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

Saswati Dhara; A. Mironov; A. Morozov; An. Morozov; P. Ramadevi; Vivek Kumar Singh; A. Sleptsov

doi:10.1007/s00023-019-00841-z

paper

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

arXiv:1805.03916 · doi:10.1007/s00023-019-00841-z

Abstract

Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $S U (N)$ 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 $U_{q} (s l_{2})$ . 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.

11 pages, 1 figure