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arxiv: 2407.04178 · v3 · submitted 2024-07-04 · 🧮 math.GT · math.CO· math.QA· math.RT

Monomial web basis for the SL(N) skein algebra of the twice punctured sphere

Tommaso Cremaschi , Daniel C. Douglas This is my paper

Pith reviewed 2026-05-23 23:03 UTC · model grok-4.3

classification 🧮 math.GT math.COmath.QAmath.RT
keywords skein algebraSL(n) webstwice punctured spherecharacter varietyquantum trace mapmonomial basispolynomial algebra
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The pith

The SL(n) skein algebra of the twice punctured sphere is a commutative polynomial algebra on n-1 explicit crossing-free webs.

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

The paper constructs an explicit basis of n-1 crossing-free SL(n) webs for the skein algebra of the twice punctured sphere. It proves this algebra is commutative and isomorphic to a polynomial ring in these generators for generic values of q. At q=1 the construction identifies the skein algebra with the coordinate ring of the SL(n) character variety. The proof uses the SL(n) quantum trace map to establish both that the webs span and that they are linearly independent.

Core claim

We construct a linear basis for the SL(n) skein algebra of the twice punctured sphere consisting of monomials in n-1 explicit SL(n) webs without crossings. This shows the skein algebra is a commutative polynomial algebra in these n-1 generators. The result holds for any non-zero complex q except finitely many roots of unity of small order, and at q=1 identifies the algebra with the coordinate ring of the SL(n) character variety of the surface. Both spanning and independence follow from properties of the SL(n) quantum trace map.

What carries the argument

The SL(n) quantum trace map, which supplies the spanning and linear-independence arguments for the monomial web basis.

If this is right

  • The quantum trace map embeds the polynomial algebra into the Fock-Goncharov quantum higher Teichmüller space of the annulus.
  • The splitting map embeds the polynomial algebra into the Lê-Sikora stated skein algebra of the annulus.
  • The construction yields a relationship with Fock-Goncharov duality.

Where Pith is reading between the lines

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

  • The explicit monomial basis may make concrete calculations of invariants on the twice punctured sphere feasible in cases previously limited to abstract descriptions.
  • The method could extend to produce bases for skein algebras on surfaces with additional punctures or different topologies.

Load-bearing premise

The SL(n) quantum trace map has the required embedding and trace properties to prove both spanning and linear independence of the proposed webs.

What would settle it

A linear dependence among the n-1 proposed basis webs whose image under the quantum trace map is nonzero for generic q would falsify linear independence.

Figures

Figures reproduced from arXiv: 2407.04178 by Daniel C. Douglas, Tommaso Cremaschi.

