Torus Knots and Mirror Symmetry

By Andrea Brini, Marcos Mariño, B. Eynard. 2011

203 citations recorded by the source corpus.

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 $${{\rm Sl}(2, \mathbb {Z})}$$ 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.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a65a9da2f4db38cdd30cec998988961f55356c15

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Citations in the corpus

Torus knots and the rational DAHA (2012, methodology): “ity of Λkhin L m/n is [m+n− k−1]q! [m]q ·[n]q ·[k]q!·[n− k− 1]q![m− k− 1]q! . This matches the formula for the coefﬁcient of (−1)kaµ+2k for the HOMFLY polynomial obtained in [Gor10] (Theorem 3.1) and [BEM11] (Equation 3.46). In particular, it is symmetric in mand n. We obtain the following result. T”
Knot Homology from Refined Chern-Simons Theory (2011, background): “15 In the unrefined case, these were considered from perspective of string theory recently in [81].”
Random matrices. (2015, background): “In the case of the torus knot KP,Q, the relevant eigenvalue integral is Z ∝ ˆ RN ∏ i j 2 sinh ( λi − λj 2P ) 2 sinh ( λi − λj 2Q ) N ∏ i=1 e− N 2PQt λ2i dλi , (3.93) = ˆ RN+ ∏ i j (xQi − xQj )(xPi − xPj ) N ∏ i=1 e− N 2PQt (log xi) 2 elog xidxi , (3.94) where λi = PQ log xi.”