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

TL;DR

This paper conjectures a deep connection between the HOMFLY homology of plane curve singularities and the geometry of Hilbert schemes, providing explicit formulas and linking knot invariants to algebraic geometry and representation theory.

Contribution

It introduces a conjectural formula relating HOMFLY homology to Hilbert scheme weight polynomials and generalizes known results to torus knots, connecting knot theory with algebraic geometry.

Findings

Conjectural expression for HOMFLY homology dimensions in terms of Hilbert scheme polynomials.

Explicit prediction of HOMFLY homology for (k,n) torus knots as a sum over diagrams.

Hilbert scheme series related to the perverse filtration on the compactified Jacobian.

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 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 (k, n) 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 (k,n) 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.