture, two punctures). We construct a triangulated polygon T δ associated with δ as follows.
Let τ 1 , . . . , τ n be the arcs of T 0 crossing γ in order of occurrence along γ (we can have τ i = τ j even if i = j). Hence γ crosses n + 1 triangles △ 0 , . . . , △ n , in this order. Suppose first that none of these triangles is self-folded. Then for i ∈ [0, n], let △ γ,i be a copy of the oriented triangle △ i , hence △ γ,i contains the sides τ i and τ i+1 (τ 1 only if i = 0, and τ n only if i = n). Then T γ is the triangulation of an (n + 3)-gon obtained by gluing these triangles along the edges τ i . Similarly, we construct T γ (p) (resp., T γ (pq) ) by adjoining to T γ copies of all triangles incident to p (resp., p and q) if none of them is self-folded. See Figure 1. If γ crosses self-folded triangles or there are self-folded triangles incident to p or q, we adapt the construction using the local transformations of Figure 2. Note that, by Assumption 2, it is not necessary to consider the case, where the end of δ in the middle of Figure 2 is tagged notched.
In this paper, we call interior arcs of each polygon T δ diagonals and non-interior arcs of T δ boundary segments. We recall a definition we introduced in [Y].
A perfect matching of angles in T δ is a selection of marked angles such that: (1) each vertex v incident to at least one diagonal is matched to one marked angle incident to v, (2) each triangle of T δ has exactly one marked angle. We denote by A(T δ ) the set of perfect matchings of angles in T δ .
It is easy to see that A(T δ ) = ∅ (e.g. see Figure 3). For a diagonal or boundary segment τ of T δ , we denote x τ = x τ ′ if τ corresponds to a non-boundary segment τ ′ of T and we denote x τ = 1 otherwise. Then, for an angle a of T δ , x a := x τ , where τ is the side opposite to a in the triangle containing a. Using Assumption 1, we define a ring homomorphism where k is the plain arc of T corresponding to j, x j otherwise, for any j ∈ [1, N ]. Our main result Theorem 1.5 gives a cluster expansion formula for cluster algebras with principal coefficients defined from triangulated surfaces. In this subsection, we specialize it to the coefficient-free case.
Theorem 1.3. Let δ be a tagged arc of (S, M ).
(1) If δ / ∈ T , we have (2) Suppose that δ ∈ T and δ / ∈ T . Let p and q be the endpoints of δ. If p (resp., q) is a puncture, we denote by ℓ p (resp., ℓ q ) the loop with endpoint q (resp., p) cutting out a monogon containing only p (resp., q). We can define triangulated polygons T ℓp and T ℓq in the same way as for plain arcs. Then, for s = p or q, we have There are two key steps to prove Theorem 1.5. The first step is the cluster expansion formula given by Musiker-Schiffler-Williams [MSW1]. A perfect matching in a graph G is a set P of edges of G such that each vertex of G is contained in exactly one edge in P . One can construct a snake graph G δ associated with T δ . Musiker-Schiffler-Williams gave a cluster expansion formula in terms of perfect matchings of G δ (see Subsection 2.4). Note that perfect matchings of G γ (p) and G γ (pq) are different from general perfect matchings of graphs, that are also called symmetric perfect matchings and compatible perfect matchings, respectively (see Definitions 2.17 and 2.20).
The second step is Theorem 1.4 below. It gives bijections between several different combinatorial objects, that we introduce now. The bipartite graph B δ associated with T δ is defined as follows: The set of black vertices consists of vertices incident to at least one diagonal of T δ and the set of white vertices consists of triangles of T δ . Edges are drawn between the white vertex corresponding to a triangle ABC and the three black vertices corresponding to A, B and C if they exist. On the other hand, we associate to δ a quiver with potential (Q δ , W δ ) in Subsection 5.1, and we define minimal cuts of (Q δ , W δ ) in Definition 5.4. Theorem 1.4. There are bijections between the following objects: (1) Perfect matchings of angles in T δ , (2) Perfect matchings of G δ , (3) Perfect matchings of B δ , (4) Minimal cuts of (Q δ , W δ ), for any tagged arc δ of (S, M ) such that δ / ∈ T .
By Theorem 1.4, we also obtain cluster expansion formulas in terms of perfect matchings of bipartite graphs and minimal cuts of quivers with potential. More precisely, the bijection between (1) and (3) in Theorem 1.4 is induced by a natural bijection ̟ between the set of angles incident to at least one diagonal of T δ and the set of edges of B δ given by the following picture: For a side e of B δ , we denote x e = x ̟ −1 (e) . For a tagged arc δ of (S, M ) with δ / ∈ T , we have 1 x 4 x 5 x 2 7 x 9 x 10 + x 4 x 5 x 7 x 2 10 + x 3 x 2 5 x 7 x 9 x 10 +x 4 x 5 x 7 x 8 x 10 + x 5 x 6 x 9 x 10 + x 3 x 2 5 x 2 10 +x 3 x 2 5 x 8 x 10 + x 5 x 6 x 2 10 + x 3 x 5 x 6 x 8 x 10 +x 5 x 6 x 8 x 10 + x 3 x 5 x 6 x 2 8 + x 2 6 x 8 x 10 +x 2 which is not affected by Φ since x 2 don't appear.
1.3. Our results in the principal coefficients case. We keep the notations of Subsection 1.1. Let ζ 1 , . . . , ζ m (resp., ξ 1 , . . . , ξ ℓ ) be the diagonals of T δ incident to p (resp., q) winding counter-clockwisely around p (resp., q) such that τ n , ζ 1 , and ζ m (resp., τ 1 , ξ 1 , and ξ ℓ ) are contained in the same triangle (see Figure 3). We define an element A − (T δ ) ∈ A(T δ ), which we call the minimal matching of T δ , satisfying the following min-condition: For each boundary vertex v of T δ that is incident to at least one diagonal of T δ , the angle a ∈ A − (T δ ) at v comes first in the counterclockwise order around v. Clearly, the minimal matching is uniquely determined (see Figure 3). We expand the ring homomorphism (1.1) into if j is plain and corresponds to the 1-notched arc k of T , y j otherwise, for any j ∈ [1, N ]. For two sets A and B, we denote by A△B the symmetric difference (A∪B)\(A∩B). An exterior angle of T δ is an angle between a boundary segment and a diagonal of T δ . Let A ∈ A(T δ ). We denote by Y ′ (A) the set of diagonals of T δ that are sides of at least one exterior angle in A − (T δ )△A. We define the set , n = 1, and A contains at least one of the four angles between ζ m or ξ ℓ and τ 1 or a boundary segment of T γ (pq) , We are ready to state the main theorem of this paper.
