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 μ_J(L) vanishes for any sequence J with length at most k, then any Milnor μ-invariant μ_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.