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

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