pith. sign in

arxiv: 2206.07856 · v7 · submitted 2022-06-16 · 🧮 math.GT · math.QA

On skein algebras of planar surfaces

Haimiao Chen This is my paper

Pith reviewed 2026-05-24 12:05 UTC · model grok-4.3

classification 🧮 math.GT math.QA
keywords skein algebraKauffman bracketplanar surfacedefining relationsgeneratorsn-holed diskpresentationsKirby problem
0
0 comments X

The pith

The ideal of relations among generators of the Kauffman bracket skein algebra of an n-holed disk is generated by relations from small subsurfaces.

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

The paper shows that presentations of the skein algebra S_n of the n-holed disk can be reduced to local relations on subsurfaces with a bounded number of holes. When q plus q inverse is invertible, the relations of degree at most 6 supported on subsurfaces diffeomorphic to the 6-holed disk generate the entire ideal among the n plus binomial n choose 2 plus binomial n choose 3 generators. When q plus q inverse is not invertible, relations of degree at most 2k plus 2 supported on subsurfaces with at most n holes generate the ideal among the 2 to the n minus 1 generators. These statements give explicit bounds on the support of relations and advance toward an explicit presentation of these algebras.

Core claim

When q+q^{-1} is invertible, the ideal of defining relations among the n + binom(n,2) + binom(n,3) generators is generated by relations of degree ≤6 supported by subsurfaces diffeomorphic to Σ_{0,k+1} with k≤6; when q+q^{-1} is not invertible, the ideal for the 2^n-1 generators is generated by relations of degree ≤2k+2 supported by subsurfaces with k≤n.

What carries the argument

The Kauffman bracket skein algebra S_n of the n-holed disk, presented via generators whose relations reduce to those supported on small planar subsurfaces.

If this is right

  • The skein algebra admits a presentation whose relations are all supported on subsurfaces with at most six holes when q+q^{-1} is invertible.
  • Any relation among the generators can be rewritten using only the relations visible inside a 6-holed subsurface.
  • The complexity of the presentation is bounded independently of n in the invertible case.
  • Verification of the presentation reduces to checking finitely many small cases for each fixed n.

Where Pith is reading between the lines

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

  • An explicit list of the small relations could yield an algorithm to multiply elements in S_n for any n by reducing via the local moves.
  • The same reduction technique might apply to skein modules of other surfaces once generating sets are known.
  • If the local relations can be classified completely for k=6, the structure constants of S_n become computable from a finite table.
  • These bounded-support results suggest that the skein algebra behaves like a local theory on the surface.

Load-bearing premise

The sets of generators identified by Przytycki-Sikora and Bullock are generating sets for the skein algebra over the given ring.

What would settle it

An explicit element of the ideal of defining relations that cannot be written as an R-linear combination of the local relations coming from the stated small subsurfaces.

Figures

Figures reproduced from arXiv: 2206.07856 by Haimiao Chen.

