Milnor invariants and the HOMFLYPT polynomial

We give formulas expressing Milnor invariants of an $n$-component link $L$ in the 3-sphere in terms of the HOMFLYPT polynomial as follows.
If the Milnor invariant $\overline{\mu}_J(L)$ vanishes for any sequence $J$ with length at most $k$,
then any Milnor $\overline{\mu}$-invariant $\overline{\mu}_I(L)$ with length between $3$ and $2k+1$ can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations.
In particular, the `first non vanishing' Milnor invariants can be always represented as such a linear combination.
This is a joint work with JB Meilhan.