1 Introduction↩︎

1.1 Backgrounds↩︎

Cluster algebras, introduced by Fomin–Zelevinsky [1], are commutative algebras generated by cluster variables which are constructed recursively via a combinatorial algorithm called mutation. They serve as an algebraic framework for studying the dual canonical basis [2], [3] [4] and the theory of total positivity [5]. Subsequently, Berenstein–Zelevinsky [6] introduced a natural quantization of this structure, giving rise to the theory of quantum cluster algebras.

A primary objective in the study of (quantum) cluster algebras is the construction of "good" bases. These bases are expected to share similar properties with the dual canonical bases of quantum groups. Important families of these bases include:

• The generic basis in the sense of Dupont [7] generalizes the dual semi-canonical basis [8] to the cluster algebra setting [9]. For explicit computations on tame quivers, see [10] [7]. Its general construction and existence can be found in [11] [12].

• The common triangular basis, introduced by the first author [13], is a Kazhdan–Lusztig type basis that extends the dual canonical basis to cluster algebras [13], [14]. In acyclic cases, it coincides with the triangular basis of Berenstein–Zelevinsky [15], see [16] [17].

• The theta basis, introduced by Gross–Hacking–Keel–Kontsevich [18], consists of theta functions arising from mirror symmetry and scattering diagrams. Its quantum analogue was subsequently developed by Davison–Mandel [19].

All three families of bases above are parameterized by tropical points, in accordance with the expectation of Fock–Goncharov [20], [21]. It is believed that all known "good" bases for cluster algebras in the literature belong to these families.

In the setting of cluster algebras from marked surfaces (via skein algebras), several bases are constructed using topological objects, such as the bangle, bracelet, and band bases [22] [23]. We refer the reader to Section 2.1 for the background on the marked surface \(\Sigma\) and its skein algebra, and to Sections 2.2 and 2.3 for the details of the bangle and band bases, respectively. These bases have been compared with the general frameworks mentioned above. For an unpunctured marked surface, Geiß–Labardini-Fragoso–Wilson [24] showed that the generic basis coincides with the bangle basis. Mandel–Qin [25] established that the bracelet basis coincides with the theta basis for general marked surfaces.

Although the bangle and bracelet bases fit into the general frameworks, the band basis remains less understood. Thurston made the following bold conjecture [23]:

Recall that the common triangular basis provides an analogue of the dual canonical basis [13], [17]. Therefore, Thurston’s conjecture follows from the conjecture below.

1.2 Main results and our approach↩︎

Our main result is a proof of Conjecture 1 for unpunctured surfaces.

Theorem 3 provides the existence of the common triangular basis for a new family of cluster algebras, which was previously known only for cluster algebras arising from Lie theory [27]. It also allows us to visualize Kazhdan–Lusztig type basis elements using topological objects.

In this paper, we often consider the following expansion in the skein algebra: \[\begin{align} \label{eq:XY-decomp} [\mathsf{X}][\mathsf{Y}]=\sum_\varepsilon q^{\alpha_\varepsilon}[\mathsf{E}_\varepsilon], \end{align}\tag{1}\] where \(\mathsf{X}\) and \(\mathsf{Y}\) are weighted simple multicurves (Section 2.2), \([\mathsf{X}]\) and \([\mathsf{Y}]\) are their homotopy classes, such that \([\mathsf{X}]\) is a cluster monomial (i.e., \(\mathsf{X}\) only consists of arc components). The sum is over the choices \(\varepsilon\) of smoothings at the crossings in \(\mathsf{X} \cap \mathsf{Y}\). These two types of local smoothings, namely the positive and negative smoothings, are illustrated in Figure 1.

For each \(\varepsilon\), we have \(\alpha_\varepsilon\in \mathbb{Z}\) and \(\mathsf{E}_\varepsilon\) is the multicurve obtained after resolving the crossings. These multicurves might contain contractible components. More details are provided in Section 3.1.

In Theorem 39, we characterize the properties of the term \([\mathsf{E}_{\varepsilon_+}]\) in the expansion of \([\mathsf{X}][\mathsf{Y}]\) as a corollary of our main result, where \(\varepsilon_+\) denotes the choice where all crossings receive positive smoothings. In particular, the following natural properties are of independent interest: \[\deg ([\mathsf{E}_{\varepsilon_+}])=\deg ([\mathsf{X}])+\deg ([\mathsf{Y}]),\] \[\Lambda(\deg ([\mathsf{X}]),\deg ([\mathsf{Y}]))=2\mu(\mathsf{X}, \mathsf{Y})+2a,\] where \(\mu(\mathsf{X}, \mathsf{Y})\) is the minimal intersection number and \(a\) is the boundary intersection number in 4 .

In our approach to Theorem 3, a key step is to understand the decomposition 1 where \([\mathsf{X}]\) is a cluster variable (i.e., \(\mathsf{X}\) is an arc). In this case, we write \(\mathsf{E}_\varepsilon=\mathbf{N}_\varepsilon \cup \mathbf{\Gamma}^0_\varepsilon \cup \widehat{\mathsf{E}}_\varepsilon\), where \(\mathbf{N}_\varepsilon\) is the set of unknots with cardinality \(|\mathbf{N}_\varepsilon|\), \(\mathbf{\Gamma}^0_\varepsilon\) consists of contractible arcs, and \(\widehat{\mathsf{E}}_\varepsilon\) contains no contractible components. By evaluating each unknot to \(-q^2-q^{-2}\) and setting contractible arcs to \(0\), we obtain \[\label{eq:decompose} [\mathsf{X}][\mathsf{Y}] = \sum_{\varepsilon: \mathbf{\Gamma}^0_\varepsilon = \emptyset} q^{\alpha_\varepsilon} (-q^2 - q^{-2})^{|\mathbf{N}_\varepsilon|} [\widehat{\mathsf{E}}_\varepsilon].\tag{2}\] It is crucial to show that the sum of the terms \((-1)^{|\mathbf{N}_\varepsilon|} q^{\alpha_\varepsilon + 2|\mathbf{N}_\varepsilon|} [\widehat{\mathsf{E}}_\varepsilon]\) with highest \(q\)-degree \(\alpha_\varepsilon + 2|\mathbf{N}_\varepsilon|\) in 2 (cf. discussion near 5 ) forms a band basis element. Let \(\mathcal{E}_{\mathrm{top}}\) denote the set of \(\varepsilon\) corresponding to these terms with highest \(q\)-degree.

First, we discover a new phenomenon: the unknots (contractible loops) of any \(\mathsf{E}_\varepsilon\) for \(\varepsilon\in \mathcal{E}_{\mathrm{top}}\) form bead-chain configurations (see Example 1 for an illustration of such a configuration), where:

• A bead is an isolated unknot.

• A chain is an unknot formed by the concatenation of multiple beads.

