  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.
