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

TL;DR

This paper computes the Racah matrices for the [3,1] representation in 3-strand braids, enabling detailed analysis of colored HOMFLY polynomials and confirming several conjectures in knot theory.

Contribution

It provides the explicit Racah matrix for [3,1], facilitating the calculation of colored HOMFLY polynomials for 3-strand knots and supporting the eigenvalue hypothesis.

Findings

Explicit Racah matrix for [3,1] obtained

Confirmed conjectures on factorizations and universality

Analyzed next-to-twist knots and verified eigenvalue hypothesis

Abstract

This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. 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.