Classical Limit of genus two DAHA

TL;DR

This paper proves the flatness of a one-parameter deformation of the genus two skein algebra, describes its classical limit as a Poisson deformation, and provides a simple presentation of the associated character variety's coordinate ring.

Contribution

It establishes the flatness of the deformation, solves the word problem, describes a monomial basis, and explicitly presents the classical limit as a Poisson deformation.

Findings

The deformation $ ilde{ ext{A}}_{q,t}$ is flat.

The classical limit $ ilde{ ext{A}}_{q=1,t}$ is a Poisson deformation.

A simple presentation of the genus two character variety's coordinate ring.

Abstract

We show that one-parameter deformation $\mathcal A_{q,t}$ of the skein algebra $Sk_q(\Sigma_2)$ of a genus two surface suggested in [AS19] is flat. We solve the word problem in the algebra and describe monomial basis. In addition, we calculate the classical limit $\mathcal A_{q=1,t}$ of the algebra and prove that it is a one-parameter flat Poisson deformation of the coordinate ring $\mathcal A_{q=t=1}$ of an $SL(2,\mathbb C)$-charater variety of a genus two surface. As a byproduct, we obtain a remarkably simple presentation in terms of generators and relations for the coordinate ring $\mathcal A_{q=t=1}$ of a genus two character variety.