arxiv: 2605.12114 · v1 · submitted 2026-05-12 · 🧮 math.QA

Recognition: 2 theorem links

· Lean Theorem

Quantum cluster algebra realization for stated {rm SL}_n-skein algebras and rotation-invariant bases for polygons

Min Huang, Peigen Cao, Zhihao Wang

Pith reviewed 2026-05-13 03:44 UTC · model grok-4.3

classification 🧮 math.QA

keywords stated skein algebraquantum cluster algebraupper cluster algebrarotation-invariant basistheta basispolygonSL_n skeinlocalization

0 comments

The pith

For polygons, the localized stated SL_n-skein algebra equals both the quantum cluster algebra and the quantum upper cluster algebra built from its fraction field.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when the surface is a polygon, localizing the stated SL_n-skein algebra at the frozen variables makes it coincide with the quantum cluster algebra and the quantum upper cluster algebra associated to its skew-field of fractions. The same holds for the projected version of the skein algebra. A reader would care because this supplies a concrete identification between skein algebras, which encode quantum topology and representations, and cluster algebras, which carry canonical bases and positivity. As a direct result the theta basis becomes a rotation-invariant basis for the skein algebra that is positive and naturally parametrized.

Core claim

We prove that for a polygon S, the localized stated skein algebra equals the quantum cluster algebra and upper cluster algebra: S_ω^fr(S) = A_ω^fr(S) = U_ω^fr(S), and likewise for the projected version. This identification is obtained by constructing a quantum cluster structure on the skew-field of fractions of the stated SL_n-skein algebra for triangulable surfaces without interior punctures and then verifying the equality specifically on polygons. Consequently the theta basis of the upper cluster algebra yields a rotation-invariant basis of the skein algebra possessing positivity and a natural parametrization.

What carries the argument

The quantum cluster algebra and upper cluster algebra associated to the skew-field of fractions of the stated SL_n-skein algebra, realized via a compatible triangulation of the polygon.

If this is right

The theta basis of the upper cluster algebra supplies a rotation-invariant basis for the skein algebra on any polygon.
This basis is positive and carries a natural parametrization by cluster variables.
The same identifications hold after projection, giving analogous bases for the projected skein algebra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The polygon case may serve as a building block for gluing arguments that extend the equality to surfaces with more boundary components.
Explicit low-rank calculations on triangles or quadrilaterals could produce closed-form expressions for the rotation-invariant bases.
The positivity inherited from the cluster side suggests new ways to compare skein relations with cluster mutations in higher rank.

Load-bearing premise

The surface is restricted to a polygon that admits a triangulation compatible with the stated skein construction, and the argument relies on prior results for the projected case.

What would settle it

An explicit basis computation for the stated skein algebra on a quadrilateral with small n that produces more or fewer elements than the corresponding cluster algebra would falsify the claimed equality.

Figures

Figures reproduced from arXiv: 2605.12114 by Min Huang, Peigen Cao, Zhihao Wang.

**Figure 1.** Figure 1: The left is C(p)ij and the right is C(p)ij . is called the reduced stated SLn-skein algebra, defined in [LY23], where I bad is the two-sided ideal of Sω(S) generated by all bad arcs. 2.2. The bigon and Oq(SLn). Here we refer to [KS12, LS24, LY23] for the definition of Oq(SLn). We call P2 the bigon. We can label the two boundary components of the bigon by el and er. A bigon with this labeling is called a di… view at source ↗

**Figure 2.** Figure 2: Barycentric coordinates ijk and a 4-triangulation with its quiver. 2.4. The n-triangulation. Consider barycentric coordinates for an ideal triangle P3 so that P3 = {(i, j, k) ∈ R 3 | i, j, k ≥ 0, i + j + k = n} \ {(0, 0, n),(0, n, 0),(n, 0, 0)}, where (i, j, k) (or ijk for simplicity) are the barycentric coordinates. Let v1 = (n, 0, 0), v2 = (0, n, 0), v3 = (0, 0, n). Let ei denote the edge on ∂P3 whose en… view at source ↗