Theorem 1.5. Let δ be a tagged arc of (S, M ).
(1) If δ / ∈ T , we have (2) Suppose that δ ∈ T and δ / ∈ T . Let r and s be the endpoints of δ. Then, for s = p or q, we have where e s (τ ) is the number of ends of τ incident to s, and Since Theorem 1.5(2) follows from [FT,Lemma 8.2,Theorem 8.6] and [MSW1,Proposition 4.21], we only prove Theorem 1.5(1) in Section 3. Theorem 1.5 is a generalization of [Y,Theorem 1.3].
In the rest of this section, we consider the bipartite graph B δ . We define the minimal matching of B δ by E − (B δ ) := ̟ −1 (A − (T δ )) ∈ P(B δ ), where P(B δ ) the set of perfect matchings of B δ . For a diagonal τ of T δ , there are exactly two triangles △, △ ′ of T δ with edge τ . We label by τ the square of B δ whose vertices are two white vertices corresponding to △, △ ′ and two black vertices corresponding to endpoints of τ .
Proposition 1.6. For E ∈ P(B δ ), the set E − (B δ )△E consists of all boundary edges of some (possibly empty or disconnected) subgraph B E of B δ that is a union of squares.
We denote by I(E) the set of the squares of B δ contained in B E . Proposition 1.7. For E ∈ P(B δ ), I(E) = Y (̟ −1 (E)) holds.
By Theorem 1.5 and Proposition 1.7, for a tagged arc δ of (S, M ) such that δ / ∈ T , we have This formula is a generalization of the cluster expansion formula in type A given by Carroll and Price [CPr] (see also [CPi, P]). We prove Propositions 1.6 and 1.7 in Section 4.
1.4. Example in the principal coefficients case. We consider the square (S, M ) with three punctures and the tagged triangulation T of (S, M ) given in Subsection 1.2. The cluster algebra A(T ) has initial cluster variables x 1 , . . . , x 10 and initial principal coefficients y 1 , . . . , y 10 . The ring homomorphism Φ : 10 , y ±1 1 , . . . , y ±1 10 ] is given by We use Theorem 1.5 to obtain the cluster expansions of δ 1 , δ 2 and δ 3 given in Subsection 1.2 with respect to the initial cluster variables x 1 , . . . , x 10 and coefficients y 1 , . . . , y 10 in A(T ).
1.5. f -vectors and intersection numbers. We keep the notations of Subsection 1.3. We recall f -vectors of cluster variables [FuG,Definition 2.6]: For a cluster variable x of A(T ), let f x,1 , . . . , f x,N be the maximal degrees of y 1 , . . . , y N in the polynomial obtained from the Laurent expression of x by substituting 1 for each of x 1 , . . . , x N . The integer vector f x := (f x,1 , . . . , f x,N ) ∈ Z N ≥0 is called the f -vector of x. For a tagged arc δ of (S, M ) such that δ / ∈ T , by Theorem 1.5(1), the f -vector (1.2) On the other hand, for tagged arcs δ and ǫ of (S, M ), Qiu and Zhou [QZ] defined the intersection number between δ and ǫ as follows: Assume that δ and ǫ intersect transversally in a minimum number of points in S \ M . Then we define the intersection number Int(δ, ǫ) = A + B + C, where • A is the number of intersection points of δ and ǫ in S \ M ; • B is the number of pairs of an end of δ and an end of ǫ that are incident to a common puncture such that their tags are different; • C = 0 unless the ideal arcs corresponding to δ and ǫ form a self-folded triangle, in which case C = −1. Note that this definition is different from the "intersection number" (δ |ǫ) defined in [FST,Definition 8.4]. We give the main result of this subsection.
Proof. Considering in each case, it is easy to show that both f x δ ,i and Int(δ, i) are equal to f ∈ Z ≥0 given as follows: If δ ∈ T , then f = 0; Suppose that δ / ∈ T . If i is a plain arc of T , then f is the number of diagonals of T δ corresponding to i. If i is a 1-notched arc of T , then f is the number of diagonals of T δ corresponding to i minus the number of diagonals of T δ corresponding to i; Suppose that δ ∈ T and T / ∈ T . We use the notations of Theorem 1.5(2).

Preliminary
For the convenience of the reader, we recall basic definitions and facts about cluster algebras, triangulated surfaces and the cluster expansion formulas of Musiker-Schiffler-Williams (e.g. [FST,FZ1,FZ2,MSW1]).
2.1. Cluster algebras with principal coefficients. To define cluster algebras with principal coefficients, we need to prepare some notations. Let F := Q(t 1 , . . . , t 2N ) be the field of rational functions in 2N variables over Q.
Definition 2.1. [FZ2, Definition 2.3] A labeled seed (or simply, seed) is a pair (x, B) consisting of the following data: Then we refer to x as the cluster, to each x i as a cluster variable, to each y i as a coefficient and to B as the exchange matrix of (x, B).
In general, one may consider skew-symmetrizable or sign-skew-symmetric matrices as exchange matrices [FZ1]. In this paper, we only study the skew-symmetric case as we focus on cluster algebras defined from triangulated surfaces.
Then it is elementary that µ k (x, B) is also a seed. Moreover, µ k is an involution, that is, we have µ k µ k (x, B) = (x, B). Now we define cluster algebras with principal coefficients. For a skew-symmetric N × N integer matrix B, we define B = (b ij ) as the 2N ×N integer matrix whose upper part (b ij ) 1≤i,j≤N is B and lower part (b ij ) N +1≤i≤2N,1≤j≤N is the N ×N identity matrix. We fix a seed (x = (x 1 , . . . , x N , y 1 , . . . , y N ), B) that we call an initial seed. We also call each x i an initial cluster variable.
Definition 2.3. [FZ2,Definition 3.1] The cluster algebra A(B) = A(x, B) with principal coefficients for the initial seed (x, B) is the Z-subalgebra of F generated by the cluster variables and coefficients obtained by all sequences of mutations from (x, B).
One of the remarkable properties of cluster algebras is the Laurent phenomenon.

2.2.
