Large N duality, lagrangian cycles, and algebraic knots

TL;DR

This paper explores the connection between knot invariants, large N duality, and topological string theory, providing explicit constructions of Lagrangian cycles for algebraic knots and linking them to HOMFLY polynomials.

Contribution

It offers explicit constructions of Lagrangian cycles for algebraic knots and relates these to HOMFLY polynomials, advancing understanding of knot invariants in topological string theory.

Findings

Explicit construction of Lagrangian cycles for algebraic knots

Verification of a conjecture linking stable pairs and HOMFLY polynomials

Direct A-model computation of HOMFLY polynomial for torus knots

Abstract

We consider knot invariants in the context of large $N$ transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicity constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.