DAHAs and skein theory

TL;DR

This paper constructs a skein-theoretic model of the $rak{gl}_n$ double affine Hecke algebra using braids and tangles, and explores its connection to the elliptic Hall algebra, providing new insights into their relationship.

Contribution

It offers a skein-theoretic realization of the DAHA and conjectures a link with the elliptic Hall algebra, advancing understanding of their interplay.

Findings

Skein-theoretic model of DAHA in the punctured torus

Evidence supporting the conjectured relationship between skein algebra and elliptic Hall algebra

New framework for studying DAHA via braids and tangles

Abstract

We give a skein-theoretic realization of the $\mathfrak{gl}_n$ double affine Hecke algebra of Cherednik using braids and tangles in the punctured torus. We use this to provide evidence of a relationship we conjecture between the classical skein algebra of the punctured torus and the elliptic Hall algebra of Burban and Schiffmann.