Iterated torus knots and double affine Hecke algebras

TL;DR

This paper provides a topological framework for the double affine Hecke algebra of type A1, constructs modules for knots, and relates Cherednik's polynomials to knot invariants, extending to iterated cables.

Contribution

It introduces a topological realization of the spherical double affine Hecke algebra and extends Cherednik's polynomials to all iterated cables of the unknot.

Findings

Topological interpretation of Cherednik's polynomials

Proof that polynomials specialize to colored Jones polynomials

Extension of Cherednik's construction to iterated cables

Abstract

We give a topological realization of the (spherical) double affine Hecke algebra $\mathrm{SH}_{q,t}$ of type $A_1$, and we use this to construct a module over $\mathrm{SH}_{q,t}$ for any knot $K \subset S^3$. As an application, we give a purely topological interpretation of Cherednik's 2-variable polynomials $P_n(r,s; q,t)$ of type $A_1$ from [Che13] (where $r,s \in \mathbb{Z}$ are relatively prime), and we give a new proof that these specialize to the colored Jones polynomials of the $r,s$ torus knot. We then generalize Cherednik's construction (for $\mathcal{sl}_2$) to all iterated cables of the unknot and prove the corresponding specialization property. Finally, in the appendix we compare our polynomials associated to iterated torus knots to the ones recently defined in [CD14], in the specialization $t=-q^2$.