Quantum Racah matrices and 3-strand braids in representation [3,3]

Abstract

This paper is a next 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 $\textit{inclusive}$ Racah matrices, i.e. the whole set of mixing matrices in channels $R^{\otimes 3}\longrightarrow Q$ with all possible $Q$, for $R=[3,3]$. The case $R=[3,3]$ is a multiplicity free case as well as $R=[2,2]$ obtained in arXiv:1605.03098. The calculation is made possible by the use of highest weight method with the help of Gelfand-Tseitlin tables. The result allows one to evaluate and investigate $[3,3]$-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. With the help of a method developed in arXiv:1605.04881 we manage to calculate {\it exclusive} Racah matrices $S$ and $\bar S$ in $R=[3,3]$. Our results confirm a calculation of these matrices in arXiv:1606.06015, which was based on the conjecture of explicit form of differential expansion for twist knots. Explicit answers for Racah matrices and $[3,3]$-colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org.