Ideal and tagged triangulations. Let S be a connected compact oriented Riemann surface with (possibly empty) boundary and M a non-empty finite set of marked points on S with at least one marked point on each boundary component if S has boundaries. We call the pair (S, M ) a marked surface. Any marked point in the interior of S is called a puncture. For technical reasons, (S, M ) is not a monogon with at most one puncture, a digon without punctures, a triangle without punctures nor a sphere with at most three punctures. An ordinary arc δ in (S, M ) is a curve in S with endpoints in M , considered up to isotopy, such that: δ does not intersect itself except at its endpoints; δ is disjoint from M and from the boundary of S except at its endpoints; δ does not cut out an unpunctured monogon or an unpunctured digon. An ordinary arc with two identical endpoints is called a loop. A curve homotopic to a boundary component between two marked points is called a boundary segment.
Two ordinary arcs are called compatible if they do not intersect in the interior of S. An ideal triangulation is a maximal collection of pairwise compatible ordinary arcs. A triangle with only two distinct sides is called self-folded (see Figure 4). For an ideal triangulation T , a flip at an ordinary arc δ ∈ T replaces δ by another arc δ ′ / ∈ T such that T \ {δ} ∪ {δ ′ } is an ideal triangulation. Notice that an ordinary arc inside a self-folded triangle can not be flipped. This problem was solved by the notion of tagged arcs introduced in [FST].
Definition 2.6. [FST,Definition 7.1] A tagged arc is an ordinary arc with each end tagged in one of two ways, plain or notched, such that the following conditions are satisfied: the tagged arc does not cut out a once-punctured monogon; an endpoint lying on the boundary of S is tagged plain; both ends of a loop are tagged in the same way.
In this paper, we assume that if (S, M ) is a closed surface with exactly one puncture, all tagged arcs are plain arcs. For an ordinary arc γ of (S, M ), we define a tagged arc ι(γ) as follows: If γ does not cut out a once-punctured monogon, ι(γ) is the tagged arc obtained from γ by tagging both ends plain: If γ cuts out a once-punctured monogon with endpoint o and puncture p, ι(γ) is the tagged arc obtained by tagging the unique arc that connects o and p and does not intersect γ, plain at o and notched at p (see Figure 4). For a tagged arc δ, we denote by δ • the ordinary arc obtained from δ by forgetting its tags.
Definition 2.7. [FST,Definition 7.4] Two tagged arcs δ and ǫ are called compatible if the following conditions are satisfied: • the two ordinary arcs δ • and ǫ • are compatible, • if δ • = ǫ • , at least one end of ǫ is tagged in the same way as the corresponding end of δ, • if δ • = ǫ • and they have a common endpoint o, the ends of δ and ǫ at o are tagged in the same way. A tagged triangulation is a maximal collection of pairwise compatible tagged arcs.
Note that it is possible to flip at any tagged arc of a tagged triangulation [FST,Theorem 7.9]. Moreover, any two tagged triangulations of (S, M ) are connected by a sequence of flips by [FST,Proposition 7.10].
2.3. Cluster algebras defined from triangulated surfaces. Let (S, M ) be a marked surface. First, we consider an ideal triangulation T of (S, M ). For an ordinary arc γ, π(γ) is defined as follows: if there is a self-folded triangle in T with non-loop side γ, π(γ) is its loop side; otherwise π(γ) = γ.
Definition 2.8. [FST,Definition 4.1] Let T be an ideal triangulation of (S, M ) and t 1 , . . . , t N be all ordinary arcs of T . For any non and π(t j ) are sides of △ with π(t j ) following π(t i ) in the clockwise order, −1, if π(t i ) and π(t j ) are sides of △ with π(t j ) following π(t i ) in the counterclockwise order, 0, otherwise.
We consider a tagged triangulation T of (S, M ). We obtain a tagged triangulationT from T by simultaneous changing all tags at some punctures, in such a way that there is an ideal triangulation T 0 satisfyingT = ι(T 0 ) (see [MSW1,Remark 3.11]). Notice thatT satisfies Assumption 1.
Definition 2.9. [FST,Definition 9.6] For a tagged triangulation T , we define the N × N matrix Since B T is skew-symmetric, we get a cluster algebra A(T ) := A(B T ) with principal coefficients for any tagged triangulation T .
Theorem 2.10. [FST,Theorem 7.11] [FT,Theorem 6.1] Let T be a tagged triangulation of (S, M ). Then the tagged arcs δ of (S, M ) correspond bijectively with the cluster variables x δ in A(T ). This induces that the tagged triangulations T ′ of (S, M ) correspond bijectively with the clusters x T ′ in A(T ). Moreover, the tagged triangulation obtained from T ′ by flipping at δ ∈ T ′ corresponds the cluster obtained from x T ′ by mutating at x δ .
For a tagged arc t and a puncture p of (S, M ), we denote by t (p) the tagged arc obtained from t by changing tags at p, where t (p) = t if p is not an endpoint of t.
Proposition 2.11. [MSW1,Proposition 3.15] Let T be a tagged triangulation of (S, M ) consisting of tagged arcs t 1 , . . . , t N . We denote by T (p) the tagged triangulation consisting of t ) be the corresponding initial seeds of A(T ) and A(T (p) ), respectively. Then for a tagged arc δ, we have In view of Proposition 2.11, since we haveT = T (p1···pr ) for some punctures p 1 , . . . , p r , it is enough to consider a tagged triangulation T satisfying T =T , that is satisfying Assumption 1. In the rest of this paper, we assume that any tagged triangulation satisfy Assumption 1. Moreover, suppose that there is a 1-notched arc t ∈ T with endpoint p tagged notched. Let s ∈ T the corresponding plain arc. Then t (p) = s and s (p) = t hold. Therefore, for a tagged arc δ, we have ΣT | xt↔xs by Proposition 2.11. Thus we can make Assumption 2.
2.4. Musiker-Schiffler-Williams cluster expansion formulas. In this subsection, we recall the cluster expansion formula given by Musiker-Schiffler-Williams [MS,MSW1]. We call it the MSW formula. Fix a marked surface (S, M ) and a tagged triangulation T of (S, M ) satisfying Assumptions 1 and 2. Let γ be a plain arc of (S, M ) such that γ / ∈ T . We use the notations of the introduction.
2.4.1. Formula for plain arcs. Recall the MSW formula for x γ . In the triangulation T γ constructed in the introduction, triangles have at most two sides that are non-boundary segments and at least one side that is a boundary segment. We construct the snake graph G γ := G Tγ from T γ by unfolding each triangle of T γ , two sides of which are non-boundary segments, along its third side (see Figure 5). We label all edges of G γ by the corresponding tagged arcs of T .