Figure 1
Figure 1. Figure 1: The surface Σ = Σ0,n+1. For 1 ≤ i1 < · · · < ir ≤ n, fix a subsurface Σ(i1, . . . , ir) ⊂ Σ homeomorphic to Σ0,r+1, punctured at pi1 , . . . , pir , and intersecting γk for k ∈ {i1, . . . , ir}. For a set Y , let #Y denote its cardinality. A 1-submanifold X ⊂ Σ × (0, 1) is always assumed to be compact and in generic position, in the sense that up to diffeomorphism, π(X) is stable under small perturbations.… view at source ↗
Figure 2
Figure 2. Figure 2: From left to right: x1, x2, x3, x4 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: These are obtained using skein relations. 1. When #supp(F) = 3, suppose supp(F) = {i1, i2, i3} with 1 ≤ i1 < i2 < i3 ≤ n. Take Σ′ F ⊂ Σ such that F ⊂ Σ ′ F and Σ′ F ∩ γj = ∅ for j ̸= i1, i2, i3. For each r ∈ {1, 2, 3}, take a sufficiently small subarc Ar ⊂ F oriented from left to right such that Ar ∩ γir = F ∩ γir . Using arcs in Σ′ F to connect ∂+A1, ∂+A2, ∂+A3 to ∂−A2, ∂−A3, ∂−A1 respectively, to cut out… view at source ↗
Figure 4
Figure 4. Figure 4: From left to right, first row: y1, y2, y3; second row: z1, z2, z3, z4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: This is obtained using skein relations. terms in the form a[C], with a ∈ Tn and |C| ≤ 2. From [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The portion near pi1 , pi2 takes the form in the rightmost of first or second row [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Without loss of generality we may just assume i1 = 1, i2 = 2, i3 = 3. at least one r ∈ {1, 2, 3}. See [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Three of the seven shorter arcs close to the arc in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: This is a special case of the formula in [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The expression for s ♯ (K, F) is given in the second to fourth rows. Example 3.5. A degree 7 knot K is given in the upper-left of [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: First row: t1232 = tr(G ∗ x232 ). Second row: s ♯ (t1232 , x232 ). For t1234 = tr(G ∗ x234), substitute x234 with su(x234) as given by (2), so as to get s ♯ (t1234, x234). The following identity in S4 is deduced: t1234 = −α −1 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: First row (from left to right): the EP (A, B); L× = tr(A)tr(B), L∞ = tr(AB), L0 = tr(AB). Second row: the EP (A, B); L× = tr(AB), L∞ = tr(A)tr(B), L0 = tr(AB). Third row: the EP (A, B); L× = tr(AB), L∞ = tr(AB), L0 = tr(A)tr(B). In each case, the region bounded by the (non-existent) thin circle should avoid γ, and the dotted lines represent the outside parts that are irrelevant. In the three cases, respec… view at source ↗
Figure 13
Figure 13. Figure 13: First row (from left to right): the EP (A1, B1); tr(A1B1); tr(A1B1); tr(A1)tr(B1). Second row: the EP (B2, A2); tr(B2A2); tr(B2A2); tr(B2)tr(A2). Let A1 ⊂ Σ × ( 1 2 , 1) be a copy of K◦ 1 , and B1 ⊂ Σ × (0, 1 2 ) be a copy of K◦ 2 ; choose orientations to build tr(A1B1),tr(A1B1),tr(A1)tr(B1)  into an EST. Let A2 ⊂ Σ × (0, 1 2 ) be a copy of K◦ 1 , and B2 ⊂ Σ × ( 1 2 , 1) be a copy of K◦ 2 ; choose orient… view at source ↗
Figure 14
Figure 14. Figure 14: From left to right: x123; x ′ 321 ; S = tr(x123 ∗ x ′ 321 ). Example 4.7. Consider the simple curve S = tr(x123 ∗ x ′ 321 ) in Σ0,4 as shown [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: First row: pull the arc F ⊂ K up to the top, then F ♯ is a convenient arc of KF . Second row: (KF ) c0∞, (KF ) c0 0 , obtained by resolving the unique crossing of F ♯ . Proof. We prove ϕm,c by induction on (m, c). It holds tautologically for m = 6. Suppose m ≥ 7 and suppose ϕm′ ,c′ has been established for all (m′ , c′ ) ≺ (m, c). Let K be an arbitrary knot with λ(K) = (m, c). Step 1. Let F ∈ A3(K). Write… view at source ↗
Figure 16
Figure 16. Figure 16: A typical situation for K is shown in the upper-left, where the horizontal line presents F, the solid curve presents F ′ , and the dotted arcs stand for the remaining part of K. Abusing the notation, denote the arc of KF ′ resulting from F ⊂ K also by F. In this example, Cr(⟨K′ F |F⟩/F) = {c 1 , c 2 }. or |C1| = |C2| = 0, |E| = 3. Since |K| ≥ 7, there exists an arc F ′ ⊂ K with |F ′ | = 3 and F ′ ∩ (F1 ∪ … view at source ↗
Figure 17
Figure 17. Figure 17: P Each minimal shortenable arc F can be replaced by a linear combination i aiCi, with ai ∈ Tn and Ci unshortenable. Moreover, each Ci can be chosen to be “close to F”. Lemma 5.1. (i) Suppose x± ∈ Σ × {0}. For each minimal shortenable F ∈ H0(x−, x+), there exist as ∈ Tn and unshortenable arcs Cs such that [F] = su(F) in S(x−, x+), where su(F) = P s as[Cs]. (ii) Suppose x± ∈ Σ × {1}. For each minimal shorte… view at source ↗
read the original abstract

Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$. Let $\mathcal{S}_n$ denote the Kauffman bracket skein algebra of the $n$-holed disk $\Sigma_{0,n+1}$ over $R$. When $q+q^{-1}$ is invertible, in 2000 Przytycki and Sikora found a set of $n+{n\choose 2}+{n\choose 3}$ generators for $\mathcal{S}_n$; we show that the ideal of defining relations among these generators is generated by relations of degree $\le6$ supported by certain subsurfaces diffeomorphic to $\Sigma_{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, a set of $2^n-1$ generators for $\mathcal{S}_n$ was known to Bullock in 1999; we show that the ideal of defining relations is generated by relations of degree $\le 2k+2$ supported by certain subsurfaces diffeomorphic to $\Sigma_{0,k+1}$ with $k\le n$. These results are substantial progresses towards answering Problem 1.92 (J) in the Kirby's list.

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.

Referee Report

2 major / 2 minor

Summary. The paper claims to determine explicit presentations for the Kauffman bracket skein algebra S_n of the n-holed disk Σ_{0,n+1} over a commutative ring R with invertible q^{1/2}. When q + q^{-1} is invertible, it asserts that the ideal of relations among the n + binom(n,2) + binom(n,3) generators identified by Przytycki-Sikora is generated by relations of degree ≤6 supported on subsurfaces diffeomorphic to Σ_{0,k+1} with k≤6. When q + q^{-1} is not invertible, it asserts that the ideal among the 2^n - 1 generators identified by Bullock is generated by relations of degree ≤2k+2 supported on subsurfaces with k≤n. These are presented as progress toward Problem 1.92(J) in Kirby's list.

Significance. If the results hold, they supply concrete, finite presentations for skein algebras of planar surfaces with explicit degree and support bounds on the relations. This would be a useful advance for explicit computations in quantum topology and for studying the algebraic structure of skein modules. The case distinction based on invertibility of q + q^{-1} is a substantive feature, and the subsurface-supported nature of the relations aligns with known locality properties of skein algebras.

major comments (2)
  1. [Introduction, §2] Introduction and §2 (generating sets): The central claims concern the ideal of defining relations among the listed generators, but the manuscript takes the surjectivity of the maps from the free algebras on the Przytycki-Sikora and Bullock sets onto S_n as given from the 2000 and 1999 citations without an independent check or self-contained outline. If either set fails to generate, the stated relations do not present the full algebra S_n. This assumption is load-bearing for interpreting the results as presentations rather than relations among a proper subset.
  2. [Main theorems] Theorem statements (e.g., the two main theorems): The degree bounds (≤6 and ≤2k+2) and the restriction to subsurfaces with k≤6 or k≤n are asserted to generate the full relation ideal, but the manuscript does not appear to contain an explicit reduction showing that all higher-degree or larger-support relations follow from these via the skein relations or the algebra structure; the argument relies on the external generating sets without deriving the completeness internally.
minor comments (2)
  1. [§1] Notation for the ring R and the element q^{1/2} should be fixed consistently at the first appearance to avoid ambiguity when q + q^{-1} is or is not invertible.
  2. [Abstract, Introduction] The abstract and introduction cite the prior generating sets but could include one-sentence reminders of the precise counts (n + binom(n,2) + binom(n,3) and 2^n - 1) for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and the recommendation for major revision. The comments correctly identify that our results build directly on the cited generation theorems, and we address each point below with plans for clarification where appropriate.

read point-by-point responses

  1. Referee: [Introduction, §2] The manuscript takes the surjectivity of the maps from the free algebras on the Przytycki-Sikora and Bullock sets onto S_n as given from the 2000 and 1999 citations without an independent check or self-contained outline. If either set fails to generate, the stated relations do not present the full algebra S_n.

