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

Understand

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.