Multi-colored Links From 3-Strand Braids Carrying Arbitrary Symmetric Representations

Abstract

Obtaining HOMFLY-PT polynomials $$H_{R_1,\ldots ,R_l}$$ for arbitrary links with l components colored by arbitrary SU(N) representations $$R_1,\ldots ,R_l$$ is a very complicated problem. For a class of rank r symmetric representations, the [r]-colored HOMFLY-PT polynomial $$H_{[r_1],\ldots ,[r_l]}$$ evaluation becomes simpler, but the general answer lies far beyond our current capabilities. To simplify the situation even more, one can consider links that can be realized as a 3-strand closed braid. Recently (Itoyama et al. in Int J Mod Phys A28:1340009, 2013. arXiv:1209.6304 ), it was shown that $$H_{[r]}$$ for knots realized by 3-strand braids can be constructed using the quantum Racah coefficients (6j-symbols) of $$U_q(sl_2)$$ , which makes easy not only to evaluate such invariants, but also to construct analytical formulas for $$H_{[r]}$$ of various families of 3-strand knots. In this paper, we generalize this approach to links whose components carry arbitrary 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. Using our results for exclusive Racah matrices, it is possible to calculate symmetric-colored HOMFLY-PT polynomials of links for the so-called one-looped links, which are obtained from arborescent links by adding a loop. This is a huge class of links that contains the entire Rolfsen table, all 3-strand links, all arborescent links, and, for example, all mutant knots with 11 intersections.