We introduce strip operations to describe how these chains merge or split when the smoothings are changed. This process is analogous to restringing beads on necklaces. Applying strip operations to \(\mathsf{E}_\varepsilon\) yields a new \(\mathsf{E}_\varepsilon'\) where \(\varepsilon'\in \mathcal{E}_{\mathrm{top}}\) still holds (Lemma 30). We refer the reader to Section 3.2 for details.

Moreover, each bead-chain configuration is constrained within an annular neighborhood \(\mathbb{A}_\ell\), which contains all loop components of \(\mathsf{E}_{\varepsilon_+}\) isotopic to \(\ell\) (Lemma 31 and Lemma 32). We prove that the configurations in distinct neighborhoods can be constructed independently (Proposition 33). Therefore, the construction of \(\mathsf{E}_{\varepsilon_+}\) reduces to separate constructions with in these annuli. Within each annulus, the problem involves only a cluster algebra associated with a quiver of type \(\mathbb{A}_{m,m}\), allowing this local construction to be completely understood (Section 3.3).

It is natural to ask what patterns the unknots of \(\mathsf{E}_\varepsilon\) might exhibit when \(\varepsilon\) is not restricted to \(\mathcal{E}_{\mathrm{top}}\). In Appendix 4, we describe one such phenomenon: certain unknots form a nested sequence.

1.3 Convention↩︎

Throughout this paper, we fix the following notations and assumptions:

• Quantum parameter: We fix a formal variable \(q^{1/2}\) and denote \(\mathbb{Z}_q := \mathbb{Z}\big[q^{\pm 1/2 }\big]\).

• Surfaces: \(\Sigma = (\mathcal{S}, \mathcal{M})\) is an unpunctured, compact, oriented surface where \(\partial \mathcal{S} \neq \emptyset\) and each boundary component contains \(\geq 1\) marked point. We exclude disks with \(\leq 2\) marked points.

• Skein elements: \([\mathrm{X}]\) is the equivalence class of a link \(\mathrm{X}\) in the skein algebra \(\mathrm{Sk}_q(\Sigma)\). For a simple multicurve \(\mathsf{X}\), we denote \(\langle \mathsf{X} \rangle \mathrel{\vcenter{:}}= \mathrm{Band}(\mathsf{X})\) and \(U(\mathsf{X}) \mathrel{\vcenter{:}}= \langle \mathsf{X} \rangle\) if \(\mathsf{X}\) consists of loops.

• Seeds: \(\Delta^+\) is the set of seeds mutation equivalent to an initial seed \(\mathbf{s}_0\).

• Bar involution: The involutive ring anti-automorphism \(\overline{(\cdot)}\) on \(\mathrm{Sk}_q(\Sigma)\) satisfies \(q^{1/2} \mapsto q^{-1/2}\) and \([\mathrm{X}] \mapsto [\overline{\mathrm{X}}]\).

• Figures: Several figures in the following sections are adapted from the TikZ source code of Muller [28].

2 Preliminaries↩︎

2.1 The skein algebra↩︎

In this section, we briefly review the fundamental concepts of skein algebras on marked surfaces. For detailed definitions and proofs, we refer the reader to [28].

Let \(\mathcal{S}\) be a compact oriented surface with boundary, and \(\mathcal{M} \subset \partial\mathcal{S}\) a finite set of marked points. The pair \(\Sigma= ( \mathcal{S}, \mathcal{M})\) is called a marked surface (unpunctured).

A curve \(\mathsf{x}\) in \(\mathcal{S}\) is an immersion \(\mathsf{x}: C \rightarrow \mathcal{S}\) from a compact connected \(1\)-manifold \(C\) to \(\mathcal{S}\) such that the boundary of \(C\) maps into \(\mathcal{M}\) and the interior of \(C\) does not map to \(\mathcal{M}\) or \(\partial \mathcal{S}\). Two curves are homotopic if they can be related by a homotopy respecting \(\mathcal{M}\) and orientation reversal. An arc is a curve with endpoints in \(\mathcal{M}\), and a loop is a closed curve with no endpoints. An arc is a boundary arc if it is homotopic to the closure of a component of \(\partial \mathcal{S}\backslash \mathcal{M}\). A multicurve \(\mathsf{X}\) is an unordered finite collection of (possibly homotopic) curves in \(\mathcal{S}\). Two multicurves are homotopic if there is a bijection between their constituent curves which takes a curve to a homotopic one.

A strand in a multicurve \(\mathsf{X}\) at a point \(p\in \mathcal{S}\) is a connected component of \(\mathsf{X} \cap \mathcal{D}_p\), where \(\mathcal{D}_p\) is a sufficiently small disk neighborhood of \(p\). A multicurve \(\mathsf{X}\) is transverse if the strands have distinct tangent directions at each intersection, and each interior intersection (a crossing) involves exactly two strands. Moreover, every multicurve is homotopic to a transverse multicurve.

A transverse multicurve is simple if it has no crossings and no contractible curves. Here, a contractible curve is either a null-homotopic loop (an unknot) or an arc that bounds a disk in \(\mathcal{S}\) (a contractible arc).

A link \(\mathrm{X}\) is a transverse multicurve \(\mathsf{X}\) equipped with an ordering (over/under) of the two strands at each crossing, as well as an equivalence relation and a total ordering on the resulting equivalence classes of strands at each \(p \in \mathcal{M}\). A simple multicurve \(\mathsf{X}\) can be regarded as a link with a simultaneous ordering at each endpoint, also denoted by \(\mathsf{X}\). Homotopies between links are performed within the class of transverse multicurves, preserving crossing data.

Let \(\mathbb{Z}_q := \mathbb{Z}\big[q^{\pm 1/2 }\big]\). Denote by \(\mathbb{Z}_q^{\mathrm{Links}}(\Sigma)\) the free \(\mathbb{Z}_q\)-module spanned by the equivalence classes of links in \(\Sigma\). The skein module is defined as the quotient \[\mathrm{Sk}_q(\Sigma) := \mathbb{Z}_q^{\mathrm{Links}}(\Sigma) \big/ I,\] where \(I\) is generated by the following relations:

• The Kauffman skein relation

  The first link on the right is called the positive smoothing of the crossing, and the second is called the negative smoothing.

• The boundary skein relation

  The boundary skein relation describes the exchange of strands at a marked point.

