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 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.

20 pages + 21 pages of knot tables

References in corpus (2)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation

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• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• Differential expansion for link polynomials
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• New symmetries for the $U_q(s{l}_N)$ 6-j symbols from the Eigenvalue conjecture
• On possible existence of HOMFLY polynomials for virtual knots
• Gaussian distribution of LMOV numbers
• Matrix integral expansion of colored Jones polynomials for figure-eight knot