Milnor invariants and the HOMFLYPT polynomial

TL;DR

This paper establishes formulas linking Milnor invariants of links in 3-sphere to the HOMFLYPT polynomial, enabling their computation via knot polynomials after specific band sum operations.

Contribution

It provides explicit formulas expressing Milnor invariants in terms of the HOMFLYPT polynomial, especially when lower-order invariants vanish, extending the computational tools for link invariants.

Findings

Milnor invariants can be expressed via HOMFLYPT polynomial after band sum operations.

First non-vanishing Milnor invariants are representable as linear combinations of HOMFLYPT polynomials.

Formulas apply to links with certain vanishing conditions on lower-order invariants.

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}_J(L) vanishes for any sequence J with length at most k, then any Milnor \bar{\mu}-invariant \bar{\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.