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

Abstract

This paper is devoted to the advance in the project of systematic description of colored knot polynomials started in [35] – explicit calculation of the inclusive Racah matrices for representation R=[3,3]. This is made possible by a powerful technique which we suggest in this paper – the use of highest weight method in the basis of Gelfand-Tseitlin tables. Our 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. Furthermore, with the help of a method developed in [45] we manage to calculate exclusive Racah matrices in R=[3,3]. Our results confirm a calculation of these matrices in [51], 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 [1]. With the help of our results for inclusive and exclusive Racah matrices, it is possible to compute [3,3]-colored HOMFLY-PT polynomial of any link for the so-called one-looped family links, which are obtained from arborescent links by adding one loop.