**Figure 3.** Figure 3: Left: Weighted graph Yv, Middle: Elongation Y ‹v, Right: Turning left Step (iii): Elongate the nonzero-weighted edges of Yv to have an embedded weighted directed graph Y ‹v as drawn in the middle of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: (A) Attaching triangles. (B) The triangulation for P ∗ 2 = P4. Define the matrix C : V ′ λ × V λ∗ → Z by C(v, v) = 1, v ∈ V ′ λ , C(v, p(v)) = −1, v ∈ V ′ λ \ V λ, C(v, v′ ) = 0, otherwise. (38) The extended matrix Pλ : V ′ λ × V ′ λ → Z is then given by Pλ := C P λ∗ C T . (39) Define the extended A-version quantum tori associated to (S, λ) by Aω(S, λ) = R⟨A ±1 v , v ∈ V ′ λ ⟩ . AvAv ′ = ω Pλ(v,v′ )Av ′Av,… view at source ↗

**Figure 5.** Figure 5: (A) The triangle τ with edges labeled e1, e2, e3, and a distinguished vertex labeled v1 (note that e1 ̸= e3). (B) The labeling of the small vertices in V λ ∩ τ for n = 4. (C) An alternative labeling of the small vertices in V λ ∩ τ for n = 4 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗

**Figure 6.** Figure 6: (A) The picture for an essential arc, in red. (B) The labeling for small vertices contained in the quadrilateral bounded by c1 ∪ c2 ∪ c3 ∪ c4 for n = 4. Remark 3.10. Our S(S), A ω(S), and U ω(S) correspond respectively to S(S), Aω(S), and Uω(S) in [HW25]. Remark 3.11. In [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗

**Figure 7.** Figure 7: Labeling of the small vertices contained in P4,e when n = 4, where e is the diagonal. V i λ,e := G 0≤s≤i V i,s λ,e ⊂ Vλ,e. (73) Each V i λ,e forms a grid of vertices inside a quadrilateral, containing (i + 1)(n − i − 1) points. Note that these sets for different i are not necessarily disjoint. The mutation sequence of Fock, Goncharov, and Shen proceeds as follows: first mutate at all vertices of V 0 λ,e in… view at source ↗

**Figure 8.** Figure 8: Labeling of small vertices contained in e2 in an attached triangle. We use ∂λ to denote the set of boundary edges of λ. Recall that for each e ∈ ∂λ, there is an attached triangle τ in S∗ (see [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗

**Figure 9.** Figure 9: The full subquiver consisting of v and vertices adjacent to v. When i = 1: Note that µ1(µ0(Av)) = µv(µ0(Av)). The full subquiver of µ0(Γλ4 ), consisting of v and vertices adjacent to v, is as shown in [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗

**Figure 10.** Figure 10: (A) The picture for an essential arc, in red. (B) The labeling for small vertices contained in the quadrilateral bounded by c1 ∪ c2 ∪ c3 ∪ c4 for n = 4. 4.4. Sω(S) = Aω(S) = Uω(S) when n = 2. Let A be an R-algebra. A prime element of A is a nonzero, nonunit element a ∈ A such that aA = Aa and A/(a) is a domain. Here (a) := Aa = aA denotes the principal ideal generated by a. Let S be a triangulable pb surf… view at source ↗

**Figure 11.** Figure 11: Labels of the vertices inside the triangle (i, i+1, i+2) in λi when n = 4. For any integers j, s with 1 < s ≤ j ≤ n − 1, define µe ♢ (j,s) = (µn−1,s−1 · · · µj+1,s−1µj,s−1)· · ·(µn−s+2,s−1 · · · µj−s+4,2µj−s+3,2)(µn−s+1,1 · · · µj−s+3,1µj−s+2,1). (116) See [PITH_FULL_IMAGE:figures/full_fig_p044_11.png] view at source ↗

