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Knots, BPS States, and Algebraic Curves

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Abstract

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.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Stavros Garoufalidis

  2. Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093, Warsaw, Poland

    Piotr Kucharski & Piotr Sułkowski

  3. Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA, 91125, USA

    Piotr Sułkowski

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  1. Stavros Garoufalidis
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  2. Piotr Kucharski
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  3. Piotr Sułkowski
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Correspondence to Piotr Sułkowski.

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Communicated by M. Salmhofer

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Garoufalidis, S., Kucharski, P. & Sułkowski, P. Knots, BPS States, and Algebraic Curves. Commun. Math. Phys. 346, 75–113 (2016). https://doi.org/10.1007/s00220-016-2682-z

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