    Authors: The generation results are classical and are cited from Przytycki-Sikora (2000) and Bullock (1999), which are standard references establishing that the indicated sets generate S_n. Our contribution is the explicit description of the relation ideal among those generators. We will add a short clarifying paragraph in the introduction that recalls these generation theorems with precise citations to make the logical structure self-contained. revision: partial

  2. Referee: [Main theorems] The degree bounds and subsurface restrictions are asserted to generate the full relation ideal, but the manuscript does not appear to contain an explicit reduction showing that all higher-degree or larger-support relations follow from these; the argument relies on the external generating sets without deriving the completeness internally.

    Authors: The proofs establish completeness by showing that any relation can be reduced, via repeated application of the skein relations and the algebra product, to linear combinations of the listed bounded-degree, small-support relations. To make this reduction step more transparent, we will expand the statements of the main theorems with a brief outline of the reduction strategy (induction on degree and support size) before the detailed arguments. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation builds on external generating sets from independent prior literature

full rationale

The paper states its results conditional on the generating sets identified in Przytycki-Sikora (2000) and Bullock (1999), which are cited as external prior work by different authors. It then proves statements about the ideal of relations among those generators. No self-citations appear in the load-bearing steps, no parameters are fitted inside the paper, and no step reduces a claimed result to a definition or ansatz internal to the manuscript. The cited generating property is an independent external benchmark, so the central claims about relation generators do not collapse by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper takes as given two external generating sets and works over an arbitrary commutative ring R containing an invertible square root of q; no new entities are postulated.

free parameters (1)
  • q^{1/2}
    Fixed invertible element of R stated in the setup; not fitted to data inside the paper.
axioms (1)
  • domain assumption R is a commutative ring with identity
    Explicitly stated at the beginning of the abstract as the coefficient ring.

pith-pipeline@v0.9.0 · 5750 in / 1335 out tokens · 28033 ms · 2026-05-24T12:05:45.601340+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Bonahon and H

    F. Bonahon and H. Wong, Quantum traces for representations of surface groups in SL 2(C), Geom. Topol. 15 (2011), 1569–1615. Doi: 10.2140/gt.2011.15.1569

    work page doi:10.2140/gt.2011.15.1569 2011

  2. [2]

    Bullock, Rings of SL2(C)-characters and the Kauffman bracket skein mod- ule

    D. Bullock, Rings of SL2(C)-characters and the Kauffman bracket skein mod- ule. Comment. Math. Helv. 72 (1997) 521–542. Doi: 10.1007/s000140050032

    work page doi:10.1007/s000140050032 1997

  3. [3]

    Bullock, A finite set of generators for the Kauffman bracket skein algebra

    D. Bullock, A finite set of generators for the Kauffman bracket skein algebra. Math. Z. 231 (1999), 91–101. Doi: 10.1007/PL00004727

    work page doi:10.1007/pl00004727 1999

  4. [4]

    Bullock, and J.H

    D. Bullock, and J.H. Przytycki. Multiplicative structure of Kauffman bracket skein module quantizations. Proc. Amer. Math. Soc. 128 (2000), no. 3, 923–

    work page 2000

  5. [5]

    Doi: 10.1090/s0002-9939-99-05043-1

    work page doi:10.1090/s0002-9939-99-05043-1

  6. [6]

    Charles and J

    L. Charles and J. March´ e, Multicurves and regular functions on the rep- resentation variety of a surface in SU(2). Comment. Math. Helv. 87 (2012), 409–431. DOI: 10.4171/cmh/258

    work page doi:10.4171/cmh/258 2012

  7. [7]

    Chen, Integral structure of the skein algebra of the 5-punctured sphere

    H.-M. Chen, Integral structure of the skein algebra of the 5-punctured sphere. arXiv:2304.06605

    work page arXiv

  8. [8]

    Chen, Presentation of Kauffman bracket skein algebras of planar sur- faces

    H.-M. Chen, Presentation of Kauffman bracket skein algebras of planar sur- faces. In preparation

    work page

  9. [9]

    Cooke and A

    J. Cooke and A. Lacabanne, Higher rank Askey-Wilson algebras as skein algebras. arXiv:2205.04414

    work page arXiv

  10. [10]

    Cooke and P

    J. Cooke and P. Samuelson, On the genus two skein algebra. J. Lond. Math. Soc. 104 (2021), no. 5, 2260–2298. Doi: 10.1112/jlms.12497

    work page doi:10.1112/jlms.12497 2021

  11. [11]

