Knots, BPS states, and algebraic curves

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.

43 pages, 6 figures

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Cited by in corpus (20)

• Knots-quivers correspondence
• BPS states, knots and quivers
• Tabulating knot polynomials for arborescent knots
• Reconstructing WKB from topological recursion
• 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
• BPS counting for knots and combinatorics on words
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Branches, quivers, and ideals for knot complements
• Checks of integrality properties in topological strings
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Refined large N duality for knots
• 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
• A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras
• Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition