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

TL;DR

This paper explores the connection between Roger-Yang skein algebras and Kauffman bracket skein algebras for small surfaces, providing new insights into their representations and structural constants.

Contribution

It establishes a surjective homomorphism from the Roger-Yang skein algebra of a punctured annulus to the Kauffman bracket skein algebra of a torus, and characterizes its irreducible representations.

Findings

Surjective homomorphism from $ ext{RY}$ to Kauffman algebra

Classification of irreducible, finite-dimensional representations

Computed structural constants for a bracelets basis, indicating positivity

Abstract

We calculate the Roger-Yang skein algebra of the annulus with two interior punctures, $ \mathcal S^{RY}(\Sigma_{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}(\Sigma_{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}(\Sigma_{0, 2, 2})$, giving evidence for positivity.