Torus knots and the rational DAHA

TL;DR

This paper proposes a conjectural algebraic framework connecting the triply graded Khovanov-Rozansky homology of torus knots to the representation theory of rational DAHA, unifying various previous conjectures in knot theory.

Contribution

It introduces a novel conjectural approach linking knot homology to rational DAHA representations, providing explicit algebraic formulas for differentials and homologies.

Findings

Conjectural link between rational DAHA and torus knot homology

Explicit algebraic expressions for differentials in knot homology

Unification of previous conjectures relating knot homology to q,t-Catalan numbers and Hilbert schemes

Abstract

We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q, t-Catalan numbers, and of the last three authors relating knot homology to Hilbert schemes on singular curves.