Milnor invariants and the HOMFLYPT Polynomial

Abstract

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 ${\bar{\mu }}_L\left(J\right)$ vanishes for any sequence $J$ with length at most $k$, then any Milnor $\bar{\mu }$–invariant ${\bar{\mu }}_L\left(I\right)$ 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 nonvanishing” Milnor invariants can be always represented as such a linear combination.