Matrix integral expansion of coloured Jones polynomials for figure-eight knot

In this note we examine a possible extension of the matrix integral representation of knot invariants beyond the class of torus knots. In particular, we study a representation of the $SU(2)$ quantum Racah coefficients by double matrix integrals. We find that the Racah coefficients are mapped to expansion coefficients in some basis of double integrals. The transformed coefficients have a number of interesting algebraic properties.