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

TL;DR

This paper explores the relationship between the geometry of plane curve singularities and knot invariants, proposing a conjecture linking Hilbert schemes to the HOMFLY polynomial, and verifies it for specific singularities and links.

Contribution

It conjectures a novel connection between the generating functions of Euler characteristics of refined punctual Hilbert schemes and the HOMFLY polynomial of the link of a singularity.

Findings

Conjecture verified for irreducible singularities y^k = x^n.

Confirmed for the singularity y^4 = x^7 - x^6 + 4 x^5 y + 2 x^3 y^2.

Links studied include torus knots and a cable of the trefoil.

Abstract

The intersection of a complex plane curve with a small three-sphere surrounding one of its singularities is a non-trivial link. The refined punctual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fixed length and whose defining ideals have a fixed number of generators. We conjecture that the generating function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularities y^k = x^n, whose links are the k,n torus knots, and for the singularity y^4 = x^7 - x^6 + 4 x^5 y + 2 x^3 y^2, whose link is the 2,13 cable of the trefoil.