Torus Knots and the Topological Vertex

TL;DR

This paper introduces a new class of toric Lagrangian A-branes on the resolved conifold to model torus knots, deriving knot invariants and establishing mirror symmetry with B-model analysis, with implications for knots in various three-manifolds.

Contribution

It proposes a novel A-brane construction for torus knots, derives colored HOMFLY polynomials, and demonstrates mirror symmetry and generalization to other geometries.

Findings

Reproduces colored HOMFLY polynomials for torus knots

Establishes mirror symmetry between A-model and B-model

Generalizes construction to other toric Calabi-Yau geometries

Abstract

We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S^3. The key role is played by the SL(2,Z) transformation, which generates a general torus knot from the unknot. Applying the topological vertex to the proposed A-branes, we rederive the colored HOMFLY polynomials for torus knots, in agreement with the Rosso and Jones formula. We show that our A-model construction is mirror symmetric to the B-model analysis of Brini, Eynard and Marino. Comparing to the recent proposal by Aganagic and Vafa for knots on S^3, we demonstrate that the disk amplitude of the A-brane associated to any knot is sufficient to reconstruct the entire B-model spectral curve. Finally, the construction of toric Lagrangian A-branes is generalized to other local toric Calabi-Yau geometries, which paves the road to study knots in other three-manifolds such as lens spaces.