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Factorization of differential expansion for antiparallel double-braid knots. (English) Zbl 1388.57013
Summary: Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution – that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations \(R=[r^s]\) we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of \(R=[33]\). The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices \(\bar S\) and \(S\) – what allows to calculate \([33]\)-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations fully described are only the contributions of the single-floor pyramids. One step still remains to be done.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81T13 Yang-Mills and other gauge theories in quantum field theory
Software:
Knot Atlas
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