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Multi-colored links from 3-strand braids carrying arbitrary symmetric representations. (English) Zbl 1426.81044
Summary: 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 [H. Itoyama et al., Int. J. Mod. Phys. A 28, No. 3–4, Paper No. 1340009, 81 p. (2013; Zbl 1259.81082)], 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.

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
22E70 Applications of Lie groups to the sciences; explicit representations
58J28 Eta-invariants, Chern-Simons invariants
57K14 Knot polynomials
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