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Torus Knots and the Topological Vertex

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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 \({SL(2, \mathbb{Z})}\) 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 Mariño. Compared to the recent proposal by Aganagic and Vafa for knots on S 3, we demonstrate that the disk amplitude of the A-brane associated with 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.

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Authors and Affiliations

  1. Bethe Center for Theoretical Physics, Physikalisches Institut der Universität Bonn, Nussallee 12, 53315, Bonn, Germany

    Hans Jockers, Albrecht Klemm & Masoud Soroush

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  1. Hans Jockers
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  2. Albrecht Klemm
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  3. Masoud Soroush
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Correspondence to Masoud Soroush.

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Jockers, H., Klemm, A. & Soroush, M. Torus Knots and the Topological Vertex. Lett Math Phys 104, 953–989 (2014). https://doi.org/10.1007/s11005-014-0687-0

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