Torus knots and mirror symmetry

TL;DR

This paper introduces a spectral curve in the B-model that encodes torus knot invariants, linking mirror symmetry, topological recursion, and matrix models to compute colored HOMFLY polynomials.

Contribution

It constructs a novel spectral curve for torus knots using SL(2,Z) symmetry and derives it from large N matrix model limits, connecting multiple theoretical frameworks.

Findings

Spectral curve encodes all colored HOMFLY invariants of torus knots.

Application of topological recursion to the curve reproduces knot invariants.

The curve is obtained as a large N limit of a matrix model for torus knots.

Abstract

We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.