Journal of High Energy Physics

, 2016:120

BPS counting for knots and combinatorics on words

Open Access
Regular Article - Theoretical Physics

DOI: 10.1007/JHEP11(2016)120

Cite this article as:
Kucharski, P. & Sułkowski, P. J. High Energ. Phys. (2016) 2016: 120. doi:10.1007/JHEP11(2016)120


We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincaré series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to the branch of mathematics referred to as combinatorics on words. We illustrate these results in the framework of colored extremal knot polynomials: among others we determine dual quantum extremal A-polynomials for various knots, present associated combinatorial models, find corresponding BPS invariants (extremal Labastida-Mariño-Ooguri-Vafa invariants) and discuss their integrality.


Chern-Simons Theories Non-Commutative Geometry Topological Field Theories Topological Strings 

Supplementary material

13130_2016_5060_MOESM1_ESM.nb (102 kb)
ESM 1(NB 102 kb)

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Faculty of PhysicsUniversity of WarsawWarsawPoland
  2. 2.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.

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