Racah matrices and hidden integrability in evolution of knots

We construct a general procedure to extract the exclusive Racah matrices S   and S¯ from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R=[1]R=[1], [2], [3] and [2,2][2,2]. The matrices S   and S¯ relate respectively the maps (R⊗R)⊗R¯⟶R with R⊗(R⊗R¯)⟶R and (R⊗R¯)⊗R⟶R with R⊗(R¯⊗R)⟶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.