**Figure 12.** Figure 12: A pictorial illustration of µe ♢ (j,s) Lemma 5.8. For any integers j, s with 1 ≤ s ≤ j ≤ n − 1 and any i with 2 ≤ i ≤ k, the following identities hold [PITH_FULL_IMAGE:figures/full_fig_p044_12.png] view at source ↗

**Figure 13.** Figure 13: A pictorial illustration of D a,a+1 ij . For a vertex jsi ∈ I(λ), we introduce the following notations: • E qc(jsi ): the cluster variable in the seed ← µ≺jsi (s qc λ ) of A qc ω (Pk+2) located at the vertex j1 1 ∈ I(λ). • E(jsi ): the cluster variable in the seed ← µ≺jsi (sλ) of A ω(Pk+2) located at the vertex j1 1 ∈ I(λ). • D(jsi ): the stated essential arc in S ω(Pk+2) defined by D(jsi ) := D (a,b) (p,… view at source ↗

**Figure 14.** Figure 14: The labeling of V λ when n = 3. description of the cluster variables (see (153)-(166)) and clusters (see Theorem 6.3) for Aω(P2). Using these results, we provide a web interpretation of the dual canonical basis of Aω(P2) = Oq(SL3) (see Theorem 6.18). We refer to [LS25] for another construction of a web interpretation of this dual canonical basis. 6.1. Cluster structure of A ω(P4) = S ω(P4) when n = 3. Rec… view at source ↗

**Figure 15.** Figure 15: Three labeled points on each boundary component of P2. Definition 6.2. Two labeled arcs C and C ′ are called compatible if one of the following three conditions, (1), (2), or (3), holds. They are called strongly compatible if one of the conditions (1), (2), or (3′ ) holds. (1) C and C ′ do not intersect, which corresponds to the condition (l(C) − l(C ′ ))(r(C) − r(C ′ )) < 0. (2) C and C ′ intersect on th… view at source ↗

**Figure 16.** Figure 16: The local configuration appearing in diagrams in Jk,t, where i, j ∈ {1, 2, 3} and the orientation of the web diagram is arbitrary. Lemma 6.7. Let k be a positive integer and 0 ≤ t ≤ k − 1. For any W ∈ Jk,t, we have W = 0 ∈ Sω(P2) [PITH_FULL_IMAGE:figures/full_fig_p056_16.png] view at source ↗

**Figure 17.** Figure 17: (B) for each i = 1, 2, 3. Denote by ∂i the set of endpoints of Se obtained from this procedure. • Finally, obtain the stated web W(S) from Se by assigning state i to each endpoint in ∂i , for each i = 1, 2, 3. The height ordering of W(S) along each component of ∂P2 is indicated by the two arrows in . (A) i• . . . •i . . . −→ i• . . . •i . . . (B) [PITH_FULL_IMAGE:figures/full_fig_p057_17.png] view at source ↗

**Figure 18.** Figure 18: Wk,t when 2 ≤ t ≤ k − 1. Lemma 6.9. For each 1 ≤ t ≤ k, we have Wk,t = Wk,t+1 ∈ Sω(P2). Proof. A direct calculation using Lemmas 6.4-6.6 gives Wk,1 = Wk,2. Lemma 6.4(b) implies = + q 4 3 (169) = q − 4 3 + . Next, we simplify each term. We have = = q − 1 3 , (170) where the first equality follows from repeated applications of Lemmas 6.5 and 6.7, and the second equality follows from (20) [PITH_FULL_IMAGE:f… view at source ↗

**Figure 19.** Figure 19: The bad arc Cp in (A), and the stated arcs Cp(+) and Cp(−) in (B) and (C), respectively. Lemma B.2. [LY22, Lemma 4.4] (a) Cp Sω(S) = Sω(S) Cp. (b) Let α ∈ B(S). In Sω(S), we have α Cp ω= Cp α ω= α ⊔ Cp. We have the following relation in Sω(S)/(Cp), where (Cp) = Cp Sω(S) = Sω(S) Cp. Lemma B.3 ([CL22]). In Sω(S)/(Cp), we have Cp(+)Cp(−) = Cp(−)Cp(+) = 1, where Cp(+) and Cp(−) are stated arcs illustrated in … view at source ↗

