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String theory and the Kauffman polynomial. (English) Zbl 1207.81129
Summary: 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.

MSC:
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
81T45 Topological field theories in quantum mechanics
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