  Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid
carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank
$r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation
becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from
3-strand braid, can be constructed using the quantum Racah coefficients
(6j-symbols) of $U_q(sl_2)$. In this paper, we generalise it to links whose
components carry different symmetric representations. We illustrate the
technique by evaluating multi-colored link polynomials $H_{[r_1],[r_2]}$ for
the two-component link L7a3 whose components carry $[r_1]$ and $[r_2]$ colors.