**Figure 20.** Figure 20: Left: the boundary edge e and the corner arc D(e). Middle: the bad arc D(e)(+, −). Right: the bad arc D(e)(+, −) when the two endpoints of e coincide. Remark B.6. Let 0 ̸= x ∈ SI ω(S). We want to show that Θe(x) ̸= 0. Since BI (S) is an R-basis, there exists a nonempty finite subset S ⊂ BI (S) such that x = X α∈S cαα, 0 ̸= cα ∈ R. Let t = maxα∈S i(α, e), where i(α, e) denotes the algebraic intersection nu… view at source ↗

read the original abstract

We construct a quantum cluster structure on the skew-field of fractions ${\rm Frac}({\mathscr S}_\omega(\mathfrak{S}))$ of the stated ${\rm SL}_n$-skein algebra ${\mathscr S}_\omega(\mathfrak{S})$, where $\mathfrak{S}$ is a triangulable pb surface without interior punctures. This work complements the construction for the projected stated skein algebra $\widetilde{\mathscr S}_\omega(\mathfrak{S})$ given by the last two authors. Let ${\mathscr S}_\omega^{\rm fr}(\mathfrak{S})$ denote the localization of ${\mathscr S}_\omega(\mathfrak{S})$ at the multiplicative set generated by all frozen variables. Let ${\mathscr A}_\omega^{\rm fr}(\mathfrak{S})$ and ${\mathscr U}_\omega^{\rm fr}(\mathfrak{S})$ (respectively $\overline{\mathscr A}_\omega(\mathfrak{S})$ and $\overline{\mathscr U}_\omega(\mathfrak{S})$) denote the quantum cluster algebra and quantum upper cluster algebra associated to ${\rm Frac}({\mathscr S}_\omega(\mathfrak{S}))$ (respectively ${\rm Frac}(\widetilde{\mathscr S}_\omega(\mathfrak{S}))$). We prove that \[ \widetilde{\mathscr S}_\omega(\mathfrak{S}) = \overline{\mathscr A}_\omega(\mathfrak{S}) = \overline{\mathscr U}_\omega(\mathfrak{S}) \quad \text{and} \quad {\mathscr S}_\omega^{\rm fr}(\mathfrak{S}) = {\mathscr A}_\omega^{\rm fr}(\mathfrak{S}) = {\mathscr U}_\omega^{\rm fr}(\mathfrak{S}) \] whenever $\mathfrak{S}$ is a polygon. As a consequence, when $\mathfrak{S}$ is a polygon, we show that the theta basis of $\overline{\mathscr U}_\omega(\mathfrak{S})$ (respectively ${\mathscr U}_\omega^{\rm fr}(\mathfrak{S})$) yields a rotation-invariant basis of $\overline{\mathscr S}_\omega(\mathfrak{S})$ (respectively ${\mathscr S}_\omega^{\rm fr}(\mathfrak{S})$) with several desirable properties, including positivity and a natural parametrization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the localized stated SL_n-skein algebra on a polygon equals its quantum cluster algebra and upper cluster algebra, yielding rotation-invariant theta bases.

