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

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.