Journal of High Energy Physics

, 2016:120 | Cite as

BPS counting for knots and combinatorics on words

  • Piotr Kucharski
  • Piotr Sułkowski
Open Access
Regular Article - Theoretical Physics


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 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Supplementary material

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


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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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