String theory and the Kauffman polynomial

TL;DR

This paper introduces a new integrality conjecture linking the colored Kauffman and HOMFLY polynomials, inspired by topological string theory, and provides tests and theoretical insights into their relationship.

Contribution

It proposes a novel integrality conjecture connecting Kauffman and HOMFLY polynomials within a topological string theory framework, supported by tests and theoretical reasoning.

Findings

The conjecture implies Rudolph's theorem.

Provides non-trivial tests supporting the conjecture.

Offers string theory explanations for the polynomial relationships.

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.