Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations

TL;DR

This paper extends methods for calculating colored HOMFLY-PT polynomials from 3-strand braids to multi-component links with different symmetric representations, simplifying evaluations using quantum Racah coefficients.

Contribution

It generalizes previous techniques to evaluate multi-colored link polynomials with components carrying different symmetric representations.

Findings

Successfully evaluated multi-colored link polynomials for a two-component link.

Demonstrated the use of quantum Racah coefficients in the computation.

Provided explicit examples for links with arbitrary symmetric representations.

Abstract

Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from 3-strand braid, can be constructed using the quantum Racah coefficients (6j-symbols) of $U_q(sl_2)$. In this paper, we generalise it to links whose components carry different symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials $H_{[r_1],[r_2]}$ for the two-component link L7a3 whose components carry $[r_1]$ and $[r_2]$ colors.