read the letter

The central result is that for a polygon the localized stated SL_n-skein algebra equals both the quantum cluster algebra and the upper cluster algebra, and the same holds for the projected version. This produces rotation-invariant bases with positivity and a parametrization coming from the theta basis construction. The work extends the authors' earlier results on the projected case to the full stated algebra via a triangulation of the surface and standard quantum cluster machinery applied to the fraction field. The equalities are stated cleanly and the rotation invariance is checked explicitly for polygons, which is a concrete payoff. The proofs rely on prior constructions for the projected algebras and on the usual Ore localization and cluster algebra inclusions, so the new technical step is adapting those to the stated setting without introducing extra relations. The limitation is the narrow scope: everything is restricted to polygons without interior punctures, so the result does not immediately apply to more general surfaces. The triangulation choices and verification of the basis properties are plausible but would need line-by-line checking in the full text to confirm there are no hidden dependencies on the choice of triangulation. No circularity or missing conditions appear in the stated argument. This paper is for researchers already working on quantum skein algebras, cluster structures in quantum topology, or related representation theory. A reader looking for explicit bases or cluster realizations in this area would get direct value from the constructions and the invariance statement. It deserves a serious referee because the claims are specific, the methods are standard in the field, and the extension from projected to stated is a natural next step even if the generality is limited.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs a quantum cluster structure on the skew-field of fractions of the stated SL_n-skein algebra for triangulable pb surfaces without interior punctures, complementing prior work on the projected case. For polygons, it proves the equalities S_ω^fr(S) = A_ω^fr(S) = U_ω^fr(S) and the analogous statement for the projected skein algebra. As a consequence, the theta basis of the upper cluster algebra yields a rotation-invariant basis of the (localized) stated skein algebra with positivity and a natural parametrization.

Significance. If the identifications hold, the work furnishes an explicit quantum cluster algebra realization of stated SL_n-skein algebras on polygons and supplies rotation-invariant theta bases with positivity. This links skein-algebra techniques to the machinery of quantum cluster algebras and theta bases, extending the authors' earlier results on projected skein algebras and offering concrete bases that are useful for representation-theoretic and topological applications.

minor comments (3)

The abstract introduces a large number of symbols (S_ω^fr, A_ω^fr, U_ω^fr, etc.) without a brief parenthetical reminder of their meanings; a short glossary sentence would aid readability.
Notation for the localization at frozen variables is introduced in the abstract but the precise multiplicative set is not restated in the introduction; repeating the definition once more would prevent confusion for readers who skip the abstract.
The manuscript relies on prior results for the projected skein algebra; a one-sentence pointer to the exact theorem numbers from that work in the introduction would make the logical dependence explicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies the construction of the quantum cluster structure on the skew-field of fractions of the stated SL_n-skein algebra for triangulable pb surfaces without interior punctures, the equalities for polygons, and the resulting rotation-invariant theta bases with positivity. We appreciate the recommendation for minor revision and the recognition of how this work complements our prior results on projected skein algebras while linking to quantum cluster algebra techniques.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs a quantum cluster structure directly on Frac(S_ω(S)) for triangulable pb surfaces, then proves the equalities S_fr = A_fr = U_fr (and the projected analogs) specifically when S is a polygon by invoking skein relations together with standard quantum cluster algebra machinery. The reference to prior work by the last two authors applies only to the complementary construction of the cluster structure on the projected version; the equalities themselves are established in this manuscript and do not reduce to that citation by definition or by construction. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing uniqueness theorems imported from overlapping-author citations appear in the stated argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions and theorems from quantum cluster algebra theory and skein algebra literature; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Standard axioms and mutation rules of quantum cluster algebras
Invoked implicitly when associating A and U to the fraction field of the skein algebra.
domain assumption Existence of triangulations for pb surfaces without interior punctures
Used to define the stated skein algebra and its localization.

pith-pipeline@v0.9.0 · 5696 in / 1287 out tokens · 102628 ms · 2026-05-13T03:44:08.451867+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We prove that ... S_ω^fr(S) = A_ω^fr(S) = U_ω^fr(S) whenever S is a polygon ... theta basis ... rotation-invariant
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
quantum seed s_λ = (Q_λ, Π_λ, M_λ) ... mutation class ... cluster variables from stated essential arcs

Reference graph

Works this paper leans on

160 extracted references · 160 canonical work pages

[1]

Galashin, Pavel and Lam, Thomas and Sherman-Bennett, Melissa , TITLE =. Invent. Math. , FJOURNAL =. 2026 , NUMBER =. doi:10.1007/s00222-025-01390-5 , URL =

work page doi:10.1007/s00222-025-01390-5 2026
[2]

arXiv preprint arXiv:2309.10920 , year=

Finiteness and dimension of stated skein modules over Frobenius , author=. arXiv preprint arXiv:2309.10920 , year=

work page arXiv
[3]

