Communications in Mathematical Physics

, Volume 346, Issue 1, pp 75–113 | Cite as

Knots, BPS States, and Algebraic Curves

  • Stavros Garoufalidis
  • Piotr Kucharski
  • Piotr Sułkowski


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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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Stavros Garoufalidis
    • 1
  • Piotr Kucharski
    • 2
  • Piotr Sułkowski
    • 2
    • 3
  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Faculty of PhysicsUniversity of WarsawWarsawPoland
  3. 3.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaUSA

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