HOMFLY polynomials, stable pairs and motivic Donaldson-Thomas invariants

TL;DR

This paper connects HOMFLY polynomials of plane curve singularities with stable pair invariants of Calabi-Yau threefolds, providing a motivic Donaldson-Thomas framework that interprets knot invariants geometrically.

Contribution

It establishes a Calabi-Yau threefold interpretation of the Oblomkov-Shende conjecture on HOMFLY polynomials and introduces motivic invariants for algebraic knots via Donaldson-Thomas theory.

Findings

Identifies Hilbert scheme invariants with stable pair invariants.

Provides a geometric interpretation of the Oblomkov-Shende conjecture.

Develops motivic invariants for algebraic knots.

Abstract

Hilbert scheme topological invariants of plane curve singularities are identified to framed threefold stable pair invariants. As a result, the conjecture of Oblomkov and Shende on HOMFLY polynomials of links of plane curve singularities is given a Calabi-Yau threefold interpretation. The motivic Donaldson-Thomas theory developed by M. Kontsevich and the third author then yields natural motivic invariants for algebraic knots. This construction is motivated by previous work of V. Shende, C. Vafa and the first author on the large $N$ duality derivation of the above conjecture.