  Using the recently proposed differential hierarchy (Z-expansion) technique,
we obtain a general expression for the HOMFLY polynomials in two arbitrary
symmetric representations of link families, including Whitehead and Borromean
links. Among other things, this allows us to check and confirm the recent
conjecture of arXiv:1304.5778 that the large representation limit (the same as
considered in the knot volume conjecture) of this quantity matches the
prediction from mirror symmetry consideration. We also provide, using the
evolution method, the HOMFLY polynomial in two arbitrary symmetric
representations for an arbitrary member of the one-parametric family of
2-component 3-strand links, which includes the Hopf and Whitehead links.
