Knots, BPS states, and algebraic curves

TL;DR

This paper explores the relationship between BPS degeneracies, algebraic curves, and knot invariants, introducing extremal A-polynomials and deriving formulas that reveal new integrality properties and connections to knot polynomials.

Contribution

It introduces extremal A-polynomials, derives formulas for BPS degeneracies, and connects these to knot invariants and integrality conjectures, advancing understanding of knot-related algebraic structures.

Findings

Derived exact and asymptotic formulas for extremal BPS degeneracies.

Established consistency of augmentation polynomial degeneracies with colored HOMFLY polynomials.

Verified integrality of refined BPS degeneracies for various knots.

Abstract

We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that 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 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.