Knots, BPS States, and Algebraic Curves

@article{Garoufalidis2015KnotsBS,
  title={Knots, BPS States, and Algebraic Curves},
  author={Stavros Garoufalidis and Piotr Kucharski and Piotr Sułkowski},
  journal={Communications in Mathematical Physics},
  year={2015},
  volume={346},
  pages={75-113}
}
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… 
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