    Frohman and J

    C. Frohman and J. Kania-Bartoszynska, The structure of the Kauffman bracket skein algebra at roots of unity. Math. Z. 289 (2018), no. 3–4, 889–

    work page 2018

  12. [12]

    Doi: 10.1007/s00209-017-1980-2

    work page doi:10.1007/s00209-017-1980-2 1980

  13. [13]

    Frohman, J

    C. Frohman, J. Kania-Bartoszynska and T.T.Q. Lˆ e, Dimension and trace of the Kauffman bracket skein algebra. Trans. Amer. Math. Soc. (Series B) 8 (2021), no. 18, 510–547. Doi: 10.1090/btran/69. 27

    work page doi:10.1090/btran/69 2021

  14. [14]

    Kirby, Problems in low-dimensional topology , AMS/IP Stud

    R. Kirby, Problems in low-dimensional topology , AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35–473, Amer. Math. Soc., Providence, RI, 1997

    work page 1993

  15. [15]

    Korinman, Finite presentations for stated skein algebras and lattice gauge field theory

    J. Korinman, Finite presentations for stated skein algebras and lattice gauge field theory. Algebr. Geom. Topol. 23 (2023), no.3 , 1249–1302. Doi: 10.2140/agt.2023.23.1249

    work page doi:10.2140/agt.2023.23.1249 2023

  16. [16]

    Lˆ e, Quantum Teichmuller spaces and quantum trace map

    T.T.Q. Lˆ e, Quantum Teichmuller spaces and quantum trace map. J. Ins. Math. Jussieu 18 (2019), no. 2, 249–291. Doi: 10.1017/S1474748017000068

    work page doi:10.1017/s1474748017000068 2019

  17. [17]

    Quantum Topol

    Muller, Skein and cluster algebras of marked surfaces. Quantum Topol. 7 (2016), no. 3, 435–503. Doi: 10.4171/QT/79

    work page doi:10.4171/qt/79 2016

  18. [18]

    Ohtsuki, Problems on invariants of knots and 3-manifolds, with an intro- duction by J

    T. Ohtsuki, Problems on invariants of knots and 3-manifolds, with an intro- duction by J. Roberts. Geom. Topol. Monogr., 4, Invariants of knots and 3- manifolds (Kyoto, 2001), i–iv, 377–572, Geom. Topol. Publ., Coventry, 2002

    work page 2001

  19. [19]

    Przytycki and A.S

    J. Przytycki and A.S. Sikora, On skein algebras and Sl 2(C)-character vari- eties. Topology 39 (2000), no. 1, 115–148. Doi: 10.1016/S0040-9383(98)00062-7

    work page doi:10.1016/s0040-9383(98)00062-7 2000

  20. [20]

    Przytycki and A.S

    J. Przytycki and A.S. Sikora, Skein algebras of surfaces.Trans. Amer. Math. Soc. 371 (2019), no. 2, 1309–1332. Doi: 10.1090/tran/7298

    work page doi:10.1090/tran/7298 2019

  21. [21]

    Sikora and B.W

    A.S. Sikora and B.W. Westbury, Confluence theory for graphs. Algebr. Geom. Topol. 7 (2007), 439–478. Doi:10.2140/agt.2007.7.439

    work page doi:10.2140/agt.2007.7.439 2007

  22. [22]

    Santharoubane, Algebraic generators of the skein algebra of a surface

    R. Santharoubane, Algebraic generators of the skein algebra of a surface. arXiv: 1803.09804

    work page arXiv

  23. [23]

    Turaev, Skein quantization of Poisson algebras of loops on surfaces.Ann

    V. Turaev, Skein quantization of Poisson algebras of loops on surfaces.Ann. Sci. ´Ecole Norm. Sup. 24 (1991), 635–704. Doi: 10.24033/asens.1639. Haimiao Chen (orcid: 0000-0001-8194-1264) chenhm@math.pku.edu.cn Department of Mathematics, Beijing Technology and Business University, Liangxiang Higher Education Park, Fangshan District, Beijing, China. 28

    work page doi:10.24033/asens.1639 1991