Stable pairs and the HOMFLY polynomial

2012/10/23 by Maulik, Davesh · 3 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.1210.6323

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.

Cited by

• Cohomology of compactified Jacobians for locally planar integral curves
• On the geometry of punctual Hilbert schemes on singular curves and their motivic zeta functions
• Motivic classes of fixed-generators Hilbert schemes of unibranch curve singularities and Igusa zeta functions

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