• The value of the unknot

• The value of a contractible arc

For a link \(\mathrm{X}\), its equivalence class in \(\mathrm{Sk}_q(\Sigma)\) is denoted by \([\mathrm{X}]\). The \(\mathbb{Z}_q\)-module \(\mathrm{Sk}_q(\Sigma)\) admits a \(\mathbb{Z}_q\)-bilinear, non-commutative superposition product. Let \(\mathrm{X}\) and \(\mathrm{Y}\) be links such that the union of the underlying multicurves \(\mathsf{X} \cup \mathsf{Y}\) is transverse. Define the superposition \(\mathrm{X}\cdot \mathrm{Y}\) to be the link with underlying multicurve \(\mathsf{X} \cup \mathsf{Y}\), where at each new crossing, the strand of \(\mathrm{X}\) crosses over that of \(\mathrm{Y}\), and all other crossings retain their original ordering from \(\mathrm{X}\) and \(\mathrm{Y}\). Since \([\mathrm{X} \cdot \mathrm{Y}]\) depends only on the homotopy classes of \(\mathrm{X}\) and \(\mathrm{Y}\), the superposition product is defined on equivalence classes by: \[[\mathrm{X}][\mathrm{Y}]:=[\mathrm{X}'\cdot \mathrm{Y}'],\] where \(\mathrm{X}' \sim \mathrm{X}\) and \(\mathrm{Y}' \sim \mathrm{Y}\) are representatives such that \(\mathrm{X}'\cup \mathrm{Y}'\) transverse.

This product endows \(\mathrm{Sk}_q(\Sigma)\) with the structure of an associative \(\mathbb{Z}_q\)-algebra with unit \([\emptyset]\) (the empty link), called the Kauffman skein algebra of \(\Sigma\). The localized skein algebra \(\mathrm{Sk}^{\circ}_q(\Sigma)\) is the Ore localization of \(\mathrm{Sk}_q(\Sigma)\) at the multiplicative set generated by the boundary arcs.

Given a link \(\mathrm{X}\), let \(\overline{\mathrm{X}}\) denote the link with the same underlying multicurve but with all crossing data reversed. There exists an involutive ring anti-automorphism \(\overline{(\cdot)}: \mathrm{Sk}_q(\Sigma) \to \mathrm{Sk}_q(\Sigma)\), called the bar involution, defined by: \[\begin{align} q^{1/2} &\mapsto q^{-1/2}, \\ [\mathrm{X}] &\mapsto [\overline{\mathrm{X}}], \end{align}\] for all links \(\mathrm{X}\). Elements fixed by the bar involution are called bar-invariant.

2.2 Weighted simple multicurves and bangles↩︎

Let \(\mathrm{SMulti}(\Sigma)\) denote the set of homotopy classes of simple multicurves. For any simple multicurve \(\mathsf{X}\), there exist finitely many pairwise non-homotopic and non-intersecting simple curves \(\mathsf{x}_1,\ldots,\mathsf{x}_k\) such that \(\mathsf{X}\) is homotopic to the union \(\bigcup_{i=1}^k w_i \mathsf{x}_i\) for some integers \(w_i \geq 1\). We write \(\mathsf{X} = \bigcup_{i=1}^k w_i \mathsf{x}_i\) to denote this decomposition.

A weighted crossingless multicurve is a collection of pairwise non-homotopic and non-intersecting curves \(\mathsf{x}_1,\ldots,\mathsf{x}_k\), each assigned an integer weight \(w_i\), such that the weights on non-boundary curves are positive. We identify weight-\(0\) curves with the empty set \(\emptyset\). It is further called a weighted simple multicurve if all components are non-contractible. Denote by \(\mathrm{SMulti}^\circ(\Sigma)\) the set of homotopy classes of weighted simple multicurves.

Any simple multicurve \(\mathsf{X} = \bigcup_{i=1}^k w_i \mathsf{x}_i\) defines an element \([\mathsf{X}] \in \mathrm{Sk}_q(\Sigma)\) given by: \[[\mathsf{X}] := q^{\lambda/2} \prod_{i=1}^k [\mathsf{x}_i]^{w_i},\] where \(q^{\lambda/2}\) is the unique power of \(q\) such that \([\mathsf{X}]\) is bar-invariant. This definition extends naturally to weighted multicurves.

Let \(\mathrm{Bang}_q(\Sigma)\) denote the set of bangles with non-negative weights, and \(\mathrm{Bang}_q^\circ(\Sigma)\) the set of all bangles.

By Lemma 6, the homotopy classes of multicurves form a \(\mathbb{Z}_q\)-basis. We shall also consider their \(q^{\mathbb{Z}}\)-shifts as a \(\mathbb{Z}\)-basis. For convenience, we refer to this \(\mathbb{Z}\)-basis as the shifted multicurve basis.

2.3 Bands↩︎

The Chebyshev polynomials of the second kind \(\{U_k(z)\}_{k \in \mathbb{N}}\) are defined recursively by: \[U_0(z)=1,\] \[U_1(z)=z,\] \[U_{k+1}(z)=zU_k(z)-U_{k-1}(z).\]

These polynomials satisfy the following identities for all \(k, l \in \mathbb{N}\): \[U_k(z)U_l(z)=U_{k+l}(z)+U_{k+l-2}(z)+\cdots+U_{|k-l|}(z),\] \[U_k(e^x+e^{-x})=e^{kx}+e^{(k-2)x}+\cdots+e^{-kx}.\]

For simplicity, we denote \(\langle \mathsf{X}\rangle:=\mathrm{Band}(\mathsf{X})\). When \(\mathsf{X}\) consists only of loop components, we also write \(U(\mathsf{X}):=\langle \mathsf{X} \rangle\).

Denote the sets of band elements in \(\mathrm{Sk}^\circ_q(\Sigma)\) by: \[\begin{align} \mathrm{Band}_q(\Sigma) & := \left\{ \langle\mathsf{X}\rangle \in \mathrm{Sk}^\circ_q(\Sigma) \mid \mathsf{X} \in \mathrm{SMulti}(\Sigma) \right\}, \\ \mathrm{Band}^\circ_q(\Sigma) & := \left\{ \langle\mathsf{X}\rangle \in \mathrm{Sk}^\circ_q(\Sigma) \mid \mathsf{X} \in \mathrm{SMulti}^\circ(\Sigma) \right\}. \end{align}\]

2.4 Quantum cluster algebras↩︎

A quantum seed of skew-symmetric type in a skew-field \(\mathcal{F}\) is a triple \((B, \Lambda, M)\), where:

• The exchange matrix \(B\) is an \(N \times \mathbf{ex}\) integer matrix for a subset \(\mathbf{ex} \subset \{1, \ldots, N \}\), such that the principal part \(\pi B\) is skew-symmetric, where \(\pi\) is the \(\mathbf{ex} \times N\) matrix projecting \(\mathbb{Z}^N\) onto \(\mathbb{Z}^{\mathbf{ex}}\).

• The compatibility matrix \(\Lambda\) is an \(N \times N\) skew-symmetric integer matrix satisfying the compatibility condition \(\Lambda B = D \iota\), where \(\iota\) is the \(N \times \mathbf{ex}\) matrix that includes \(\mathbb{Z}^{\mathbf{ex}}\) into \(\mathbb{Z}^N\), and \(D= \mathrm{diag}(d_{11}, \ldots, d_{NN})\) is a diagonal matrix with entries \(d_{ii} > 0\).

• \(M: \mathbb{Z}^N \to \mathcal{F} \setminus \{0\}\) satisfies the multiplicative relation \[M(\alpha) M(\beta) =q^{\frac{1}{2} \Lambda (\alpha, \beta )} M(\alpha + \beta)\] such that the \(\mathbb{Z}_q\)-subalgebra generated by \(M(\mathbb{Z}^N)\subset \mathcal{F}\) forms a quantum torus \(\mathbb{T}_\Lambda\) defined by \(\Lambda\), whose skew-field of fractions is \(\mathcal{F}\).

A quantum seed \((B', \Lambda', M')\) in \(\mathcal{F}\) is the mutation at \(i \in \mathbf{ex}\) of \((B, \Lambda, M)\) if:

• The exchange relation holds: \[b'_{jk} = \begin{cases} -b_{jk} &\text{if}\;i =j\;\text{or}\;i=k, \\ b_{jk} + \frac{1}{2} (|b_{ji}| b_{ik} + b_{ji} |b_{ik}|) & \text{otherwise}. \end{cases}\]

• \(M(\alpha) =M'(\alpha)\) for all \(\alpha \in \mathbb{Z}^N\) satisfying \(\alpha_i=0\).

• The quantum cluster relation holds: \[M'(e_i) = M \left( -e_i + \sum_{j:b_{ji}>0} b_{ji} e_j \right)+ M \left(-e_i - \sum_{j:b_{ji}<0} b_{ji}e_j \right).\]

Two quantum seeds \((B, \Lambda, M)\) and \((B', \Lambda', M')\) are mutation equivalent if they can be related by a finite sequence of mutations and permutations of indices. An element of the form \(M'(e_i)\in \mathcal{F}\) is called a cluster variable of \((B, \Lambda, M)\) if \((B', \Lambda', M')\) is mutation equivalent to \((B, \Lambda, M)\). For such a variable \(M'(e_i)\):

• It is mutable if \(i\in \mathbf{ex}\).

• It is frozen if \(i \in \{1, \dots, N \} \setminus \mathbf{ex}\).

The quantum cluster algebra \(\mathcal{A}_q\) associated with \((B, \Lambda, M)\) is the \(\mathbb{Z}_q\)-subalgebra of \(\mathcal{F}\) generated by the cluster variables, together with the inverses of the frozen variables. The quantum upper cluster algebra \(\mathcal{U}_q\) of a quantum seed \((B, \Lambda, M)\) is defined as the intersection of the quantum tori over all quantum seeds \((B', \Lambda', M')\) mutation equivalent to \((B, \Lambda, M)\), i.e., \[\mathcal{U}_q =\bigcap _{(B', \Lambda', M') \sim (B, \Lambda, M)} \mathbb{Z}_q M'(\mathbb{Z}^N).\]

For an \(n \times n\) integral skew-symmetric matrix \(A\), a mutation at an index \(i\in \{1,\dots, n \}\) is defined by the exchange relation given earlier. The exchange type of \(A\) is its equivalence class under the equivalence relation generated by mutations and conjugations by permutation matrices.

For a quantum seed \(\mathbf{s}=(B, \Lambda, M)\), the exchange type of \(\pi B\) consists of matrices of the form \(\pi B'\) for all quantum seeds \((B', \Lambda', M')\) mutation equivalent to \((B, \Lambda, M)\). The exchange type of a quantum seed \((B, \Lambda, M)\) is the exchange type of \(\pi B\), and the exchange type of a cluster algebra \(\mathcal{A}_q\) is defined as the exchange type of any of its quantum seeds.

An \(n\times n\) skew-symmetric matrix \(A\) is acyclic if there is no sequence of indices \(i_1, i_2,\dots, i_k=i_1\) with \(k \geq 3\) such that \(a_{i_{j+1}i_j} >0\) for all \(j \in \{1, \dots, k-1\}\). An exchange type is acyclic if it contains at least one acyclic matrix.

2.5 Quantum cluster algebras of marked surfaces↩︎

A marked surface \(\Sigma= ( \mathcal{S}, \mathcal{M})\) is triangulable if it satisfies the following conditions:

• \(\partial \mathcal{S} \neq \emptyset\);

• Each boundary component of \(\mathcal{S}\) contains at least one marked point;

• No connected component of \(\mathcal{S}\) is a disk with \(\leq 2\) marked points on its boundary.

An ideal triangulation of a triangulable \(\Sigma\) is a maximal collection \(\Delta\) of pairwise non-homotopic simple arcs in \(\Sigma\) that are mutually non-intersecting in the interior of \(\mathcal{S} \setminus \mathcal{M}\). In the following, we assume that \(\Sigma\) is triangulable (i.e., it admits an ideal triangulation).

An end of an arc \(\mathsf{x}\) is defined as a strand of \(\mathsf{x}\) within a sufficiently small neighborhood of its endpoint. For any arc \(\mathsf{x}\), we denote its two ends by \(\partial_1(\mathsf{x})\) and \(\partial_2(\mathsf{x})\).

Given a triangulation \(\Delta\) of \(\Sigma\), we associate it with a quantum seed \[\mathbf{s}_\Delta = (B^\Delta, \Lambda^\Delta, M^\Delta),\] where:

• \(\mathbf{ex} \subset \{ 1, \ldots, N\} \simeq \Delta\) is the subset of non-boundary arcs in \(\Delta\).

• \(B^\Delta = \mathsf{Q}^\Delta \iota\), where \(\mathsf{Q}^\Delta\) is the skew-adjacency matrix of \(\Delta\), defined for \(\mathsf{x}_i, \mathsf{x}_j \in \Delta\) by: \[\mathsf{Q}^{\Delta}_{ij}:= \sum_{a,b \in \{ 1, 2 \}} \begin{cases} -1 & \text{if } \partial_a(\mathsf{x}_i) \text{ is immediately clockwise to } \partial_b(\mathsf{x}_j) \text{ at a common endpoint}, \\ 1 & \text{if } \partial_a(\mathsf{x}_j) \text{ is immediately clockwise to } \partial_b(\mathsf{x}_i) \text{ at a common endpoint}, \\ 0 & \text{otherwise}. \end{cases}\]

• \(\Lambda^\Delta\) is the orientation matrix of \(\Delta\) defined component-wise for \(\mathsf{x_i}, \mathsf{x_j} \in \Delta\) by: \[\Lambda^\Delta_{ij} := \sum_{a,b \in \{ 1, 2 \}} \begin{cases} -1 & \text{if } \partial_a(\mathsf{x}_i) \text{ is clockwise to } \partial_b(\mathsf{x}_j) \text{ at a common endpoint}, \\ 1 & \text{if } \partial_a(\mathsf{x}_j) \text{ is clockwise to } \partial_b(\mathsf{x}_i) \text{ at a common endpoint}, \\ 0 & \text{otherwise}. \end{cases}\]

• \(M^\Delta : \mathbb{Z}^N \to \mathcal{F}\) is defined by \(M^\Delta(\alpha)=[ \Delta^\alpha]\). In particular, \(M^\Delta(e_i)=[\Delta^{e_i}]=[\mathsf{x}_i]\), where \(\mathsf{x}_i\) is the \(i\)-th arc in \(\Delta\).

