The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link

Abstract

We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting $t=-1$. By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a $\left(k,n\right)$ torus knot as a certain sum over diagrams.

The Hilbert scheme series corresponding to the summand of the HOMFLY homology with minimal “$a$” grading can be recovered from the perverse filtration on the cohomology of the compactified Jacobian. In the case of $\left(k,n\right)$ torus knots, this space furnishes the unique finite-dimensional simple representation of the rational spherical Cherednik algebra with central character $k∕n$. Up to a conjectural identification of the perverse filtration with a previously introduced filtration, the work of Haiman and Gordon and Stafford gives formulas for the Hilbert scheme series when $k=mn+1$.