Figure 1
Figure 1. Figure 1: The permutation ( 3 1 2 4 ) and its corresponding positive permu￾tation braid. These multi-curves are considered up to homotopy fixing endpoints on the box boundary. In particular, the composition σ2 ◦ σ1 corresponds, up to homotopy, to placing the picture for σ2 above the picture for σ1. The length ℓ(σ) becomes the number of crossings in the picture for σ represented by a multi-curve in minimal position … view at source ↗
Figure 2
Figure 2. Figure 2: Monogon and bigon. 1. Setting 1.1. Webs. Consider an edge oriented graph with finitely many vertices, possibly with mul￾tiple edges between vertices and possibly with more than one component, that is properly embedded in R 3 such that, in addition: (1) the graph is n-valent, namely, there are n edges incident to each vertex; (2) each vertex is either a sink or source, namely, the orientations of the edges … view at source ↗
Figure 3
Figure 3. Figure 3: Defining relations for the based skein algebra. classes of webs W in A × I by the relations shown in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Defining relations for the unbased skein algebra. Proposition 1.2 ([49]). Assume W and W′ are webs such that W and W′ agree except at a single vertex where the distinguished half-edge of W′ is just after, or before, that of W with respect to the cyclic order. Then W′ = (−1)n+1W in S q n . □ We say such a W and W′ differ by a cilia change relation. Consequently, for all intents and purposes, one can forget … view at source ↗
Figure 5
Figure 5. Figure 5: More relations for the based skein algebra. (2) For n even, and for a choice of spin structure s on A × I, the assignment W 7→ s(W)u #sinks(W)W◦ determines an isomorphism of algebras S q n ∼→ (S q n ) ◦ , with inverse sending W◦ to s(W)(u −1 ) #sinks(W)W for any web W such that W◦ underlies W. □ In this paper, we prefer to work with the based skein algebra S q n . However, by Proposition 1.3, we immediatel… view at source ↗
Figure 6
Figure 6. Figure 6: The i-th irreducible basis web. The following statement is the main result of the paper. Theorem 2.1. Let Q ⊂ C ∗ be the set of q 1/n such that q = (q 1/n) n is a (2m)-root of unity for m = 2, 3, . . . , n − 1 and q ̸= ±1. Then for all q 1/n ∈ C ∗ \ Q, the skein algebra S q n is a [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Permutation multi-curve, braid link, and permutation knot. Note that the third permutation is different from the first two [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Positive and negative position of the i-th irreducible basis web. Starting in the positive position, we can write Bi = q n(n−1) nY−1 k=i q −k [n − k] ! X σ∈Symi [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Algebraic intersection number. Later we will use the following refinements. Lemma 3.6. Let L be a link in A × I and let γ be the projection of L on A. If γ has at most i positive intersections and j negative intersections with an arc α oriented out of the puncture and cutting the annulus, colored blue in [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Local square relation. A strand with a triple arrow indicates n − 2 parallel strands oriented according to the direction of the arrow. Lemma 3.12. Let q 1/n ∈ C ∗ such that q is not a (2k)-root of unity for any k = 2, 3, . . . , n−2 but allowing q = ±1. Assuming γi ∈ B for all |i| ≤ n − 1, then γi ∈ B for all |i| ≥ n [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The relation γi = cx + dy. A triple arrow indicates n − 2 parallel strands. Proof. As a preliminary calculation, let i ≥ n. Using the planarizing relation, which is where the restriction on q comes in, write γi = cx + dy ∈ Sq n , where c = q (n−1)/n and d = −q −n(n−1)/2−1/n/[n − 2]!, and where x and y are the webs appearing in [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The relation y = cx′ + dy′ . We see that x ′ = B2γi−2 ∈ Sq n or, in the particular case n = 2, we have x ′ ∝ γi−2 ∈ Sq n . Note that y ′ is isotopic to the web appearing on the left hand side of [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The relation y ′ = s1 P σ∈Symn−2 b ℓ(σ)x ′′ σ + s2 P σ∈Symn−1 b ℓ(σ) y ′′ σ . Using Lemma 3.11, write y ′ = s1 X σ∈Symn−2 b ℓ(σ)x ′′ σ + s2 X σ∈Symn−1 b ℓ(σ) y ′′ σ ∈ Sq n where s1 = a 2an−1an−2 Qn−1 k=2 ak and s2 = a 2an−1b Qn−1 k=2 ak with a, b, ak as in Lemma 3.11, and where x ′′ σ and y ′′ σ are the webs appearing in [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Defining boundary relations for the stated skein algebra, includ￾ing reversing all strand orientations. An ideal monoangle D1, resp. ideal biangle D2 or ideal triangle D3 (or just monoangle, resp. biangle or triangle) is the punctured oriented surface obtained by removing one, resp. two or three, points from the boundary of a closed disk with the stan￾dard orientation. Theorem 4.4 ([10, 49]). The stated s… view at source ↗
Figure 15
Figure 15. Figure 15: Definition of the co-unit. For k ∈ {0, 1, . . . , n} and ij ∈ {1, 2, . . . , n} for j = 1, 2, . . . , k and lj ′ ∈ {1, 2, . . . , n} for j ′ = 1, 2, . . . , n − k, let W◦ source(k,(ij )j ,(lj ′)j ′) and W◦ sink(k,(ij )j ,(lj ′)j ′) be the source and sink stated webs in the biangle D2 appearing in [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Co-unit for source and sink webs in the biangle. S is strictly less than d/2. Note that the only punctured surfaces excluded in this way are the monoangle, the biangle, and the sphere with one or two punctures. For simplicity, assume ideal triangulations do not contain any self-folded triangles, in which case the once punctured monoangle should also be excluded. The 1-cells, namely the ideal arcs, often c… view at source ↗
Figure 17
Figure 17. Figure 17: Fock–Goncharov quiver [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Quantum left and right matrices. 4.4.6. Digression: an open problem originating in the physical point of view. There is also a different construction of quantum traces from spectral networks [25, 57, 58], motivated by the physical problem of counting ‘framed BPS states’. At least for simple oriented loops L ◦ , this different construction appears, experimentally, to agree with the construction of τ q L (L… view at source ↗
Figure 19
Figure 19. Figure 19: Good position of the k-th irreducible basis web with respect to the split ideal triangulation of the annulus. right is the source web W◦ source(k). The calculations will be in terms of the compatibly stated restricted webs A◦ L,j′(lj ′, l′ j ′), W◦ sink(n − k,(l ′ j ′)j ′,(i ′ j )j ), A◦ R,j (ij , i′ j ), W◦ source(k,(ij )j ,(lj ′)j ′) in the restrictions DL 3 × I, DL 2 × I, DR 3 × I, DR 2 × I respectivel… view at source ↗
read the original abstract

We give a new proof of a slightly modified version of a result of Queffelec--Rose, by constructing a linear basis for the $\mathrm{SL}(n)$ skein algebra of the twice punctured sphere for any non-zero complex number $q$, excluding finitely many roots of unity of small order. In particular, the skein algebra is a commutative polynomial algebra in $n-1$ generators, where each generator is represented by an explicit $\mathrm{SL}(n)$ web, without crossings, on the surface. This includes the case $q=1$, where the skein algebra is identified with the coordinate ring of the $\mathrm{SL}(n)$ character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called $\mathrm{SL}(n)$ quantum trace map, due originally to Bonahon--Wong in the case $n=2$. Two consequences of our method are that the quantum trace map and the so-called splitting map embed the polynomial algebra into the Fock--Goncharov quantum higher Teichm\"uller space and the L\^{e}--Sikora stated skein algebra, respectively, of the annulus. We end by discussing the relationship with Fock--Goncharov duality.

