On the genus two skein algebra

TL;DR

This paper investigates the genus two skein algebra, explicitly computes its action on the handlebody skein module, and reveals its isomorphism to a specialized double affine Hecke algebra, advancing understanding of algebraic structures in low-dimensional topology.

Contribution

It provides an explicit computation of the genus two skein algebra's action and establishes its isomorphism with a specialized double affine Hecke algebra.

Findings

Explicit action of the genus two skein algebra on the handlebody skein module

Decomposition of the module over subalgebras via polynomial representations

Isomorphism to the $t=q$ specialization of the genus two spherical DAHA

Abstract

We study the skein algebra of the genus 2 surface and its action on the skein module of the genus 2 handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the $t=q$ specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.