A note on double affine Hecke algebra for skein algebra on twice-punctured torus

TL;DR

This paper generalizes the double affine Hecke algebra to the skein algebra on a twice-punctured torus, linking algebraic automorphisms to geometric Dehn twists and using cluster algebra techniques for the classical limit.

Contribution

It introduces a new algebraic structure for the skein algebra on a twice-punctured torus and connects automorphisms to geometric Dehn twists.

Findings

Automorphisms correspond to Dehn twists on the surface.

Constructs the classical limit via cluster algebra mutations.

Provides a new algebraic framework for skein algebras on punctured surfaces.

Abstract

We construct a generalization of the $C^\vee C_1$-type double affine Hecke algebra for the skein algebra on the twice-punctured torus $\Sigma_{1,2}$ using the Heegaard dual of the Iwahori--Hecke operator recently introduced in our previous article. We show that the automorphisms of our algebra correspond to the Dehn twists about the curves on $\Sigma_{1,2}$. We also give the cluster algebraic construction of the the classical limit of the skein algebra, where the Dehn twists are given in terms of the cluster mutations.