Given a triangulation \(\Delta\) and an interior arc \(\mathsf{x} \in \Delta\), there is a unique quadrilateral \(R\) in \(\Delta\) with \(\mathsf{x}\) as a diagonal. The flip of \(\mathsf{x}\) in \(\Delta\) is the triangulation obtained by replacing \(\mathsf{x}\) with the other diagonal \(\mathsf{x}'\) of \(R\) while keeping the remaining arcs of \(\Delta\) unchanged (see Figure 9).

2.6 Dominance order, degree and support↩︎

Let \(\mathbf{s} = (B, \Lambda, M)\) be a quantum seed. Recall that the compatibility condition \(\Lambda B = D \iota\) implies that \(B\) has full rank, i.e., the column vectors \(\{ \mathrm{col}_k B \}_{k \in \mathbf{ex}}\) are linearly independent. For simplicity, we denote \(x_i(\mathbf{s}):= M^\mathbf{s}(e_i)\) for \(i \in \{1, \dots, N\}\), \(y_k(\mathbf{s}):=x(\mathbf{s})^{\mathrm{col}_k B^\mathbf{s}}\) for \(k\in \mathbf{ex}\), and \(y(\mathbf{s})^n:= x(\mathbf{s})^{B^\mathbf{s}n}\) for \(n\in \mathbb{N}^{\mathbf{ex}}\). In the following, we shall omit the symbol \(\mathbf{s}\) if it does not cause any ambiguity.

Let \(\mathfrak{M}(\mathbf{s}):=\mathbb{Z}^N\) be the lattice with standard basis vectors \(\{e_k\}_{k=1}^N\). The dominance order \(\prec_\mathbf{s}\) on \(\mathfrak{M}(\mathbf{s})\) is defined as follows: \(g'\prec_\mathbf{s} g\) if and only if there exists a non-zero vector \(n \in \mathbb{N}^{\mathbf{ex}}\) such that \(g'=g+B^{\mathsf{s}}n\).

Denote by \(\mathbb{Z}_q[\widehat{\mathbb{N}^{\mathbf{ex}}}]\) the completion of the monoid algebra \(\mathbb{Z}_q[\mathbb{N}^{\mathbf{ex}}] := \mathbb{Z}_q[y^n]_{n \in \mathbb{N}^{\mathbf{ex}}}\) with respect to the maximal ideal generated by \(\{ y^n \mid n \in \mathbb{N}^{\mathbf{ex}} \setminus \{0\} \}\). We define the ring of formal Laurent series as \(\widehat{\mathbb{T}}_{\Lambda} := \mathbb{T}_{\Lambda} \otimes_{\mathbb{Z}_q[\mathbb{N}^{\mathbf{ex}}]} \mathbb{Z}_q[\widehat{\mathbb{N}^{\mathbf{ex}}}].\)

Consider a formal sum \(Z = \sum_{d \in \mathfrak{M}(\mathbf{s})} c_d x^d\) with \(c_d \in \mathbb{Z}_q\). If the set \(\{d \mid c_d \neq 0\}\) has a unique \(\prec_{\mathbf{s}}\)-maximal element \(g\), we define its degree by \(\deg^{\mathbf{s}}(Z) = g\). We say that \(Z\) is \(g\)-pointed if \(c_g = 1\). Similarly, if the set \(\{d \mid c_d \neq 0\}\) has a unique \(\prec_{\mathbf{s}}\)-minimal element \(h\), we define its codegree by \(\mathrm{codeg}^{\mathbf{s}}(Z) = h\), and we say \(Z\) is \(h\)-copointed if \(c_h = 1\). \(Z\) is said to be bipointed if it is both \(g\)-pointed and \(h\)-copointed.

For any vector \(n = \sum_{k \in \mathbf{ex}} n_k f_k \in \mathbb{N}^{\mathbf{ex}}\), where \(\{f_k\}_{k \in \mathbf{ex}}\) denotes the standard basis vectors of \(\mathbb{Z}^{\mathbf{ex}}\), we define its support as \[\mathrm{supp}(n) := \{ k \in \mathbf{ex} \mid n_k \neq 0 \}.\] For any \(Z = x^g \sum_{n \in \mathbb{N}^{\mathbf{ex}}} c_n y^n \in \widehat{\mathbb{T}}_{\Lambda}\) with \(c_0 \neq 0\), its support is defined as \[\mathrm{supp}(Z) := \bigcup_{c_n \neq 0} \mathrm{supp}(n) \subseteq \mathbf{ex}.\] If \(Z\) is a Laurent polynomial, we further define its support dimension \(\mathrm{suppDim}(Z) \in \mathbb{N}^{\mathbf{ex}}\) by \[(\mathrm{suppDim}(Z))_k := \max \{ n_k \mid c_n \neq 0 \}, \quad \forall k \in \mathbf{ex}.\]

2.7 Tropical transformation↩︎

Fix a seed \(\mathbf{s}\) and let \(\mathbf{s}' = \mu_i \mathbf{s}\) be the seed obtained by mutation at \(i \in \mathbf{ex}\). The tropical transformation \(\phi_{\mathbf{s}', \mathbf{s}} : \mathfrak{M}(\mathbf{s}) \to \mathfrak{M}(\mathbf{s}')\) is a piecewise linear map defined as follows:

For \(m = \sum_{k=1}^N m_k e_k^{\mathbf{s}} \in \mathfrak{M}(\mathbf{s})\), its image \(m' = \phi_{\mathbf{s}', \mathbf{s}}(m) = \sum_{k=1}^N m'_k e_k^{\mathbf{s}'} \in \mathfrak{M}(\mathbf{s}')\) is given by \[m'_k = \begin{cases} -m_i & \text{if } k = i, \\ m_k + b_{ki}^{\mathbf{s}} [m_i]_+ & \text{if } b_{ki}^{\mathbf{s}} \geq 0 \text{ and } k \neq i, \\ m_k + b_{ki}^{\mathbf{s}} [-m_i]_+ & \text{if } b_{ki}^{\mathbf{s}} \leq 0 \text{ and } k \neq i, \end{cases}\] where \([x]_+ := \max(x, 0)\) denotes the positive part.

