The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra

Abstract

We give a generators and relations presentation of the HOMFLYPT skein algebra $H$ of the torus $T^2$, and we give an explicit description of the module corresponding to the solid torus. Using this presentation, we show that $H$ is isomorphic to the $\sigma ={\overline{\sigma }}^{-1}$ specialization of the elliptic Hall algebra of Burban and Schiffmann.

As an application, for an iterated cable $K$ of the unknot, we use the elliptic Hall algebra to construct a 3-variable polynomial that specializes to the $\lambda$-colored HOMFLYPT polynomial of $K$. We show that this polynomial also specializes to one constructed by Cherednik and Danilenko using the ${\mathfrak{gl}}_N$ double affine Hecke algebra. This proves one of the connection conjectures in their recent work.