On the block structure of the quantum R-matrix in the three-strand braids

TL;DR

This paper investigates the block structure of quantum R-matrices in three-strand braids, revealing conditions under which these matrices can be simplified into block-diagonal form, aiding calculations of colored HOMFLY polynomials.

Contribution

It demonstrates that R2 matrices can be transformed into block-diagonal form when eigenvalues of R1 matrices coincide, with a specific rotation angle of ±π/4, advancing the simplification of braid invariant computations.

Findings

R2 matrices can be block-diagonalized under certain eigenvalue conditions

The rotation angle for basis change is ±π/4

Simplifies calculations of colored HOMFLY polynomials

Abstract

Quantum $\mathcal{R}$-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation $T$ of $SU_q(N)$ associated with each strand one needs two matrices: $\mathcal{R}_1$ and $\mathcal{R}_2$. They are related by the Racah matrices $\mathcal{R}_2 = \mathcal{U} \mathcal{R}_1 \mathcal{U}^{\dagger}$. Since we can always choose the basis so that $\mathcal{R}_1$ is diagonal, the problem is reduced to evaluation of $\mathcal{R}_2$-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that $\mathcal{R}_2$-matrices could be transformed into a block-diagonal ones. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of $\mathcal{R}_1$-matrix. The angle of the rotation in the sectors corresponding to accidentally coinciding eigenvalues from the basis defined by the Racah matrix to the basis in which $\mathcal{R}_2$ is block-diagonal is $\pm \frac{\pi}{4}$.