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.
Highlights
One of the most surprising consequences of the Gopakumar–Vafa duality [21] is that Chern–Simons invariants of knots and links in the three-sphere can be described by A-model open topological strings on the resolved conifold [45]
An equivalent description should exist in terms of open strings in the B-model, where the boundary conditions are set by holomorphic submanifolds
As usual in local mirror symmetry, the mirror is an algebraic curve in C∗ × C∗, and the invariants of the framed unknot can be computed as open topological string amplitudes in this geometry using the formalism of [8,40]
Summary
One of the most surprising consequences of the Gopakumar–Vafa duality [21] is that Chern–Simons invariants of knots and links in the three-sphere can be described by A-model open topological strings on the resolved conifold [45] (see [41] for a recent review). An equivalent description should exist in terms of open strings in the B-model, where the boundary conditions are set by holomorphic submanifolds This conjectural equivalence between knot theory and Gromov–Witten theory has been implemented and tested in detail for the (framed) unknot and the Hopf link. Their colored U (N ) invariants can be computed systematically by applying the topological recursion of [19] to the spectral curve, exactly as in [8] In this description, the (P, Q) torus knot comes naturally equipped with a fixed framing of QP units, just as in Chern–Simons theory [31]. 4, we study the matrix model representation of torus knots and we show that it leads to the spectral curve proposed in Sect. In the Appendix we derive that the loop equations satisfy by the torus knot matrix model
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