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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33 Citations
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33 Citations

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References

SHOWING 1-10 OF 77 REFERENCES
Polynomial Invariants for Torus Knots¶and Topological Strings
Abstract: We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic…
Homological algebra of knots and BPS states
It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves…
Knot Invariants from Topological Recursion on Augmentation Varieties
Using the duality between Wilson loop expectation values of SU(N) Chern–Simons theory on S3 and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on S3…
Super-A-polynomial for knots and BPS states
Quadruply-graded colored homology of knots
We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of…
Knot Invariants and Topological Strings
3d analogs of Argyres-Douglas theories and knot homologies
A bstractWe study singularities of algebraic curves associated with 3d $ \mathcal{N}=2 $ theories that have at least one global flavor symmetry. Of particular interest is a class of theories TK…
Super-A-polynomials for twist knots
A bstractWe conjecture formulae of the colored superpolynomials for a class of twist knots Kp where p denotes the number of full twists. The validity of the formulae is checked by applying…
Large N Duality, Lagrangian Cycles, and Algebraic Knots
We consider knot invariants in the context of large N transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show…
Torus Knots and the Topological Vertex
We propose a class of toric Lagrangian A-branes on the resolved conifold that is suitable to describe torus knots on S3. The key role is played by the $${SL(2, \mathbb{Z})}$$SL(2,Z) transformation,…
...
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