From Cherednik algebras to knot homology via cuspidal D-modules

TL;DR

This paper connects knot homology of torus knots with Cherednik algebra representations, revealing a new geometric approach and proving the rational master conjecture.

Contribution

It introduces a novel method linking Cherednik algebra modules with knot homology via cuspidal D-modules, confirming the rational master conjecture.

Findings

Khovanov-Rozansky homology recovered from Cherednik algebra representations

Hodge filtration identified with algebraic filtrations on modules

Proof of the rational master conjecture

Abstract

We show that the triply-graded Khovanov-Rozansky homology of the $(m,n)$ torus knot can be recovered from the finite-dimensional representation $\mathrm{L}_{m/n}$ of the rational Cherednik algebra at slope $m/n$, endowed with the Hodge filtration coming from the cuspidal character D-module. Our approach involves expressing the associated graded of the cuspidal character D-module in terms of a dg module closely related to the action of the shuffle algebra on the equivariant K-theory of the Hilbert scheme of points on the plane, thereby proving the rational master conjecture. As a corollary, we identify the Hodge filtration with the inductive and algebraic filtrations on $\mathrm{L}_{m/n}$.