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A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces

Chloe Marple, Helen Wong

TL;DR

This work analyzes the Roger-Yang skein algebra of the twice-punctured annulus, establishing a finite presentation and a surjective map to the torus skein algebra to transfer structural insights. It characterizes irreducible finite-dimensional representations at odd roots of unity via a 5-tuple of central data (the classical shadow) and provides an explicit construction with a uniqueness criterion under polynomial conditions, linking these representations to torus data. The paper also develops a framework to study positivity by pulling back torus product-to-sum formulas, builds geometric and torus-preimage bases, and derives infinite-family product formulas that provide strong evidence for the bracelet-positivity conjecture in the presence of interior punctures.

Abstract

We calculate the Roger-Yang skein algebra of the annulus with two interior punctures, $ \mathcal S^{RY}(Σ_{0, 2, 2})$, and show there is a surjective homomorphism from this algebra to the Kauffman bracket skein algebra of the closed torus. Using this homomorphism, we characterize the irreducible, finite-dimensional representations of $ \mathcal S^{RY}(Σ_{0, 2, 2})$, showing that they can be described by certain complex data and that the correspondence is unique if certain polynomial conditions are satisfied. We also use the relationship with the skein algebra of the torus to compute structural constants for a bracelets basis for $ \mathcal S^{RY}(Σ_{0, 2, 2})$, giving evidence for positivity.

A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces

TL;DR

This work analyzes the Roger-Yang skein algebra of the twice-punctured annulus, establishing a finite presentation and a surjective map to the torus skein algebra to transfer structural insights. It characterizes irreducible finite-dimensional representations at odd roots of unity via a 5-tuple of central data (the classical shadow) and provides an explicit construction with a uniqueness criterion under polynomial conditions, linking these representations to torus data. The paper also develops a framework to study positivity by pulling back torus product-to-sum formulas, builds geometric and torus-preimage bases, and derives infinite-family product formulas that provide strong evidence for the bracelet-positivity conjecture in the presence of interior punctures.

Abstract

We calculate the Roger-Yang skein algebra of the annulus with two interior punctures, , and show there is a surjective homomorphism from this algebra to the Kauffman bracket skein algebra of the closed torus. Using this homomorphism, we characterize the irreducible, finite-dimensional representations of , showing that they can be described by certain complex data and that the correspondence is unique if certain polynomial conditions are satisfied. We also use the relationship with the skein algebra of the torus to compute structural constants for a bracelets basis for , giving evidence for positivity.

Paper Structure

This paper contains 10 sections, 27 theorems, 71 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Definitions and Background
  4. Presentations of $\mathcal{S}^{RY}(\Sigma_{0,2,2})$
  5. Relationships between skein algebras of small surfaces
  6. Classification of Representations of $\mathcal{S}^{RY}(\Sigma_{0,2,2})$
  7. Evidence for Positivity Conjecture of $\mathcal{S}^{RY}(\Sigma_{0, 2, 2})$
  8. A geometric basis for $S^{RY}(\Sigma_{0,2,2})$
  9. Description of pre-image of torus knots
  10. Structural constants for infinite families of products

Key Result

Proposition 3.3

For $A$ a primitive root of unity of odd order $N$, the center of the Roger-Yang skein algebra $Z(\mathcal{S}_{A}^{RY}(\Sigma))$ is the $\mathbb C[v_{i}^{\pm}]$-subalgebra generated by the following elements.

Figures (4)

  • Figure 1: Framed links that are equivalent up to regular isotopy
  • Figure 2: Generators for $\mathcal{S}^{RY}(\Sigma_{0,2,2})$
  • Figure 3: The geometric basis for $\mathcal{S}^{RY}(\Sigma_{0,2,2})$. Here, $\varepsilon=n-k\mod 2$ and $h=(n-k+\varepsilon)/2$.
  • Figure 4: The intersections of $\omega$ and $\theta'$ from the proof of Lemma \ref{['lem:intersection-nb']}.

Theorems & Definitions (65)

  • Definition 3.1
  • Example 3.2
  • Proposition 3.3: KaruoMoonWongCenter
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Definition 4.3
  • Corollary 4.4
  • proof
  • Theorem 5.1
  • ...and 55 more