Torus Knots and Mirror Symmetry

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.arXiv:1105

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

“…The present work generalizes the analysis of the model r = 2 [1,18,31] relevant to study lens spaces, and the invariant of fiber knots in lens spaces is equal to invariants of torus knots in S 3 . Chern-Simons theory on M at large N is dual to type A open topological strings on T * M [49], and through geometric transitions, this can sometimes be related to closed topological strings on another target space X M .…”

“…Chern-Simons theory on M at large N is dual to type A open topological strings on T * M [49], and through geometric transitions, this can sometimes be related to closed topological strings on another target space X M . This program has been completed for M = S 3 [29] and the lens spaces Z p 2 \S 3 /Z p 1 [1,18,31] and X M is obtained by cyclic quotient of the resolved conifold in both cases. At the level of the spectral curve, this just amounts to perform a fractional framing transforming on the mirror curve of the resolved conifold.…”

Formal multidimensional integrals, stuffed maps, and topological recursion

“…. ), and has appeared provably or experimentally in many problems of 2D enumerative geometry: the two hermitian matrix model [EO08] and the chain of hermitian matrices [CEO06], topological string theory and Gromov-Witten invariants [BKMP09, BEMS10, EMS09, MP12, NS11, EO12], integrable systems [BE09,BE10,BE11], intersection numbers on the moduli space of curves [EO07b,Eyn11b,Eyn11a], asymptotic of knot invariants [DFM11,BE12,BEM12], . .…”

On universal knot polynomials

“…The purpose of this paper is to demonstrate that for the adjointcolored polynomials this is indeed possible, at least for some classes of knots. Namely, we find universal knot polynomials for 2− and 3-strand torus knots, when Rosso-Jones formula [42][43][44][45][46] is available for any representation of any simple Lie algebra, moreover, in this case the Rosso-Jones formula itself can be made universal. We also do so for the figure eight knot 4 1 , where this provides a new set of colored HOMFLY polynomials and continuation to exceptional groups is a new result of its own value.…”