Racah matrices and hidden integrability in evolution of knots

Abstract

We construct a general procedure to extract the exclusive Racah matrices S and \bar 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 \bar S relate respectively the maps (R\otimes R)\otimes \bar R\longrightarrow R with R\otimes (R \otimes \bar R) \longrightarrow R and (R\otimes \bar R) \otimes R \longrightarrow R with R\otimes (\bar 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.

arXiv comment16 pages

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Cited by in corpus (23)

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• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
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