2000 , publisher=

Foundations of quantum group theory , author=. 2000 , publisher=

work page 2000
[4]

Huang, Min and Wang, Zhihao , journal=

work page
[5]

Chang, Wen and Zhu, Bin , TITLE =. J. Algebra , FJOURNAL =. 2016 , PAGES =

work page 2016
[6]

Huang, Min and Li, Fang and Yang, Yichao , TITLE =. Sci. China Math. , FJOURNAL =. 2018 , NUMBER =

work page 2018
[7]

Fu, Changjian and Keller, Bernhard , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2010 , NUMBER =

work page 2010
[8]

Chang, Wen and Huang, Min and Li, Jian-Rong , TITLE =. J. Algebra , FJOURNAL =. 2024 , PAGES =. doi:10.1016/j.jalgebra.2023.09.036 , URL =

work page doi:10.1016/j.jalgebra.2023.09.036 2024
[9]

2025 , publisher=

Wang, Zhihao , journal=. 2025 , publisher=

work page 2025
[10]

Wang, Zhihao , journal=

work page
[11]

2020 , pages=

Rhoades, Brendon , journal=. 2020 , pages=

work page 2020
[12]

Forum Math

Shen, Linhui and Weng, Daping , TITLE =. Forum Math. Sigma , FJOURNAL =. 2021 , PAGES =

work page 2021
[13]

SIGMA Symmetry Integrability Geom

Kimura, Yoshiyuki and Qin, Fan and Wei, Qiaoling , TITLE =. SIGMA Symmetry Integrability Geom. Methods Appl. , FJOURNAL =. 2023 , PAGES =. doi:10.3842/SIGMA.2023.105 , URL =

work page doi:10.3842/sigma.2023.105 2023
[14]

Fomin, Sergey and Zelevinsky, Andrei , TITLE =. Compos. Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1112/S0010437X06002521 , URL =

work page doi:10.1112/s0010437x06002521 2007
[15]

Kim, Hyun Kyu and Wang, Zhihao , journal=

work page
[16]

Huang, Min , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2023 , NUMBER =

work page 2023
[17]

Selecta Math

Huang, Min , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2022 , NUMBER =

work page 2022
[18]

Rupel, Dylan , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2011 , NUMBER =

work page 2011
[19]

arXiv preprint arXiv:2503.12319 , year=

Cluster algebras and skein algebras for surfaces , author=. arXiv preprint arXiv:2503.12319 , year=

work page arXiv
[20]

An expansion formula for type. J. Combin. Theory Ser. A , FJOURNAL =. 2020 , PAGES =

work page 2020
[21]

arXiv preprint arXiv:2406.03362 , year=

Positivity for quantum cluster algebras from orbifolds , author=. arXiv preprint arXiv:2406.03362 , year=

work page arXiv
[22]

Schrader, Gus and Shapiro, Alexander , journal=

work page
[23]

arXiv preprint arXiv:2310.06189 , year=

Frohman, Charles and Kania-Bartoszynska, Joanna and L. arXiv preprint arXiv:2310.06189 , year=

work page arXiv
[24]

Muller, Greg , journal=

work page
[25]

Transactions of the American Mathematical Society , volume=

Degenerations of skein algebras and quantum traces , author=. Transactions of the American Mathematical Society , volume=

work page
[26]

2023 , publisher=

Ishibashi, Tsukasa and Yuasa, Wataru , journal=. 2023 , publisher=

work page 2023
[27]

2018 , publisher=

Frohman, Charles and Kania-Bartoszynska, Joanna , journal=. 2018 , publisher=

work page 2018
[28]