For an arbitrary seed \(\mathbf{s}' = \mu_{\underline{i}} \mathbf{s}\) obtained by a sequence of mutations \(\underline{i} = (i_1, \dots, i_k)\), the tropical transformation \(\phi_{\mathbf{s}', \mathbf{s}}\) is defined as the composition of the tropical transformations along the mutation sequence \(\underline{i}\). This composition is well-defined, in the sense that it is independent of the choice of the mutation sequence connecting \(\mathbf{s}\) to \(\mathbf{s}'\).

We define a linear map \(\psi_{\mathbf{s}', \mathbf{s}}: \mathfrak{M}(\mathbf{s}) \to \mathfrak{M}(\mathbf{s}')\) by: \[\psi_{\mathbf{s}', \mathbf{s}}\left( \sum_{k \in N} m_k e_k^{\mathbf{s}} \right) := \sum_{k \in N} m_k \phi_{\mathbf{s}', \mathbf{s}}(e_k^{\mathbf{s}}), \quad \forall (m_k)_{k \in N} \in \mathbb{Z}^N.\]

Let \(Z \in \widehat{\mathbb{T}}_{\Lambda^{\mathbf{s}}}\) be a \(g\)-pointed element. We say that \(Z\) is compatibly pointed at seeds \(\mathbf{s}\) and \(\mathbf{s}'\) if \(Z\) is also \(\phi_{\mathbf{s}', \mathbf{s}}(g)\)-pointed in \(\widehat{\mathbb{T}}_{\Lambda^{\mathbf{s}'}}\). We say \(Z\) is compatibly pointed if it is compatibly pointed at any pair of mutation-equivalent seeds \(\mathbf{s}, \mathbf{s}'\).

A seed \(\mathbf{s}\) is said to be injective-reachable if there exists a mutation sequence \(\underline{i}\) and a permutation \(\sigma\) of \(\mathbf{ex}\) such that in the seed \(\mathbf{s}[1] := \mu_{\underline{i}} \mathbf{s}\), the \(g\)-vectors of the cluster variables satisfy: \[\pi \left( g_{\sigma(k)}^{\mathbf{s}[1]} \right) = -f_k, \quad \forall k \in \mathbf{ex},\] where \(\pi : \mathbb{Z}^N \to \mathbb{Z}^{\mathbf{ex}}\) is the canonical projection map. In this case, we may also denote the original seed by \(\mathbf{s} = \mu_{\underline{i}} \mathbf{s}[-1]\).

Given an injective-reachable seed \(\mathbf{s}\) and \(g \in \mathfrak{M}(\mathbf{s})\), let \(h := \psi^{-1}_{\mathbf{s}[-1], \mathbf{s}} \phi_{\mathbf{s}[-1], \mathbf{s}}(g)\). If there exists a vector \(n \in \mathbb{N}^{\mathbf{ex}}\) such that \[h = g + B^{\mathbf{s}} n,\] we then define the support dimension of \(g\) as \(\mathrm{suppDim}(g) := n\).

2.8 Common triangular basis↩︎

Throughout this paper, we assume that the injective-reachable condition holds for all seeds under consideration.

Let \(\Delta^+\) be the set of seeds mutation equivalent to an initial seed \(\mathbf{s}_0\).

3 Main Results↩︎

3.1 Attachment Points and Angles↩︎

We first introduce the preliminary concepts and constructions. Let \(\mathsf{x}\) be a simple arc and \(\mathsf{Y}\) a weighted simple multicurve such that the superposition \(\mathsf{x}\cdot \mathsf{Y}\) is in minimal position with transverse crossings. Let \(\mathsf{x}\cap \mathsf{Y}\) denote the set of crossings. For any sign function \(\varepsilon: \mathsf{x} \cap \mathsf{Y} \to \{+, -\}\), the weighted crossingless multicurve \(\mathsf{E}_\varepsilon\) is obtained by applying:

• the \(+\) smoothing at each crossing \(c \in \varepsilon^{-1}(+)\) (as defined in Section 2.1);

• the \(-\) smoothing at each crossing \(c \in \varepsilon^{-1}(-)\).

The multicurve \(\mathsf{E}_\varepsilon\) admits the decomposition: \[\begin{align} \label{eq:decompose-E} \begin{aligned} \mathsf{E}_\varepsilon=&\mathbf{\Gamma}_\varepsilon \cup \mathbf{L}_\varepsilon \cup \mathbf{N}_\varepsilon \cup \mathbf{\Gamma}^0_\varepsilon\\ :=&\big(\bigcup_i {w_i} \gamma_i\big)\cup \big(\bigcup_j {w_j}\ell_j \big) \cup \big( \bigcup_k \mathsf{n}_k\big) \cup \big(\bigcup_h \gamma_h\big), \end{aligned} \end{align}\tag{3}\] where \(\gamma_i\) are pairwise non-homotopic simple arcs, \(\ell_j\) are pairwise non-homotopic simple loops, \(\mathsf{n}_k\) are unknots, and \(\gamma_h\) are contractible arcs. The exponents satisfy \(w_i, w_j > 0\), except that \(w_i\) may be negative if \(\gamma_i\) is a boundary arc. Furthermore, the components \(\gamma_i, \ell_j, \mathsf{n}_k\), and \(\gamma_h\) are pairwise disjoint.

Resolving the crossings in \(\mathsf{x}\cdot \mathsf{Y}\) via the skein relations yields: \[\label{eq:decompose-bd-intersection} [\mathsf{x}][\mathsf{Y}]=q^a \sum_{\varepsilon: \mathsf{x} \cap \mathsf{Y} \to \{+, - \}} q^{|{\varepsilon^{-1}(+)}|-|{\varepsilon^{-1}(-)}|} [\mathsf{E_{\varepsilon}}],\tag{4}\] where \(a \in \frac{1}{2}\mathbb{Z}\) is the boundary intersection number, an exponent that makes the endpoints of \(\mathsf{x}\cdot \mathsf{Y}\) simultaneous via the boundary skein relations.

For each crossing \(c\in \mathsf{x}\cap \mathsf{Y}\), let \(\mathcal{D}_c\) be a small disk neighborhood at \(c\) that contains only the two intersecting strands. The smoothing operation at \(c\) modifies \(\mathsf{x} \cdot \mathsf{Y}\) locally within \(\mathcal{D}_c\). The points resulting from the smoothing at \(c\) are called attachment points, labeled \(\varepsilon(c)c'\) and \(\varepsilon(c)c''\) respectively (see Figure 10). These points trace the original crossing locations and their neighborhoods. Thus, the choice of labels \(c'\) and \(c''\) is arbitrary.

3.2 Unknots and contractible arcs↩︎

Recall that \(\mathsf{E}_\varepsilon\) is a term appearing in the expansion of \([\mathsf{x}][\mathsf{Y}]\).

To describe the merging and splitting of components across different \(\mathsf{E}_\varepsilon\), we introduce the following notions.