Example 2.12. We construct the snake graph G δ 1 for the tagged arc δ 1 given in Subsection 1.2 as follows: Note that G γ consists of n squares with diagonals τ i for 1 ≤ i ≤ n. We call these squares tiles of G γ . Let G γ := G Tγ be the graph obtained from G γ by removing the diagonal of each tile. It is easy to see that the following special perfect matching is uniquely determined.
Definition 2.13. [MSW1,Definition 4.7] Let e 0 be the edge of G γ corresponding to the boundary segment of T γ that follows τ 1 in the clockwise direction in the triangle T 0 . The minimal matching P − (G γ ) is the perfect matching of G γ containing e 0 and consisting only of boundary edges.
In Example 2.12, e 0 is the bottom edge of G δ1 .
Theorem 2.14. [MS,Theorem 5.1] For P ∈ P(G γ ), the set P − (G γ )△P consists of all boundary edges of some (possibly empty or disconnected) subgraph G P of G γ that is a union of tiles.
We denote by J(P ) the set of the diagonals of all tiles of G γ that are contained in G P . The following cluster expansion formula is obtained by using perfect matchings of G γ .

2.4.2.
Formula for 1-notched arcs. Recall the MSW formula for x γ (p) . Let q = p be the other endpoint of γ (p) . In the same way as above, for the ordinary loop ℓ p defined in Theorem 1.3, we get the snake graph G ℓp which is denoted by G γ (p) in the introduction. By construction, G ℓp contains two disjoint subgraphs G 1 ℓp and G 2 ℓp with same form as G γ . Moreover, we consider the subgraph H i ℓp of G i ℓp obtained by removing the vertex p and the two edges ζ 1 , ζ m .
Example 2.16. Let ℓ p be the ordinary loop such that ι(ℓ p ) = δ 2 in given Subsection 1.2. We have the triangulated polygon T ℓp , the snake graph G ℓp and the subgraphs G i ℓp and H i ℓp of G ℓp as follows: T ℓp We denote by P(G γ (p) ) the set of γ-symmetric perfect matchings of G ℓp . We also refer to elements of P(G γ (p) ) as perfect matchings of G γ (p) .

2.4.3.
Formula for 2-notched arcs. Recall the MSW formula for x γ (pq) . As above, we get ordinary loops ℓ p and ℓ q and the snake graphs G ℓp and G ℓq . Note that the pair (G ℓp , G ℓq ) is denoted by G γ (pq) in the introduction. Remark that γ may be a loop. Then we denote by ℓ p and ℓ q the loops as in Figure  6 although they are not ordinary loops. Figure 6. Analogues of ℓ p and ℓ q for a 2-notched loop Definition 2.20. [MSW1,Definition 4.18] Let P p and P q be γ-symmetric perfect matchings of G ℓp and G ℓq , respectively. The pair (P p , P q ) is γ-compatible if res(P p ) ≃ res(P q ). We denote by P(G γ (pq) ) the set of γ-compatible pairs of P(G γ (p) ) × P(G γ (q) ). We also refer to elements of P(G γ (pq) ) as perfect matchings of G γ (pq) .

3.
Proof of Theorem 1.5 In this section, we keep the notations of the previous sections. We prove the bijection between (1) and (2) in Theorem 1.4 and Theorem 1.5 in the three cases of δ = γ, γ (p) and γ (pq) . Notice that the same notations Φ and cross(T, δ) appear in Theorems 1.5, 2.15, 2.18 and 2.21. So we only need to consider x(A) and y(A) for A ∈ A(T δ ). Let A(T δ ) be the set of angles incident to at least one diagonal of T δ , and let A ex (T δ ) be the set of exterior angles of T δ which are angles between boundary segments and diagonals of T δ . In particular, A ex (T δ ) is contained in A(T δ ). For a set S, we denote by #S the cardinality of S.
3.1. The case of plain arcs. Recall the result of our previous paper [Y]. For a plain arc γ, we denote by (G γ ) 1 (resp., (G γ ) b ) the set of edges (resp., boundary edges) of G γ . Let A(G γ ) be the set of angles between a diagonal τ i and a side of the square with diagonal τ i in G γ , and ϕ : A(G γ ) → (G γ ) 1 the surjective map sending a ∈ A(G γ ) to the side that is opposite to a. By the unfolding process (see Subsection 2.4), there is a canonical surjection π : A(G γ ) → A(T γ ) compatible with the construction of G γ .
Theorem 3.1. [Y, Lemma 3.2, Proposition 3.4] There exists a bijection ϕ : A(T γ ) → (G γ ) 1 making the following diagram commutative: Moreover the map ϕ induces a bijection ϕ : Theorem 3.1 clearly gives the bijection between (1) and (2) in Theorem 1.4 for plain arcs. We only need to show that y(A) = y(ϕ(A)) for A ∈ A(T γ ) to prove Theorem 1.5 for plain arcs.
Proof. By Theorem 3.1 and Lemma 3.2, ϕ(A − (T γ )) is a perfect matching of G γ consisting only of boundary edges. In particular, since e 0 ∈ ϕ(A − (T γ )), where e 0 was defined in Definition 2.13, ϕ(A − (T γ )) = P − (G γ ) holds. Thus we have ϕ(A − (T γ )△A) = P − (G γ )△ϕ(A). On the other hand, ϕ maps the four angles incident to τ i in T γ to sides of the square with diagonal τ i in G γ . Therefore, (A − (T γ )△A) ∩ A ex (T γ ) contains an angle incident to τ i , which is equivalent to τ i ∈ Y (A), if and only if (P − (G γ )△ϕ(A)) ∩ (G γ ) b contains an edge of the square with diagonal τ i in G γ , which is equivalent to τ i ∈ J(ϕ(A)) by the definition.
Proof of Theorem 1.5 for plain arcs. The assertion follows from Theorems 2.15 and 3.1 and Proposition 3.3.
Finally, we prepare the following lemma to use later.
Lemma 3.4. For A ∈ A(T γ ), if A − (T γ )△A contains an exterior angle incident to τ i in T γ , it contains all exterior angles incident to τ i in T γ .
Proof. By Theorem 2.14, for P ∈ P(G γ ), if P − (T γ )△P contains a boundary sides of the square with diagonal τ i in G γ , it contains all boundary sides of the square with diagonal τ i in G γ . Since ϕ maps the four angles incident to τ i in T γ to sides of the square with diagonal τ i in G γ , the assertion follows from Lemma 3.2.
