1209.6304 (H. Itoyama et al.)

Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for
3-strand knots in any symmetric and antisymmetric representations    [PDF]

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

Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this conjecture is sufficient to explicitly calculate HOMFLY polynomials for all the 3-strand braids in arbitrary (anti)symmetric representations. We list the examples of so obtained polynomials for V=[3] and V=[4], and they are in accordance with the known answers for torus and figure-eight knots, as well as for the colored special and Jones polynomials. This provides an indirect evidence in support of our conjecture.

View original: http://arxiv.org/abs/1209.6304