HOMFLY polynomials in representation [3, 1] for 3-strand braids

A bstract This paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R ⊗3 −→ Q with all possible Q , for R = [3 , 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3 , 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family ( n, −1 | 1 , −1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.