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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.

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
Full Text: DOI
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