Torus Knots and the Topological Vertex
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.
Keywords
Cite
@article{arxiv.1212.0321,
title = {Torus Knots and the Topological Vertex},
author = {Hans Jockers and Albrecht Klemm and Masoud Soroush},
journal= {arXiv preprint arXiv:1212.0321},
year = {2014}
}
Comments
harvmac, 42 pages, v2: comments and references added