We conjecturally extract the triply graded Khovanov–Rozansky homology of the torus knot from the unique finite-dimensional simple representation of the rational DAHA of type A, rank , and central character . 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 -Catalan numbers and to previous conjectures of the last three authors relating knot homology to Hilbert schemes on singular curves.
Eugene Gorsky. Alexei Oblomkov. Jacob Rasmussen. Vivek Shende. "Torus knots and the rational DAHA." Duke Math. J. 163 (14) 2709 - 2794, 1 November 2014. https://doi.org/10.1215/00127094-2827126