Knots-quivers correspondence

Abstract

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossings, and some knots with thick homology. Moreover, based on these properties, we derive previously unknown expressions for colored HOMFLY-PT polynomials and superpolynomials for various knots. For all knots, for which we identify the corresponding quivers, the LMOV conjecture for all symmetric representations (i.e. integrality of relevant BPS numbers) is automatically proved.

43 pages, 3 figures

References in corpus (9)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• BPS states, knots and quivers
• Checks of integrality properties in topological strings
• Integrality structures in topological strings I: framed unknot
• Refined large N duality for knots
• Topological strings, quiver varieties and Rogers-Ramanujan identities
• Integrality of Framing and Geometric Origin of 2-functions

Cited by in corpus (30)

• 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
• Differential expansion for link polynomials
• Stable maps to Looijenga pairs
• Branches, quivers, and ideals for knot complements
• Rational links and DT invariants of quivers
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Branes, quivers and wave-functions
• Tangle addition and the knots-quivers correspondence
• A new symmetry of the colored Alexander polynomial
• Harer-Zagier formulas for knot matrix models
• Learning knot invariants across dimensions
• Nahm sums, quiver A-polynomials and topological recursion
• Stable maps to Looijenga pairs: orbifold examples
• Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d $\mathcal{N}=2$ theories
• Quiver diagonalization and open BPS states
• Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again
• Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition
• 3d $\mathcal{N}=2$ Brane Webs and Quiver Matrices
• Open/Closed BPS Correspondence and Integrality
• Refined open topological strings revisited
• The colored Jones polynomials as vortex partition functions
• Three Dimensional Topological Field Theories and Nahm Sum Formulas
• On knots, complements, and 6j-symbols
• Permutohedra for knots and quivers
• Inverted state sums, inverted Habiro series, and indefinite theta functions
• Enumerative geometry of surfaces and topological strings
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants