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Torus knots and mirror symmetry. (English) Zbl 1256.81086
Summary: 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 \(\mathrm {Sl}(2,\mathbb Z)\) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated with 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.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J33 Mirror symmetry (algebro-geometric aspects)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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