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Knots, BPS states, and algebraic curves

Garoufalidis, Stavros and Kucharski, Piotr and Sułkowski, Piotr (2016) Knots, BPS states, and algebraic curves. Communications in Mathematical Physics, 346 (1). pp. 75-113. ISSN 0010-3616. doi:10.1007/s00220-016-2682-z.

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We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.

Item Type:Article
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URLURL TypeDescription Paper ReadCube access
Kucharski, Piotr0000-0002-9599-5658
Sułkowski, Piotr0000-0002-6176-6240
Additional Information:© 2016 Springer-Verlag Berlin Heidelberg. Received: 29 May 2015; Accepted: 10 April 2016. Published online: 4 July 2016. We thank Estelle Basor, Brian Conrey, Sergei Gukov,Maxim Kontsevich, Satoshi Nawata, and Marko Stošić for insightful discussions. We greatly appreciate the hospitality of American Institute of Mathematics, Banff International Research Station, International Institute of Physics in Natal, and Simons Center for Geometry and Physics, where parts of this work were done. This work is supported by the ERC Starting Grant no. 335739 “Quantum fields and knot homologies” funded by the European Research Council under the European Union’s Seventh Framework Programme, and the Foundation for Polish Science.Framework Programme, and the Foundation for Polish Science.
Group:Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
European Research Council (ERC)335739
Foundation for Polish ScienceUNSPECIFIED
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Issue or Number:1
Record Number:CaltechAUTHORS:20150814-093559642
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:59531
Deposited By: Joy Painter
Deposited On:14 Aug 2015 16:42
Last Modified:10 Nov 2021 22:22

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