Stable pairs and the HOMFLY polynomial

TL;DR

This paper proves a conjecture linking the geometry of Hilbert schemes of singularities to the HOMFLY polynomial of algebraic links, extending to stable pair invariants and employing wall-crossing and skein techniques.

Contribution

It establishes a geometric-quantum link conjecture and extends it to stable pair invariants, using novel wall-crossing and skein-theoretic methods.

Findings

Proved the Oblomkov-Shende conjecture relating Hilbert schemes to HOMFLY polynomials.

Extended the conjecture to stable pair invariants on the conifold.

Established identities for colored HOMFLY polynomials using skein theory.

Abstract

Given a planar curve singularity, we prove a conjecture of Oblomkov-Shende, relating the geometry of its Hilbert scheme of points to the HOMFLY polynomial of the associated algebraic link. More generally, we prove an extension of this conjecture, due to Diaconescu-Hua-Soibelman, relating stable pair invariants on the conifold to the colored HOMFLY polynomial of the algebraic link. Our proof uses wall-crossing techniques to prove a blowup identity on the algebro-geometric side. We prove a matching identity for the colored HOMFLY polynomials of a link using skein-theoretic techniques.