Stated Skeins and DAHAs

TL;DR

This paper explores the connections between stated skein algebras and DAHAs, constructing embeddings and modules, and analyzing the algebraic structure of skein algebras of punctured tori.

Contribution

It introduces a new embedding of the universal $SL_2$ spherical DAHA into a quantum torus using stated skein algebras and constructs modules over DAHA from marked 3-manifolds.

Findings

Embedding of DAHA into a rank 6 quantum torus

Construction of modules over DAHA from marked 3-manifolds

Generated set and relations for the skein algebra of the punctured torus

Abstract

Skein algebras of surfaces quantize character varieties of topological surfaces, and in low genus, these quantizations are often related to algebras arising in representation theory. For example, Terwilliger defined a universal $SL_2$ spherical double affine Hecke algebra $A$; a combination of results in the literature shows $A$ is isomorphic to the $SL_2$ skein algebra of the punctured torus. Stated skein algebras are a generalization which quantize decorated character varieties. In this paper, we used stated skein algebras to construct a new embedding of $A$ into a rank 6 quantum torus, and we show that each marked 3-manifold with a torus boundary produces a module over $A$. We also determine a generating set for the stated skein algebra of $T^2\setminus D^2$, and we find many relations; however, finding a complete list of relations is still an open problem.