# zbMATH — the first resource for mathematics

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)
Full Text: