HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

TL;DR

This paper derives explicit formulas for HOMFLY and superpolynomials of the figure eight knot in all symmetric and antisymmetric representations, generalizing previous results and confirming the Ooguri-Vafa conjecture.

Contribution

It provides a comprehensive formula for HOMFLY polynomials in arbitrary symmetric and antisymmetric representations, including a difference equation and symmetry transformations.

Findings

Explicit formulas for HOMFLY polynomials in all symmetric representations

A difference equation ('non-commutative A-polynomial') in the representation variable p

A deformation from HOMFLY to superpolynomials

Abstract

Explicit answer is given for the HOMFLY polynomial of the figure eight knot $4_1$ in arbitrary symmetric representation R=[p]. It generalizes the old answers for p=1 and 2 and the recently derived results for p=3,4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the \sigma_R = \sigma_{[1]}^{|R|} identity for the "special" polynomials (they define the leading asymptotics of HOMFLY at q=1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation ("non-commutative A-polynomial") in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R=[1^p], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.