Figure 12 illustrates these operations, where the hatched regions represent the disks or monogons bounded by contractible components. Note that if \(\mathsf{e}\) is a component of \(\mathsf{E}_{\varepsilon}\), then \(\mathsf{e}_1\) and \(\mathsf{e}_2\) are components of the multicurve \(\mathsf{E}_{\varepsilon'}\) obtained by switching the smoothing at \(c\).

Let \(\mathsf{e}\) be a contractible component of \(\mathsf{E}_{\varepsilon}\), and let \(D\) be the disk or monogon bounded by \(\mathsf{e}\). For ease of exposition, we recall the tubular neighborhood decomposition of \(D\) in [28]. This decomposition serves as a convenient framework; alternatively, the arguments below could be carried out by only analyzing the attachment points. Specifically, we choose a thin tubular neighborhood \(\mathcal{T}\) of \(\mathsf{x}\) satisfying:

1. \(\mathsf{Y}\) intersects \(\mathcal{T}\) a minimal number of times.

2. \(\mathcal{T}\) contains the neighborhoods \(\mathcal{D}_c\) for all crossings \(c \in \mathsf{x} \cap \mathsf{Y}\).

We define a graph \(\Gamma\) as follows:

• Vertices correspond to connected components of \(D \setminus \partial\mathcal{T}\).

• Edges connect vertices whose corresponding components share a common boundary in \(D \cap \partial\mathcal{T}\).

It follows that \(\Gamma\) is a tree ([28]). Since \(\mathsf{e} \cap \mathcal{T} \not= \emptyset\) but \(\mathsf{e} \not\subset \mathcal{T}\), the following properties hold:

1. There is at least one \(\mathcal{T}\)-intersecting component.

2. There is at least one \(\mathcal{T}\)-disjoint component.

3. \(\Gamma\) contains at least two leaves (vertices of degree \(1\)).

Each leaf component \(D_0 \subset D \setminus \partial\mathcal{T}\) is either Type A or Type B, as illustrated in Figure 14. In the figure, \(\mathcal{T}\) is shown in gray and \(D_0\) is the hatched region:

Type A

\(D_0 \subset \mathcal{T}\) and \(\partial D_0\) contains no marked points. This implies that \(\partial D_0\) contains a segment from \(\mathsf{x} \setminus \mathsf{Y}\) whose endpoints are attachment points with opposite signs. These are the only attachment points on \(\partial D_0\), and both associated angles are inward. Furthermore, in the clockwise order of \(\partial D_0\), these signs appear in the order \((+, -)\).

Type B

\(\partial D_0\) contains a marked point. This implies that \(\mathsf{e}\) is a contractible arc. When such a component exists, it is unique.

The tubular neighborhood decomposition yields the following constraint on the components of \(\mathsf{E}_\varepsilon\):

The proof of Lemma 25 also yields the following result.

Consequently, we obtain the following:

• If an unknot \(\mathsf{e}\) possesses exactly two attachment points, then \(\mathsf{x}\) and \(\mathsf{Y}\) are not in minimal position.

• If a contractible arc \(\mathsf{e}\) possesses exactly one attachment point, then \(\mathsf{x}\) and \(\mathsf{Y}\) are not in minimal position.

Building upon the definitions and properties above, we characterize the bead-chain configuration as follows:

For a weighted crossingless multicurve \(\mathsf{E}_\varepsilon\), we have: \[[\mathsf{E}_\varepsilon] = (-q^2 - q^{-2})^{|\mathbf{N}_\varepsilon|} \delta_\varepsilon [\widehat{\mathsf{E}}_\varepsilon],\] where, via the decomposition 3 :

• \(\widehat{\mathsf{E}}_\varepsilon\) is the weighted simple multicurve \(\mathbf{\Gamma}_\varepsilon\cup \mathbf{L}_\varepsilon\) obtained by removing all contractible components from \(\mathsf{E}_\varepsilon\).

• \(|\mathbf{N}_\varepsilon|=| \{ \text{unknot components in} \mathsf{E}_\varepsilon \} |\)

• \(\delta_\varepsilon\) is the indicator function: \[\delta_\varepsilon := \delta_{|\mathbf{\Gamma}^0_\varepsilon|, 0} = \begin{cases} 1 & \text{if } \mathsf{E}_\varepsilon \text{ contains no contractible arcs}, \\ 0 & \text{if } \mathsf{E}_\varepsilon \text{ contains a contractible arc}. \end{cases}\]

Therefore, the superposition product expands to: \[\label{eq:detail-dec} [\mathsf{x}][\mathsf{Y}] =q^a \sum_{\substack{\varepsilon: \mathsf{x} \cap \mathsf{Y} \to \{+, -\}}} q^{|\varepsilon^{-1}(+)| - |\varepsilon^{-1}(-)|} (-q^2 - q^{-2})^{|\mathbf{N}_\varepsilon|} \delta_\varepsilon [\widehat{\mathsf{E}}_\varepsilon].\tag{5}\]

Let \(\varepsilon_\pm\) denote the constant sign assignment mapping every crossing to \(\pm\). For any \(\mathsf{E}_\varepsilon\) containing no contractible arcs, Lemma 24 yields the inequality: \[\begin{align} \label{eq:q-deg-inequality} |\varepsilon^{-1}(+)| - |\varepsilon^{-1}(-)| + 2|\mathbf{N}_\varepsilon| \leq |\varepsilon^{-1}(+)| + |\varepsilon^{-1}(-)| = |\varepsilon_+^{-1}(+)|. \end{align}\tag{6}\] The right-hand side \(|\varepsilon_+^{-1}(+)|\) represents the total number of crossings \(|\mathsf{x} \cap \mathsf{Y}|\). We say \(\mathsf{E}_\varepsilon\) contributes to the top \(q\)-degree term if equality holds in 6 , which is equivalent to the condition: \[\label{eq:q-deg-top} |\mathbf{N}_\varepsilon| = |\varepsilon^{-1}(-)|.\tag{7}\]

We refer to the set \(\mathbf{N}_\varepsilon\) of unknot components as the bead-chain configuration of \(\mathsf{E}_\varepsilon\). Let \(\mathcal{E}_{\mathrm{top}}\) denote the set of sign assignments \(\varepsilon\) for which \([\mathsf{E}_\varepsilon]\) contributes to the top \(q\)-degree term in 5 . We define \[\mathbf{N}_{\mathrm{top}} := \bigsqcup_{\varepsilon \in \mathcal{E}_{\mathrm{top}}} \mathbf{N}_\varepsilon,\] and make the following observations:

