Torus knots and the rational DAHA

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 to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.