  It is well-known that the SU(2) quantum Racah coefficients or the Wigner $6j$
symbols have a closed form expression which enables the evaluation of any knot
or link polynomials in SU(2) Chern-Simons field theory. Using isotopy
equivalence of SU(N) Chern-Simons functional integrals over three balls with
one or more $S^2$ boundaries with punctures, we obtain identities to be
satisfied by the SU(N) quantum Racah coefficients. This enables evaluation of
the coefficients for a class of SU(N) representations. Using these
coefficients, we can compute the polynomials for some non-torus knots and
two-component links. These results are useful for verifying conjectures in
topological string theory.
