BPS counting for knots and combinatorics on words

TL;DR

This paper explores the relationship between quantum BPS invariants, difference equations, and combinatorial models, linking knot invariants to combinatorics on words and providing explicit examples for colored extremal knot polynomials.

Contribution

It introduces a novel combinatorial framework for understanding quantum BPS invariants and their relation to difference equations and knot polynomials, with explicit constructions and invariants.

Findings

Constructed combinatorial models encoding difference equations

Derived dual quantum extremal A-polynomials for various knots

Identified BPS invariants and discussed their integrality

Abstract

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-Poincar\'e 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-Mari\~no-Ooguri-Vafa invariants) and discuss their integrality.