Quantum Racah matrices up to level 3 and multicolored link invariants

TL;DR

This paper systematically computes inclusive Racah matrices up to level 3, enabling the evaluation of colored knot and link invariants for 3-strand knots and links, and verifies related conjectures.

Contribution

It provides explicit inclusive Racah matrices for all channels with representations up to size 3, advancing the calculation of colored knot invariants.

Findings

Explicit Racah matrices for |R| ≤ 3

Colored polynomials for 3-strand knots up to 10 crossings

Validation of the eigenvalue conjecture

Abstract

This paper is a next step in the project of systematic description of colored knot and link invariants started in previous papers. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3\longrightarrow Q$ with all possible $Q$, for $|R|\leq 3$. 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.