Knots, BPS states, and algebraic curves
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.
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.Attached Files
Submitted - 1504.06327v1.pdf
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Additional details
- Eprint ID
- 59531
- Resolver ID
- CaltechAUTHORS:20150814-093559642
- European Research Council (ERC)
- 335739
- Foundation for Polish Science
- Created
-
2015-08-14Created from EPrint's datestamp field
- Updated
-
2021-11-10Created from EPrint's last_modified field
- Caltech groups
- Walter Burke Institute for Theoretical Physics
- Other Numbering System Name
- CALT-TH
- Other Numbering System Identifier
- 2015-021