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

TL;DR

This work provides an explicit closed-form expression for the HOMFLY polynomial of the figure-eight knot 4_1 in arbitrary symmetric representations [p], and, via a Z2 symmetry, in antisymmetric representations [1^p]. It connects this main result to a first-order difference equation in the representation parameter p (a non-commutative A-polynomial) and to a beta-deformed superpolynomial that preserves positivity under standard refinements. The authors justify the formula by specializations (q→1, A=q^N, A=q^2, A=1) and by compatibility with the Ooguri-Vafa conjecture, and they derive a 3-strand braid-based derivation, as well as a structured superpolynomial reformulation as a sum over boxes. They also outline generalization paths to arbitrary representations and broader knot families, highlighting both the potential and the tests needed for full verification of these conjectures and extensions.

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 σ_R = σ_{[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.