EIGENVALUE HYPOTHESIS FOR RACAH MATRICES AND HOMFLY POLYNOMIALS FOR 3-STRAND KNOTS IN ANY SYMMETRIC AND ANTISYMMETRIC REPRESENTATIONS

Abstract

Character expansion expresses extended HOMFLY polynomials through traces of products of finite-dimensional [Formula: see text]- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding [Formula: see text]-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 R = [3] and R = [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.