String theory and the Kauffman polynomial

Abstract

We propose a new, precise integrality conjecture for the colored Kauffman polynomial of knots and links inspired by large N dualities and the structure of topological string theory on orientifolds. According to this conjecture, the natural knot invariant in an unoriented theory involves both the colored Kauffman polynomial and the colored HOMFLY polynomial for composite representations, i.e. it involves the full HOMFLY skein of the annulus. The conjecture sheds new light on the relationship between the Kauffman and the HOMFLY polynomials, and it implies for example Rudolph's theorem. We provide various non-trivial tests of the conjecture and we sketch the string theory arguments that lead to it.

36 pages, many figures; references and examples added, typos corrected, final version to appear in CMP

References in corpus (4)

• BPS Microstates and the Open Topological String Wave Function
• The Real Topological String on a local Calabi-Yau
• Universal Character and Large N Factorization in Topological Gauge/String Theory
• Composite Invariants and Unoriented Topological String Amplitudes

Cited by in corpus (27)

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• Colored knot polynomials. HOMFLY in representation [2,1]
• Colored Kauffman Homology and Super-A-polynomials
• Orientifolds and the Refined Topological String
• Exact probes of orientifolds
• Colored HOMFLY polynomial via skein theory
• Hopf link invariants and integrable hierarchies
• Tangle blocks in the theory of link invariants
• Knot Invariants from Topological Recursion on Augmentation Varieties
• Instanton counting and O-vertex
• Distinguishing Mutant Knots
• Microstates of a $2d$ Black Hole in string theory
• Checks of integrality properties in topological strings
• On Hopf-induced deformation of topological locus
• Kerov functions for composite representations and Macdonald ideal
• Matrix models for classical groups and Toeplitz$\pm$Hankel minors with applications to Chern-Simons theory and fermionic models
• A non-torus link from topological vertex
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Refined large N duality for knots
• Composite Representation Invariants and Unoriented Topological String Amplitudes
• Composite Invariants and Unoriented Topological String Amplitudes
• $S$-Functions, Spectral Functions of Hyperbolic Geometry, and Vertex Operators with Applications to Structure for Weyl and Orthogonal Group Invariants
• Invariants of knots and links at roots of unity
• Hopf superpolynomial from topological vertices
• Torus Knots in Lens Spaces & Topological Strings
• New structure for orthogonal quantum group invariants
• Topics in Open Topological Strings