3.2. The case of 1-notched arcs. In this subsection, we show the following theorem.
Theorem 3.5 clearly gives the bijection between (1) and (2) in Theorem 1.4 for 1-notched arcs. To prove Theorem 3.5, we prepare the following notations as in Figure 7. By construction of the triangulated polygon T ℓp , it contains two disjoint subgraphs T 1 ℓp and T 2 ℓp with same form as T γ , where T 1 ℓp has the boundary segment ζ m of T ℓp . The subgraph U i ℓp of T i ℓp is obtained by removing the vertex p and the two sides ζ 1 , ζ m . For i ∈ {1, 2}, let v i (resp., v ′ i ) be the common endpoint of τ n and ζ m (resp., ζ 1 ) in T i ℓp . Let a i (resp., a ′ i ) be the angle at v i (resp., v ′ i ) that comes first in the counterclockwise (resp., clockwise) order around v i (resp., v ′ i ). We denote by a • i an angle between τ n and the boundary segment of the triangle with sides τ n−1 and τ n of T i ℓp . If n > 1, it is uniquely determined, that is τn−1 τn ζ m ζ 1 ζ 1 ζ m τn τn−1 τ1 By Theorem 3.1 and Proposition 3.3, there exists a bijection ϕ p : A(T ℓp ) → (G ℓp ) 1 which induces a bijection ϕ p : A(T ℓp ) → P(G ℓp ) satisfying x(A) = x(ϕ p (A)) and y(A) = y(ϕ p (A)) for A ∈ A(T ℓp ).
Lemma 3.6. The restrictions of ϕ p induce bijections for i ∈ {1, 2}. Moreover, the map ϕ p | A(T i ℓp ) induces a bijection between A(T i ℓp ) and P(G i ℓp ).
Proof. The first assertion follows immediately from the unfolding process. The second assertion follows from T i ℓp ≃ T γ , G i ℓp ≃ G γ , and Theorem 3.1.
Definition 3.7. We say that A ∈ A(T ℓp ) is γ-symmetric if the restrictions of A satisfies A| A(U 1 We denote by A sym (T ℓp ) the set of γ-symmetric perfect matchings of angles in T ℓp . Let A ∈ A(T ℓp ). It follows from Theorem 2.18 and Lemma 3.6 that A| A(T i ℓp ) ∈ A(T i ℓp ) for some i ∈ {1, 2}. Since it is uniquely determined up to isomorphism, we denote it by res(A).
Proof. It follows from Lemma 3.6 that A ∈ A(T ℓp ) is γ-symmetric if and only if ϕ p (A) ∈ P(G γ (p) ). Since ϕ p is a bijection between A(T ℓp ) and P(G ℓp ), it induces a bijection between A sym (T ℓp ) and P(G γ (p) ). On the other hand, Theorem 3.1 and Proposition 3.3 imply that x(A) = x(ϕ p (A)) and y(A) = y(ϕ p (A)) for A ∈ A(T ℓp ), and also x(res(A)) = x(ϕ p (res(A))) and y(res(A)) = y(ϕ p (res(A))) for A ∈ A sym (T ℓp ) since T i ℓp ≃ T γ . Since ϕ p is compatible with res, we have similarly, y(A) = y(ϕ p (A)) for A ∈ A sym (T ℓp ).
All that is left is to give the following proposition for the proof of Theorem 3.5.
To prove Proposition 3.9, we prepare some lemmas. We denote by T ℓp \ T 2 ℓp the subgraphs obtained from T ℓp by removing U 2 ℓp and ζ 1 of T 2 ℓp . Similarly, we define the notation T ℓp \ T 1 ℓp . For i ∈ {1, 2}, let c i and c ′ i be the angles as in Figure 7.
Proof. Suppose that c i ∈ A. Since T i ℓp has n + 1 triangles, it follows from c i ∈ A that #A| A(T i ℓp ) = n. Thus c ′ i ∈ A since T i ℓp has n + 1 vertices incident to at least one diagonal in T i ℓp . The proof of the converse assertion is similar.
For A ∈ A sym (T ℓp ), the γ-symmetry implies that a • 1 ∈ A if and only if a • 2 ∈ A. It is consistent to use the notations a • i ∈ A and a • i / ∈ A. Let b 1 (resp., b 2 ) be the angles as in Figure 7.
The assertion (1) follows from Lemma 3.10. Consequently, we have a decomposition A = A| A(T ℓp \T 2 ℓp ) ⊔ A| A(T 2 ℓp ) . Since #A| A(T 2 ℓp ) = n + 1 and T 2 ℓp has n + 1 triangles, then A| A(T 2 ℓp ) ∈ A(T 2 ℓp ). Thus res(A) = A| A(T 2 ℓp ) holds. The proof of (2) is similar Next, we consider the triangulated polygon T γ (p) with one puncture p. We prepare the following notations as in Figure 8. Let v (resp., v ′ ) be the common endpoint of τ n and ζ m (resp., ζ 1 ) in T γ (p) . Let d (resp., d ′ ) be the angle at v (resp., v ′ ) that comes first in the counterclockwise (resp., clockwise) order around v (resp., v ′ ). We denote by d • an angle between τ n and the boundary segment of the triangle with sides τ n−1 and τ n of T γ (p) . If n > 1, it is uniquely determined, that is d Let e 1 (resp., e 2 ) be the angle between ζ 1 (resp., ζ m ) and a boundary segment of T γ (p) .
τn−1 τn Proof. We only prove (1) since the proof of (2) is similar. Suppose that e 2 ∈ A. For k ∈ [2, m], we denote by α k the angle between ζ k and the boundary segment of the triangle with sides ζ k−1 and ζ k . An easy induction shows that α k ∈ A for all k ∈ [2, m] since α m = e 2 ∈ A. Thus A has the angle between ζ 1 and ζ m , and d The graph T γ (p) is obtained from T ℓp \ T 2 ℓp by identifying the two edges ζ m along the direction from p to the other endpoint of ζ m . Similarly, it is also obtained from T ℓp \ T 1 ℓp by identifying the two edges ζ 1 from p to the other endpoint of ζ 1 . These constructions induce bijections given by a → g 1 (a) if a = c ′ 2 and c ′ 2 → e 2 , and A ex (T ℓp \ T 1 ℓp ) \ {b 2 , c 2 , the angle between ζ 1 and ζ given by a → g 2 (a) if a = c ′ 1 and c ′ 1 → e 1 . Finally, we give one lemma for a general δ. For k ∈ [1, n], let T −;k δ and T +;k δ be the two subpolygons of T δ obtained by cutting T δ along τ k , where T −;k δ contains q. We denote by A ′ (T ±;k δ ) the restriction A(T δ )| T ±;k δ . We also define that T −;n+1 δ (resp., T +;0 δ ) is the subgraph obtained from T −;n δ (resp., T +;1 δ ) by adding the triangle with sides τ n , ζ 1 and ζ m (resp., τ 1 , ξ 1 and ξ ℓ ).
Proof. Since the equality #A| A ′ (T ±;k δ ) = #{triangles of T ±;k δ } = #{vertices of T ±;k δ incident to at least one diagonal} holds, there is exactly one endpoint v of τ k such that A| A ′ (T ±;k δ ) has no angle incident to v. Therefore, there is exactly one angle a v of A(T ±;k∓1 δ ) \ A ′ (T ±;k δ ) incident to v, and we have a unique completion For {i, j} = {1, 2} and A ∈ A(T ℓp \ T i ℓp ), there exists a unique symmetric completion We are ready to prove Proposition 3.9.
3.3. The case of 2-notched arcs. In this subsection, we show the following theorem.
Definition 3.15. The pair (A p , A q ) ∈ A(T γ (p) ) × A(T γ (q) ) is called γ-compatible if A c p = A c q and A q p ⊔ A cp q ∈ A(T γ ), where we view A q p ⊔ A cp q as a subset of A(T γ ). We denote by A com (T γ (p) , T γ (q) ) the set of γ-compatible pairs of A(T γ (p) ) × A(T γ (q) ).
Lemma 3.16. If n = 1, (A p , A q ) ∈ A com (T γ (p) , T γ (q) ) if and only if A q p ⊔ A p q ∈ A(T γ ). If n > 1, (A p , A q ) ∈ A com (T γ (p) , T γ (q) ) if and only if A c p = A c q . Proof. If n = 1, the assertion follows from A c p = ∅ = A c q . Suppose n > 1 and A c p = A c q . Since A p and A q satisfy the condition (2) in Definition 1.2, so does A q p ⊔ A cp q . Therefore, we only show that A q p ⊔ A cp q satisfies the condition (1) in Definition 1.2 on T γ , which is equivalent that any two distinct angles a and b in A q p ⊔ A cp q are not incident to a common vertex. If a, b ∈ A qc p or a, b ∈ A cp q , the assertion holds since A qc p ⊂ A p , A cp q ⊂ A q , and A p and A q satisfy the condition (1) in Definition 1.2. Suppose that a ∈ A q p and b ∈ A p q are incident to a common vertex. Then τ 1 , . . . , τ n must be incident to the vertex. Since A p ∈ A(T γ (p) ), A c p contains the angle between τ i and a boundary segment of the triangle with sides τ i and τ i+1 for i ∈ [1, n − 1]. Similarly, since A q ∈ A(T γ (q) ), A c q contains the angle between τ i and a boundary segment of the triangle with sides τ i−1 and τ i for i ∈ [2, n]. It contradicts A c p = A c q . Thus the assertion holds.
We define the maps r : Proof. By the γ-compatibility, r = A q p ⊔ A cp q ∈ A(T γ ). If n > 1, in the same as the proof of Lemma 3.16, i(A p , A q ) ∈ A(T γ (pq) ) holds. Suppose that n = 1. If i(A p , A q ) / ∈ A(T γ (pq) ), each of A q q and A p p has an angle incident to one endpoint of τ 1 . Thus each of A q p and A p q must have an angle incident to the other endpoint of τ 1 , so it contradicts A q p ⊔ A p q ∈ A(T δ ).
Lemma 3.18. Let n = 1 and A = (A p , A q ) ∈ A com (T γ (p) , T γ (q) ). Then the following conditions are equivalent: Proof. In this case, r(A) = A q p ⊔ A p q has exactly two angles. Each of the conditions (1)-(3) is equivalent that the angle between τ 1 and ξ ℓ is contained in A p . Moreover, it is equivalent that A p contains either the angle between τ 1 and ζ m or the angle between ζ m and a boundary segment of T γ (pq) , that is, the condition (4) holds. Therefore, the conditions (1)-(4) are equivalent.
Proposition 3.19. The map i is a bijection between A com (T γ (p) , T γ (q) ) and A(T γ (pq) ) satisfying x(A) = x(i(A)) and y(A) = y(i(A)) for A = (A p , A q ) ∈ A com (T γ (p) , T γ (q) ), where Proof. First of all, we construct the inverse map of i.
holds by the proof of Lemma 3.17. Thus (C τ1 (B cp ), C τn (B qc )) ∈ A com (T γ (p) , T γ (q) ) by Lemma 3.16. We define the map ω : Then it is easy to show that ω i and i ω are identities. Thus i : A com (T γ (p) , T γ (q) ) → A(T γ (pq) ) is a bijection.
All that is left is to give the following proposition for the proof of Theorem 3.14.
If n > 1, by construction of ψ p and ψ q , Thus it is the same as res(S p ) = res(S q ), that is res(P p ) = res(P q ). By Lemma 3.16, A ∈ A com (T γ (p) , T γ (q) ) if and only if (P p , P q ) ∈ P(G γ (pq) ).
If n = 1 and res(P p ) = res(P q ), then A q p ⊔ A p q corresponds to res(S p ) = res(S q ). Thus A q p ⊔ A p q ∈ A(T γ ). Conversely, suppose that A q p ⊔ A p q ∈ A(T γ ). The each angle of S p which is contained in the triangles U 1 ℓp and U 2 ℓp corresponds to the angle of A q p . Thus A q p ⊔ A p q corresponds to res(S p ) since A q p ⊔ A p q ∈ A(T γ ). Similarly, A q p ⊔ A p q corresponds to res(S q ). Therefore, we have res(S p ) = res(S q ). So, by Lemma 3.16, A ∈ A com (T γ (p) , T γ (q) ) if and only if (P p , P q ) ∈ P(G γ (pq) ), also in this case.
Consequently, we have a bijection On the other hand, we have r(A) ≃ res(S p ). As in the proof of Proposition 3.8, we also have x(res(S p )) = x(res(P p )) and y(res(S p )) = y(res(P p )). Therefore, we have and, similarly, y(ϕ pq (A)) = y(A).
Proof of Theorem 1.5 for 2-notched arcs. The assertion follows immediately from Theorems 2.21 and 3.14.

Proofs of our results for bipartite graphs
We refer the necessary notations in this section to the introduction. First, we prove the bijection between (1) and (3) in Theorem 1.4 and Proposition 1.6. (1) and (3) in Theorem 1.4. Angles incident to each vertex in A(T δ ) correspond bijectively with edges incident to the corresponding black vertex in B δ . Angles in each triangle in A(T δ ) correspond bijectively with edges incident to the corresponding white vertex in B δ . The assertion immediately follows from the definitions of perfect matchings of angles and perfect matchings of graphs.

Proof of the bijection between
Proof of Proposition 1.6. Let E ∈ P(B δ ). For any vertex v of B δ , v is incident to exactly zero or two edges in E − (B δ )△E. As a consequence, E − (B δ )△E is a disjoint union of non-crossing cycles. Thus the assertion holds.
Second, we have to be careful of the following special case to prove Proposition 1.7.
Proof. Since A − (T γ (pq) ) contains the angle between ξ 1 and a boundary segment of T γ (pq) , the assertion immediately follows from Proposition 1.6.
Finally, we prove Proposition 1.7 and give an example for the results of this section.
Proof of Proposition 1.7. It is trivial that ̟ induce a bijection between A ex (T δ ) and the set of boundary edges of B δ . Therefore, for A ∈ A(T δ ) and τ ∈ T δ , τ ∈  Then there are nine perfect matchings of B δ2 as follows: It is easy to check that these correspond bijectively with perfect matchings of angles in T δ 2 given in Subsection 1.2(2). Moreover, for each E ∈ P(B δ 2 ), the subgraph B E in Proposition 1.6 is given as follows: By comparing with Subsection 1.4(2), we can check that Proposition 1.7 holds in this case.

Minimal cuts of quivers with potential
In this section, we show that perfect matchings of angles in T δ coincide with minimal cuts of quiver with potential obtained from T δ , that is the bijection between (1) and (4) in Theorem 1.4. 5.1. Quivers with potential and cuts. We recall the definitions of quivers with potential [DWZ] and of their cuts [BFPPT, HI]. We denote by Z Q the path algebra of a quiver Q over the ring Z of integers.
Definition 5.1. (1) A quiver with potential (QP for short) is a pair (Q, W ) of a quiver Q and an element W ∈ Z Q which is a linear combination of cyclic paths.
(2) A cut of a QP (Q, W ) is a subset C of Q 1 such that any cyclic path appearing in W contains precisely one arrow in C.
We define a quiver Q δ as follows: the set of vertices consists of diagonals and boundary segments of T δ ; the set of arrows consists of arrows from i to j, where i and j are in the common triangle of T δ and j follows i in the counterclockwise order. We denote by Q δ the quiver obtained from Q δ by adding arrows from i to j, where i and j are boundary segments which are not in the common triangle of T δ and i is a predecessor of j with respect to clockwise order.
To define a potential W δ of Q δ , we consider the following cycles of Q δ . A triangle cycle is a cycle of length 3 inside a triangle of T δ . An exterior cycle is a cycle winding around a vertex (possibly a puncture) of T δ . We define Note that this extends QPs for triangulated polygons without punctures defined in [DL] to QPs for triangulated polygons with punctures.
Lemma 5.2. The number of triangle cycles in Q δ and the number of exterior cycles in Q δ coincide.
Proof. By construction, the number of triangle cycles in Q δ and the number of triangles in T δ coincide. Similarly, the number of exterior cycles in Q δ and the number of vertices incident to at least one diagonal in T δ . So all these numbers coincide.
We denote by n(δ) the number in Lemma 5.2.

5.2.
Minimal cuts of QPs and Perfect matchings of angles. We have a natural injection ρ : A(T δ ) → (Q δ ) 1 given by the following picture: Cuts of (Q δ , W δ ) have the following property using the map ρ. Proof. Since there are n(δ) triangle cycles (resp., n(δ) exterior cycles) not sharing arrows with each other, (a) holds. There is an exterior cycle sharing arrows with each triangle cycle. Since the shared arrows are contained in ρ(A(T δ )), the sufficiency of (b) holds. Since ρ(A(T δ )) is contained in the set of arrows appearing in a triangle cycle of Q δ , then |C| ≤ n(δ) for C ⊂ ρ(A(T δ )). Thus the necessity of (b) holds.
By Theorem 1.4, (Q δ , W δ ) always has minimal cuts. (1) and (4) in Theorem 1.4. Let A ⊆ A(T δ ) and C := ρ(A) ⊆ (Q δ ) 1 . Then there is exactly one element a of A in any triangle of T δ (resp., incident to any vertex of T δ ) if and only if the corresponding triangle cycle (resp., exterior cycle) contains precisely one arrow ρ(a) in C. Thus A ∈ A(T δ ) if and only if C is a cut. Since minimal cuts are precisely cuts contained in ρ(A(T δ )) by Lemma 5.3(b), the assertion follows.

Proof of the bijection between
Consequently, we can give another cluster expansion formula in terms of minimal cuts.
Proof. The assertion follows immediately from Theorems 1.4 and 1.5.
Example 5.6. For the tagged arc δ 2 given in Subsection 1.2(2), we have Then there are nine minimal cuts of (Q δ 2 , W δ 2 ) as follows: It is easy to check that these correspond bijectively with perfect matchings of angles in T δ 2 given in Subsection 1.2(2).

Essential loops
Recall the definition of essential loops [MSW2]. Throughout this section, we suppose that a marked surface (S, M ) has no punctures. An essential loop ζ in (S, M ) is a closed curve in S, considered up to isotopy, such that: ζ is disjoint from M and the boundary of S; ζ does not intersect itself; ζ is not a contractible loop.
Choose a triangle △ of T that ζ crosses. Let p be a point in the interior of △ that lies on ζ. Let α and β be the two sides of △ crossed ζ immediately before and following its travel through p, and let τ be the third side of △. Let ζ be the curve whose starting and ending points are p that exactly follows ζ. We can construct the triangulated polygon T ζ associated with ζ in the same way as for plain arcs. Also, we obtain the snake graph G ζ from T ζ . Let v (resp., w) be the endpoint of τ and α (resp., β) in the first triangle of T ζ or G ζ , and let v ′ (resp., w ′ ) be the endpoint of τ and β (resp., α) in the last triangle of T ζ or G ζ (see Figure 10). Figure 10. T ζ and G ζ associated with an essential loop ζ Definition 6.1. [MSW2,Definition 3.4,3.8] The band graph G ζ associated with the essential loop ζ is the graph obtained from G ζ by identifying the edges τ in the first and last squares such that v corresponds to v ′ . That is, the band graph lies on an annulus or a Möbius strip. A perfect matching P of G ζ is called good either if τ ∈ P or if both edges incident to v and incident to w in P lie on the same square. We denote by P g ( G ζ ) the set of good perfect matchings of G ζ .
Viewing P ∈ P g ( G ζ ) as a subset of (G ζ ) 1 , we can obtain P ∈ P(G ζ ) from P by adding either the edge τ in the first square or in the last square in G ζ . Then it is easy to show that there is a bijection P g ( G ζ ) and the set P g (G ζ ) := {P ∈ P(G ζ ) | P contains τ in the first or the last triangle of G ζ } given by sending P to P. In particular, there is a unique good perfect matching P − ( G ζ ) such that P − ( G ζ ) = P − (G ζ ), called the minimal matching (see [MSW2,Remark 3.9]). Definition 6.2. [MSW2,Definition 3.14] For an essential loop ζ in (S, M ), we define a Laurent polynomial x(P )y(P ).
One reason to consider x ζ is that they give rise to a base for the cluster algebra with principal coefficients obtained from a triangulated surface without punctures. Let T be a triangulation of (S, M ). A collection of arcs and essential loops in (S, M ) is C • -compatible if they do not intersect each other. In this case, we study perfect matchings of angles. For an essential loop ζ in (S, M ), we can construct a triangulated polygon T ζ in the same way as for plain arcs, that is, it is a triangulated annulus (see Figure 11). In particular, it is not twisted unlike band graphs. Since T ζ has the same numbers of triangles and of vertices, then A(T ζ ) = ∅. We define max-condition as the dual min-condition.
Definition 6.4. Let ζ be an essential loop in (S, M ). We say that a perfect matching of angles in T ζ is bad if all angles incident to one boundary component satisfy min-condition and all angles incident to the other boundary component satisfy max-condition (see Figure 12). A non-bad perfect matching of angles in T ζ is called good. We denote by A g (T ζ ) the set of good perfect matchings of angles in T ζ . Then we have the following result.

x(A)y(A).
To prove Theorem 6.5, we need some preparations. By rotational symmetry of order two, we can assume that T ζ is the above case in Figure 10. Since there is a bijection between P g ( G ζ ) and P g (G ζ ), Theorem 3.1 induces a bijection between P g ( G ζ ) and the set where c (resp., c ′ ) is the angle between α and β in the first (resp., last) triangle of T ζ (see Figure 13). In particular, this bijection preserves the values of x(−) and y(−) by Theorem 3.1 and Proposition 3.3. We denote by c A an angle c or c ′ contained in A ∈ A g (T ζ ). If both c and c ′ are contained in A, we define c A = c. We only need to construct a bijection ψ ′ ζ : A g (T ζ ) → A g (T ζ ) satisfying x(A \ {c A }) = x(ψ ′ ζ (A)) and y(A \ {c A }) = y(ψ ′ ζ (A)) for A ∈ A g (T ζ ). Let a (resp., b) be the angle between α (resp., β) and τ in the first triangle of T ζ (see Figure 13). We denote by A g (T ζ ) ∈b (resp., A g (T ζ ) / ∈b ) the subset of elements in A g (T ζ ) containing (resp., not containing) b, in particular, A g (T ζ ) = A g (T ζ ) ∈b ⊔ A g (T ζ ) / ∈b . Let A ∈ A g (T ζ ) ∈b . Then c ′ ∈ A follows from the definition of A g (T ζ ), that is c A = c ′ . The triangulated annulus T ζ is obtained from T ζ by removing the last triangle in T ζ and by identifying the edges α in the first triangle and in the last triangle. It is easy to show that this construction induces a natural map ψ ∈b ζ : A g (T ζ ) ∈b → A(T ζ ). Abusing notation, let a (resp., b, c) be the angle between τ and α (resp., τ and β, α and β) in T ζ . We denote by u the common endpoint of α and β in T ζ . Let α s , . . . , α 1 = α, α 0 = β = β 1 , . . . , β t be all arcs incident to u winding counter-clockwisely around u (see Figure 13).
Lemma 6.6. For A ∈ A g (T ζ ) ∈b , then ψ ∈b ζ (A) ∈ A g (T ζ ). Moreover, the map ψ ∈b ζ induces a bijection between A g (T ζ ) ∈b and the set Proof. Since c ′ ∈ A, then ψ ∈b ζ (A) does not contain the angle between α s and a boundary segment incident to u. Thus the angle incident to u in ψ ∈b ζ (A) does not satisfy min-condition. Since b ∈ ψ ∈b ζ (A) satisfies max-condition, ψ ∈b ζ (A) is good. By construction, there is a bijection between A g (T ζ ) ∈b and the set {A ′ ∈ A g (T ζ ) ∈b | A ′ contains the angle between β i and β i+1 for some i ∈ [1, t − 1]}.
(6.1) Let A ′ ∈ A g (T ζ ) ∈b . If A ′ contains the angle between α j and α j+1 for some j ∈ [1, s − 1], A ′ must contain the angle between α j and the boundary segment of the triangle with sides α j and α j−1 . Continuing this process, A ′ contains a, and it contradicts b ∈ A ′ . If A ′ contains the angle between α s and a boundary segment incident to u, then A ′ must contain the angle between α i and the boundary segment of the triangle with sides α i and α i+1 for [1, s − 1]. Then by the same argument for the other endpoint u ′ = u of α s , the angle of A ′ incident to u ′ satisfies max-condition. Continuing this process, A ′ consists only of exterior angles whose angles incident to the boundary with u satisfy mincondition and angles incident to the boundary with τ satisfy max-condition. It contradicts that A ′ is good. Therefore any A ′ ∈ A g (T ζ ) ∈b satisfies the condition of (6.1), thus the set (6.1) and A g (T ζ ) ∈b coincide.
Let A ∈ A g (T ζ ) / ∈b . Then c ∈ A follows from the definition of perfect matchings of angles, that is c A = c. The triangulated annulus T ζ is obtained from T ζ by removing the first triangle in T ζ and by identifying the edges β in the first triangle and in the last triangle. In particular, c ′ in T ζ corresponds to c in T ζ . It is easy to show that this construction induces a natural map ψ / ∈b ζ : A g (T ζ ) / ∈b → A(T ζ ).