For any curve component \(\gamma \subset \mathsf{E}_{\varepsilon}\), let \(\mathcal{C}_{\gamma} \subset \mathsf{x} \cap \mathsf{Y}\) denote the set of crossings whose smoothing provides the attachment points on \(\gamma\). For a sign function \(\varepsilon \in \mathcal{E}_{\mathrm{top}}\) and a chain \(\mathsf{n}\in \mathbf{N}_\varepsilon\), we capture its combinatorial skeleton via the generating set \(\mathcal{G}_\mathsf{n} \subseteq \mathcal{C}_\mathsf{n}\): \[\mathcal{G}_\mathsf{n} := \mathrm{Str}_\mathsf{n} \sqcup \mathcal{C}^-_\mathsf{n}.\] Here, \(\mathrm{Str}_\mathsf{n}\) denotes the subset of crossings where \(\mathsf{n}\) admits a strip (see Figure 12). The subset \(\mathcal{C}^-_\mathsf{n}:=\varepsilon^{-1}(-)\cap \mathcal{C}_\mathsf{n}\) consists of exactly two negative attachment points of \(\mathsf{n}\) (see 8 ), which consists of a single crossing if \(\mathsf{n}\) is cyclic, and two crossings if \(\mathsf{n}\) is linear. Globally, we define: \[\begin{align} \label{eq:generating95set} \mathrm{Str}_\varepsilon &:= \bigcup_{\mathsf{n}\in \mathbf{N}_\varepsilon} \mathrm{Str}_\mathsf{n}, \quad \text{and} \quad \mathcal{G}_\varepsilon := \bigcup_{\mathsf{n}\in \mathbf{N}_\varepsilon} \mathcal{G}_\mathsf{n} = \mathrm{Str}_\varepsilon \cup \varepsilon^{-1}(-). \end{align}\tag{9}\] The set \(\mathcal{G}_\varepsilon\) is invariant under strip operations. Removing all strips from \(\mathbf{N}_\varepsilon\) yields a pure bead configuration \(\mathbf{N}_{\varepsilon'}\) (Lemma 30(2)), whose induced sign function \(\varepsilon'\) satisfies: \[(\varepsilon')^{-1}(-) = \mathcal{G}_{\varepsilon}.\]

On the other hand, for \(\varepsilon_+\), let \(\mathcal{L}_{\varepsilon_+}\) be a maximal collection of pairwise non-homotopic simple loops in \(\mathsf{E}_{\varepsilon_+}\). For each \(\ell \in \mathcal{L}_{\varepsilon_+}\), let \(\mathcal{H}_\ell\) denote the set of simple loops in \(\mathsf{E}_{\varepsilon_+}\) freely homotopic to \(\ell\). We define its associated crossing set as: \[\mathcal{C}({\mathbb{A}_\ell}) := \bigcup_{\gamma \in \mathcal{H}_\ell} \mathcal{C}_{\gamma}.\] The transitions between bead-chain configurations via strip operations induce a regular neighborhood system governed by these homotopy classes, which provides the topological foundation for localizing the proof of Theorem 3:

Via an ambient isotopy, the superposition \(\mathsf{x} \cdot \mathsf{Y}\) is assumed to be in a position where all crossings are contained within \(\bigcup \mathcal{D}_c \subset \bigcup \mathbb{A}_\ell\), and the complement \((\mathsf{x} \cdot \mathsf{Y}) \setminus \bigcup \mathbb{A}_\ell\) contains no segments that can be homotoped into \(\bigcup \mathbb{A}_\ell\) relative to their endpoints. Such a configuration is obtained by translating any crossing outside the neighborhoods along \(\mathsf{x}\) into the nearest \(\mathbb{A}_\ell\), and retracting all such redundant segments.

Let \(\mathcal{G}(\mathbb{A}_{\ell})\) denote the set of crossings where both strands resulting from the \(+\) smoothing belong to loops homotopic to \(\ell\). We refer to \(\mathcal{G}(\mathbb{A}_{\ell})\) as the generating set in \(\mathbb{A}_{\ell}\), or simply the generating set of type \(\ell\), denoted by \(\mathcal{G}(\ell)\). By definition, these sets are mutually disjoint: \(\mathcal{G}(\mathbb{A}_{\ell}) \cap \mathcal{G}(\mathbb{A}_{\ell'}) = \emptyset\) whenever \(\ell\) and \(\ell'\) are non-homotopic.

For any \(\varepsilon \in\mathcal{E}_{\mathrm{top}}\), attaching strips transforms the configuration \(\mathbf{N}_\varepsilon\) into a new configuration \(\mathbf{N}_{{\tilde{\varepsilon}}}\) consisting entirely of cyclic chains (see Lemma 30(2)). Recall that the global generating set is invariant under strip operations, yielding \(\mathcal{G}_\varepsilon = \mathcal{G}_{{\tilde{\varepsilon}}}\). As established previously, we naturally have \(\varepsilon^{-1}(-) \subseteq \mathcal{G}_\varepsilon\). Moreover, Lemma 32 guarantees that \(\mathcal{G}_{{\tilde{\varepsilon}}} \subseteq \bigcup_{\ell\in \mathcal{L}_{\varepsilon_+}} \mathcal{G}(\mathbb{A}_{\ell})\). We thus have the following observations:

3.3 Marked annulus↩︎

Let \(\mathbb{A}_{m,m}\) denote an annulus with \(m\) marked points on each boundary component, where \(m\geq 1\). We first establish our main result in the local model \(\mathbb{A}_{m,m}\), beginning with the base case \(m=1\).

3.4 General cases↩︎

Fix an ideal triangulation \(\Delta\) of a marked surface \(\Sigma\). Let \(\mathsf{x}\) be a simple arc such that \([\mathsf{x}]\) is an initial cluster variable in the cluster algebra associated to \(\Delta\), and let \(\mathsf{Y}\) be a multicurve with \([\mathsf{Y}] \neq 0\) in \(\mathrm{Sk}_q^{\circ}(\Sigma)\).

Based on the neighborhood system \(\{\mathbb{A}_\ell\}\) from Lemma 31, locality of skein relations allows us to restrict the top \(q\)-degree terms in the expansion of \([\mathsf{x}] \langle \mathsf{Y} \rangle\) to these regions. In each \(\mathbb{A}_\ell\), the configuration is locally equivalent to the \(\mathbb{A}_{m,m}\) case, where \(m\) is the number of arcs traversing the neighborhood (see Proposition 36). Summing terms that are identical outside the neighborhood recovers a band element \(\mathrm{Band}^{w_\ell}(\ell)\). Iterating this localization, the leading term of the global expansion is \([\mathsf{F}] \prod_{\ell} \mathrm{Band}^{w_\ell}(\ell),\) where \(\mathsf{F}\) is the remaining arc component of \(\mathsf{E}_+\). Since all other terms have strictly lower \(q\)-degrees, this product is the leading term of the triangular basis.

By Theorem 38, all conclusions of Proposition 36 generalize to arbitrary unpunctured surfaces.

4 Nested contractible sequence↩︎

We provide a refinement of Proposition 26 concerning contractible components. Although this description is not required for the main results of this paper, it may be of independent interest.

[35]

References↩︎