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 gives a new proof of a slightly modified version of a result of Queffelec--Rose by exhibiting an explicit set of n-1 crossing-free SL(n) webs that form a linear basis for the SL(n) skein algebra of the twice-punctured sphere (for q not a small root of unity). It shows that the skein algebra is the commutative polynomial algebra on these generators and identifies the q=1 case with the coordinate ring of the SL(n) character variety. Both spanning and linear independence are deduced from properties of the SL(n) quantum trace map (an extension of the Bonahon--Wong construction); two corollaries are embeddings of the polynomial algebra into the Fock--Goncharov quantum higher Teichmüller space and the Lê--Sikora stated skein algebra of the annulus.

Significance. If the central claim holds, the work supplies an explicit monomial basis for a family of skein algebras that had previously been known only abstractly, together with concrete embeddings into quantum Teichmüller spaces. The explicit web generators and the identification at q=1 are concrete contributions to the study of higher Teichmüller theory and quantum cluster algebras.

major comments (2)
  1. [sections developing the quantum trace map and the spanning/independence proofs] The spanning and linear-independence arguments (stated in the abstract and developed in the sections on the quantum trace map) both rest on the claim that the SL(n) quantum trace map is an algebra homomorphism that is injective on the polynomial subalgebra generated by the proposed n-1 webs and lands inside the appropriate completion of the skein algebra. The manuscript must supply or cite a self-contained verification that these three properties hold for the generalized map when n>2; without that verification the basis theorem is not established.
  2. [introduction and statement of main theorem] The paper announces a 'slightly modified version' of the Queffelec--Rose result. The precise statement of the modification (e.g., any change in the allowed roots of unity or in the definition of the skein algebra) should be stated explicitly before the proof, so that the reader can see exactly which prior claim is being reproved.
minor comments (2)
  1. Notation for the n-1 generators (e.g., how they are indexed and drawn) should be fixed early and used consistently in all subsequent figures and statements.
  2. A short table or diagram summarizing the n-1 explicit webs for small n (n=3,4) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on the manuscript. We address each major comment below and will revise the paper accordingly to strengthen the exposition and completeness of the arguments.

read point-by-point responses

  1. Referee: [sections developing the quantum trace map and the spanning/independence proofs] The spanning and linear-independence arguments (stated in the abstract and developed in the sections on the quantum trace map) both rest on the claim that the SL(n) quantum trace map is an algebra homomorphism that is injective on the polynomial subalgebra generated by the proposed n-1 webs and lands inside the appropriate completion of the skein algebra. The manuscript must supply or cite a self-contained verification that these three properties hold for the generalized map when n>2; without that verification the basis theorem is not established.

    Authors: We agree that the three properties of the generalized SL(n) quantum trace map (algebra homomorphism, injectivity on the polynomial subalgebra, and landing in the completion) require explicit verification for n>2 to fully support the basis theorem. In the revised manuscript we will add a self-contained verification of these properties in the sections on the quantum trace map, extending the Bonahon--Wong construction with the necessary details for the higher-rank case. revision: yes

  2. Referee: [introduction and statement of main theorem] The paper announces a 'slightly modified version' of the Queffelec--Rose result. The precise statement of the modification (e.g., any change in the allowed roots of unity or in the definition of the skein algebra) should be stated explicitly before the proof, so that the reader can see exactly which prior claim is being reproved.

    Authors: We will revise the introduction and the paragraph immediately preceding the main theorem to state the precise modifications explicitly. This will include the precise range of q (any nonzero complex number excluding finitely many roots of unity of small order) and confirmation that the skein algebra is defined in the standard way, so the differences from the Queffelec--Rose result are clear to the reader. revision: yes

Circularity Check

0 steps flagged

No circularity; spanning and independence rest on external quantum trace map from prior literature

full rationale

The paper constructs an explicit monomial web basis and proves it spans and is linearly independent for the SL(n) skein algebra by invoking the SL(n) quantum trace map (originally Bonahon-Wong for n=2). This map is cited as an independent external tool whose embedding and homomorphism properties supply the required arguments; the authors of the cited map do not overlap with Cremaschi-Douglas. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The result is therefore not equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the embedding and trace properties of the SL(n) quantum trace map from prior literature; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The SL(n) quantum trace map is well-defined for generic q and supplies both spanning and linear independence for the proposed web basis.
    Abstract states that both spanning and independence depend on this map.

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