Racah matrices and hidden integrability in evolution of knots

TL;DR

This paper develops a method to derive Racah matrices from mixing matrices in knot theory, revealing an unexpected integrability property that enhances understanding of colored HOMFLY polynomials for complex knots.

Contribution

The authors introduce a general procedure to extract Racah matrices from mixing matrices using the evolution method, applicable to simple representations, and uncover hidden integrability in the process.

Findings

Successfully derived Racah matrices for simple representations.

Revealed an unexpected integrability property in evolution matrices.

Enhanced computational methods for colored HOMFLY polynomials.

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.