Mathematische Zeitschrift , volume=

Hoste, Jim and Przytycki, J. Mathematische Zeitschrift , volume=. 1995 , publisher=

work page 1995
[29]

arXiv preprint arXiv:2310.13116 , year=

On Frobenius algebras obtained from stated skein algebras , author=. arXiv preprint arXiv:2310.13116 , year=

work page arXiv
[30]

2016 , publisher=

Frohman, Charles and Abdiel, Nel , journal=. 2016 , publisher=

work page 2016
[31]

arXiv preprint arXiv:2211.13700 , year=

Azumaya loci of skein algebras , author=. arXiv preprint arXiv:2211.13700 , year=

work page arXiv
[32]

arXiv preprint arXiv:2309.14713 , year=

Center of the stated skein algebra , author=. arXiv preprint arXiv:2309.14713 , year=

work page arXiv
[33]

Transactions of the American Mathematical Society , volume=

Skein modules and the noncommutative torus , author=. Transactions of the American Mathematical Society , volume=

work page
[34]

arXiv preprint arXiv:2303.09433 , year=

Classification of semi-weight representations of reduced stated skein algebras , author=. arXiv preprint arXiv:2303.09433 , year=

work page arXiv
[35]

2022 , publisher=

Karpenkov, Oleg N , booktitle=. 2022 , publisher=

work page 2022
[36]

Journal of algebra , volume=

On vanishing sums of roots of unity , author=. Journal of algebra , volume=. 2000 , publisher=

work page 2000
[37]

The American Mathematical Monthly , volume=

How small can a sum of roots of unity be? , author=. The American Mathematical Monthly , volume=. 1986 , publisher=

work page 1986
[38]

Matthew Conder , title=

work page
[39]

Murakami, Hitoshi and Ohtsuki, Tomotada and Okada, Masae , year=

work page
[40]

2011 , publisher=

A primer on mapping class groups (pms-49) , author=. 2011 , publisher=

work page 2011
[41]

Takenov, Nurdin , journal=

work page
[42]

Bullock, Doug and Przytycki, Jozef , journal=

work page
[43]

2009 , publisher=

Liu, Xiaobo , journal=. 2009 , publisher=

work page 2009
[44]

Bulletin of the Polish Academy of Sciences; Mathematics , volume=

Skein modules of 3-manifolds , author=. Bulletin of the Polish Academy of Sciences; Mathematics , volume=

work page
[45]

Fock, Vladimir and Goncharov, Alexander , journal=

work page
[46]

Selecta Mathematica , volume=

Quantum traces and embeddings of stated skein algebras into quantum tori , author=. Selecta Mathematica , volume=. 2022 , publisher=

work page 2022
[47]

Journal of Knot Theory and Its Ramifications , volume=

Hoste, Jim and Przytycki, J. Journal of Knot Theory and Its Ramifications , volume=. 1993 , publisher=

work page 1993
[48]

Topology , volume =

Przytycki, J. Topology , volume =. 2000 , issn =

work page 2000
[49]

Saito, Toshio and Scharlemann, Martin and Schultens, Jennifer , journal=

work page
[50]

Journal of the London Mathematical Society , volume=

Infinite families of hyperbolic 3-manifolds with finite-dimensional skein modules , author=. Journal of the London Mathematical Society , volume=. 2021 , publisher=

work page 2021
[51]

Advances in mathematics , volume=

L. Advances in mathematics , volume=. 2006 , publisher=

work page 2006
[52]

Proceedings of the American Mathematical Society , volume=

L. Proceedings of the American Mathematical Society , volume=

work page
[53]

2005 , publisher=

Bullock, Doug and Faro, Walter Lo , journal=. 2005 , publisher=

work page 2005
[54]

Talk by Joanna Kania-Bartoszynska at Quantum Topology and Geometry conference , year=

Frohman, Charles and Kania-Bartoszynska, Joanna and L. Talk by Joanna Kania-Bartoszynska at Quantum Topology and Geometry conference , year=

work page
[55]

Karuo, Hiroaki and Wang, Zhihao , journal=

work page
[56]

