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

TL;DR

This paper presents a detailed algebraic description of the HOMFLYPT skein algebra of the torus, establishes its isomorphism with a specialization of the elliptic Hall algebra, and applies this to construct and relate knot invariants.

Contribution

It provides a generators and relations presentation of the HOMFLYPT skein algebra of the torus and links it explicitly to the elliptic Hall algebra, confirming a key conjecture.

Findings

HOMFLYPT skein algebra is isomorphic to the t=q specialization of the elliptic Hall algebra.

Constructed a 3-variable polynomial for iterated cables of the unknot that generalizes known invariants.

Proved the Connection Conjecture relating different knot invariants.

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 $t=q$ specialization of the elliptic Hall algebra of Burban and Schiffmann [BS12]. 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 [CD14] using the $\mathfrak{gl}_N$ double affine Hecke algebra. This proves one of the Connection Conjectures in [CD14].