DAHAs and skein theory

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.
Comment: Preliminary version, comments welcome! 25 pages, many figures. V2: added some details, diagrams, comments, and subtracted some typos. V3: simplified arguments in Section 3.3, added some clarifying remarks