2007 , publisher=

Bonahon, Francis and Liu, Xiaobo , journal=. 2007 , publisher=

work page 2007
[57]

Quantum Topology , volume=

Triangular decomposition of skein algebras , author=. Quantum Topology , volume=

work page
[58]

Journal of knot theory and its ramifications , volume=

Bullock, Doug and Frohman, Charles and Kania-Bartoszy. Journal of knot theory and its ramifications , volume=. 1999 , publisher=

work page 1999
[59]

Fomin, Sergey and Zelevinsky, Andrei , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2002 , NUMBER =

work page 2002
[60]

Duke Math

Berenstein, Arkady and Fomin, Sergey and Zelevinsky, Andrei , TITLE =. Duke Math. J. , FJOURNAL =. 2005 , NUMBER =

work page 2005
[61]

Gross, Mark and Hacking, Paul and Keel, Sean and Kontsevich, Maxim , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =

work page 2018
[62]

Lee, Kyungyong and Schiffler, Ralf , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2015 , NUMBER =

work page 2015
[63]

Casals, Roger and Gorsky, Eugene and Gorsky, Mikhail and Le, Ian and Shen, Linhui and Simental, Jos\'e , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2025 , NUMBER =

work page 2025
[64]

Quantum Topol

Muller, Greg , TITLE =. Quantum Topol. , FJOURNAL =. 2016 , NUMBER =

work page 2016
[65]

SIGMA Symmetry Integrability Geom

Muller, Greg , TITLE =. SIGMA Symmetry Integrability Geom. Methods Appl. , FJOURNAL =. 2014 , PAGES =

work page 2014
[66]

Geiss, Christof and Leclerc, Bernard and Schr\"oer, Jan , TITLE =. J. Algebra , FJOURNAL =. 2020 , PAGES =

work page 2020
[67]

Geiss, Christof and Leclerc, Bernard and Schr\"oer, Jan , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2008 , NUMBER =

work page 2008
[68]

, TITLE =

Scott, Joshua S. , TITLE =. Proc. London Math. Soc. (3) , FJOURNAL =. 2006 , NUMBER =

work page 2006
[69]

and Sherman-Bennett, M

Serhiyenko, K. and Sherman-Bennett, M. and Williams, L. , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2019 , NUMBER =

work page 2019
[70]

Ishibashi, Tsukasa and Yuasa, Wataru , TITLE =. Adv. Math. , FJOURNAL =. 2025 , PAGES =

work page 2025
[71]

Galashin, Pavel and Lam, Thomas , TITLE =. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , FJOURNAL =. 2023 , NUMBER =

work page 2023
[72]

Bucher, Eric and Machacek, John and Shapiro, Michael , TITLE =. Sci. China Math. , FJOURNAL =. 2019 , NUMBER =

work page 2019
[73]

Muller, Greg , TITLE =. Adv. Math. , FJOURNAL =. 2013 , PAGES =

work page 2013
[74]

Cao, Peigen and Keller, Bernhard and Qin, Fan , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2024 , PAGES =

work page 2024
[75]

2019 , PAGES =

Ingermanson, Grace , TITLE =. 2019 , PAGES =

work page 2019
[76]

arXiv preprint arXiv:2207.10184 , year=

Cao, Peigen and Keller, Bernhard , TITLE =. arXiv preprint arXiv:2207.10184 , year=

work page arXiv
[77]

, TITLE =

Leclerc, B. , TITLE =. Adv. Math. , FJOURNAL =. 2016 , PAGES =

work page 2016
[78]

Quantum cluster algebra structures on quantum nilpotent algebras , author=. Mem. Amer. Math. Soc. , volume=

work page
[79]

Journal of the American Mathematical Society , volume=

Monoidal categorification of cluster algebras , author=. Journal of the American Mathematical Society , volume=

work page
[80]

Skein Algebras of Surfaces and Quantum Groups , year =

L. Skein Algebras of Surfaces and Quantum Groups , year =

work page

Showing first 80 references.