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.

本文致力于从[35]开始的彩色结多项式的系统描述项目的进展–显式计算包含在内的Racah矩阵以进行表示$\mathrm{[R}=[3，3]$。我们在本文中建议使用一种强大的技术来实现这一点-在Gelfand-Tseitlin表的基础上使用最高权重方法。我们的结果使人们可以进行评估和调查$[3，3]$任意三链结的多色多项式，这证实了许多关于各种因式分解，通用性和微分展开的猜想。此外，在开发了一种方法的帮助下[45]我们设法计算出专属的拉卡矩阵$\mathrm{[R}=[3，3]$。我们的结果证实了在[51]中对这些矩阵的计算，这是基于对扭结的微分展开的显式形式的猜想。Racah矩阵和的显式答案$[3，3]$[1]中提供了最多10个交叉的3线结的彩色多项式。借助我们的包含性和排他性Racah矩阵的结果，可以计算$[3，3]$所谓的单环族链接的任何链接的彩色HOMFLY-PT多项式，可通过添加一个环从树状链接获得。