Eigenvalue hypothesis for multi-strand braids

TL;DR

This paper extends the eigenvalue hypothesis to multi-strand braids, enabling explicit computation of $ ext{R}$-matrices and Racah matrices for complex knots, advancing the understanding of colored HOMFLY-PT polynomials.

Contribution

It generalizes the eigenvalue hypothesis to higher strand numbers, incorporating non-neighbouring relations, and explicitly computes relevant matrices for up to 6x6 sizes.

Findings

Derived explicit forms of $ ext{R}$-matrices and Racah matrices for up to 6x6 matrices.

Compared eigenvalue hypothesis results with highest weight method for four-strand braids.

Provided all Racah matrices for representation [2] and validated the approach.

Abstract

Computing polynomial form of the colored HOMFLY-PT for non-arborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the ${\cal R}$-matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighbouring $\mathcal{R}$ matrices are also incorporated. By solving these equations, we determine the explicit form for $\mathcal{R}$-matrices and the inclusive Racah matrices in terms of braiding eigenvalues (for matrices of size up to 6 by 6). For comparison, we briefly discuss the highest weight method for four-strand braids carrying fundamental and symmetric rank two $SU_q(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation $[2]$ and compare with the matrices obtained from eigenvalue hypothesis.