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

TL;DR

The paper proposes a conjecture that Racah matrices can be expressed solely in terms of R-matrix eigenvalues, enabling explicit calculation of HOMFLY polynomials for 3-strand knots in symmetric and antisymmetric representations.

Contribution

It introduces a new eigenvalue hypothesis for Racah matrices that simplifies the computation of HOMFLY polynomials for complex knots.

Findings

Explicit HOMFLY polynomials for V=[3] and V=[4] match known results.

The conjecture aligns with calculations for torus, figure-eight, and colored polynomials.

Provides indirect evidence supporting the eigenvalue-based expression of Racah matrices.

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.