Quantum Racah matrices up to level 3 and multicolored link invariants

Abstract

This paper is a next step in the project of systematic description of colored knot and link invariants started in Mironov and Morozov (2015) and Mironov et al. (2017). In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels R 1 ⊗ R 2 ⊗ R 3 ⟶ Q with all possible Q , for | R | ≤ 3 . The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org . Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.