Quantum Racah matrices and evolution of knots

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 knots.
The talk is based on the joint work with A.Mironov, A.Morozov and An.Morozov [1]. The author is partially supported by the Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z50.31.0020), RFBR grant mol-a-dk 16-31-60082 and MK-8769.2016.1
References:

• A. Mironov, A. Morozov, An. Morozov, A. Sleptsov, Racah matrices and hidden integrability in evolution of knots. arXiv:1605.04881