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arXiv · 1508.02870

Colored knot polynomials. HOMFLY in representation [2,1]

A. Mironov·A. Morozov·An. Morozov·A. Sleptsov
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Abstract

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in various representations a la arXiv:1112.2654, (iii) tabulating and collecting the results at www.knotebook.org. In this paper we discuss only representation R=[2,1] and construct all necessary ingredients that allow one to evaluate knot/links represented by three strand closed parallel braids with inserted double-fat fingers. In particular, it is used to evaluate knots from a 7-parametric family: this family contains over 80% of knots with up to 10 intersections, but does not include mutants.

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A. Mironov, A. Morozov, An. Morozov, A. Sleptsov. 2015-08-12. Colored knot polynomials. HOMFLY in representation [2,1]. https://doi.org/10.1142/s0217751x15501699

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