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HOMFLY polynomials in representation \([3, 1]\) for 3-strand braids. (English) Zbl 1388.57011
Summary: This paper is a new step in the project of systematic description of colored knot polynomials started in [A. Mironov and A. Morozov, Nucl. Phys., B 899, 395–413 (2015; Zbl 1331.81264)]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels \(R^3\to Q\) with all possible \(Q\), for \(R=[3,1]\). The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate \([3,1]\)-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family \((n,-1|1,-1)\) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
81T13 Yang-Mills and other gauge theories in quantum field theory
Software:
Knot Atlas
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References:
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