BPS counting for knots and combinatorics on words

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-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.

41 pages, 1 figure, a supplementary Mathematica file attached

References in corpus (7)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• Sequencing BPS Spectra
• Integrality of instanton numbers and p-adic B-model
• Integrality of instanton numbers
• Quantum curves and conformal field theory
• Integrality theorems in the theory of topological strings

Cited by in corpus (16)

• Knots-quivers correspondence
• BPS states, knots and quivers
• Topological strings, strips and quivers
• Donaldson-Thomas invariants, torus knots, and lattice paths
• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories
• Checks of integrality properties in topological strings
• BPS operators in $\mathcal{N}=4$ $SO(N)$ super Yang-Mills theory: plethysms, dominoes and words
• Integrality structures in topological strings I: framed unknot
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Gaussian distribution of LMOV numbers
• Nahm sums, quiver A-polynomials and topological recursion
• On explicit formulae of LMOV invariants
• Quiver diagonalization and open BPS states
• Topological strings, quiver varieties and Rogers-Ramanujan identities
• Integrality of Framing and Geometric Origin of 2-functions