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Racah matrices and hidden integrability in evolution of knots. (English) Zbl 1398.57025
Summary: We construct a general procedure to extract the exclusive Racah matrices \(S\) and \(\overline{S}\) from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations \(R = [1]\), [2], [3] and \([2, 2]\). The matrices \(S\) and \(\overline{S}\) relate respectively the maps \((R \otimes R) \otimes \overline{R} \longrightarrow R\) with \(R \otimes(R \otimes \overline{R}) \longrightarrow R\) and \((R \otimes \overline{R}) \otimes R \longrightarrow R\) with \(R \otimes(\overline{R} \otimes R) \longrightarrow R\). They are building blocks for the colored HOMFLY polynomials of arbitrary arborescent (double fat) knots. Remarkably, the calculation realizes an unexpected integrability property underlying the evolution matrices.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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