1203.5978 (H. Itoyama et al.)

HOMFLY polynomial for figure eight knot in any symmetric representations    [PDF]

H. Itoyama, A. Mironov, A. Morozov, An. Morozov

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. Further generalizations seem possible to arbitrary Young diagrams R and to entire series of non-torus links having 3-strand braid representation like (1,-1)^n.

View original: http://arxiv.org/abs/1203.5978