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A Proof of the Pentagon Relation for Skeins

Mingyuan Hu

TL;DR

The paper proves a topological pentagon relation for skein dilogarithms on the punctured torus, establishing the identity $Q_{\mathbf{x}}(v) \cdot Q_{\mathbf{y}}(w) = Q_{\mathbf{y}}(w) \cdot Q_{\mathbf{x}+\mathbf{y}}(vw) \cdot Q_{\mathbf{x}}(v)$ in $\widehat{\mathrm{Sk}}(T-D)$ and showing its compatibility with the positive elliptic Hall algebra via a surjection from $\mathrm{Sk}^+(T-D)$ to $\mathcal{E}_{q,t}^+$. The approach reduces the problem to a genus-2 handlebody using a $\pi/4$ twist and Dehn twists, and employs a sliding lemma together with an explicit Ad-action computation to verify the required relation. Consequently, the pentagon in skein theory generalizes prior results in the elliptic Hall and Ding–Iohara–Miki frameworks and provides a skein-theoretic instance of a 5-term dilogarithm identity with connections to cluster theory and Open Gromov–Witten invariants. The work also notes an independent confirmation of the result in a concurrent paper. The construction yields a local pentagon for any surface via the neighborhood isomorphism to the punctured torus.

Abstract

In \cite{HSZ23}, with Gus Schrader and Eric Zaslow we developed a skein-theoretic version of cluster theory, and made a conjecture on the pentagon relation for the skein dilogarithm. Here we give a topological proof of this conjecture. Combining \cite{MS21} and \cite{BCMN23}, we get a surjection from the skein algebra $\mathrm{Sk}^+(T - D)$ to the positive part of the elliptic Hall algebra $\mathcal{E}_{q, t}^+$. Hence our pentagon relation generalizes the ones in \cite{Z23} and \cite{GM19}.

A Proof of the Pentagon Relation for Skeins

TL;DR

The paper proves a topological pentagon relation for skein dilogarithms on the punctured torus, establishing the identity in and showing its compatibility with the positive elliptic Hall algebra via a surjection from to . The approach reduces the problem to a genus-2 handlebody using a twist and Dehn twists, and employs a sliding lemma together with an explicit Ad-action computation to verify the required relation. Consequently, the pentagon in skein theory generalizes prior results in the elliptic Hall and Ding–Iohara–Miki frameworks and provides a skein-theoretic instance of a 5-term dilogarithm identity with connections to cluster theory and Open Gromov–Witten invariants. The work also notes an independent confirmation of the result in a concurrent paper. The construction yields a local pentagon for any surface via the neighborhood isomorphism to the punctured torus.

Abstract

In \cite{HSZ23}, with Gus Schrader and Eric Zaslow we developed a skein-theoretic version of cluster theory, and made a conjecture on the pentagon relation for the skein dilogarithm. Here we give a topological proof of this conjecture. Combining \cite{MS21} and \cite{BCMN23}, we get a surjection from the skein algebra to the positive part of the elliptic Hall algebra . Hence our pentagon relation generalizes the ones in \cite{Z23} and \cite{GM19}.
Paper Structure (5 sections, 6 theorems, 35 equations, 7 figures)

This paper contains 5 sections, 6 theorems, 35 equations, 7 figures.

Table of Contents

  1. introduction
  2. HOMFLYPT Skein Modules and pentagon relation
  3. HOMFLYPT Skein Modules
  4. Skein Dilogarithms and the Pentagon Relation
  5. proof of the main theorem

Key Result

Theorem 2.2

For a surface $\Sigma$, Poincare duality gives a pairing on $H_1(M, \mathbb{Z})$. We can consider the induced quantum torus $\mathcal{T}_{H_1(M,\mathbb{Z})}^q$. There is a surjective map:

Figures (7)

  • Figure 2.1.1: A skein element in $\mathrm{Sk}(S^1 \times D^2)$. The red edges are identified so we get a link in a solid torus. We take the convention that going upward is the positive direction. Hence, the blue part lives in $\mathrm{Sk}^+(S^1 \times D)$, while the green part lives in $\mathrm{Sk}^-(S^1 \times D)$.
  • Figure 2.2.1: Loops $a_i$ on $\Sigma_g$
  • Figure 3.0.1: $a_i$ and $b_i$ on $\mathcal{H}_2$
  • Figure 3.0.2: $T-D$ (left) and $(T-D) \times I$ (right)
  • Figure 3.0.3: The homeomorphism between $(T-D) \times I$ and $\mathcal{H}_2$
  • ...and 2 more figures

Theorems & Definitions (15)

  • Remark 1.1
  • Example 2.1
  • Theorem 2.2: Przy98
  • Example 2.3: The skein module of the solid torus
  • Example 2.5: MS17 The skein algebra of the torus
  • Example 2.6: The skein algebra of the punctured torus
  • Theorem 2.7: MS21BCMN23
  • Example 2.8
  • Definition 2.9: HSZ23
  • Proposition 2.11: HSZ23
  • ...and 5 more