Milnor invariants of length $2k+2$ for links with vanishing Milnor invariants of length $\leq k$

TL;DR

This paper extends the formula for Milnor invariants of links to length $2k+2$, incorporating correction terms involving HOMFLYPT polynomials, especially for links with vanishing invariants of length $ extless= k$.

Contribution

The authors improve existing formulas for Milnor invariants of length $2k+2$ by adding correction terms based on HOMFLYPT polynomials, enhancing understanding of link invariants.

Findings

New formula for Milnor invariants of length $2k+2$ with correction terms

Explicit computation for 4-component links where invariants of length 1 vanish

Extension of previous results to higher-length invariants

Abstract

J.-B. Meilhan and the second author showed that any Milnor $\bar{\mu}$-invariant of length between 3 and $2k+1$ can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all $\bar{\mu}$-invariants of length $\leq k$ vanish. They also showed that their formula does not hold for length $2k+2$. In this paper, we improve their formula to give the $\bar{\mu}$-invariants of length $2k+2$ by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined by $\bar{\mu}$-invariants of length $k+1$. In particular, for any 4-component link the $\bar{\mu}$-invariants of length 4 are given by our formula, since all $\bar{\mu}$-invariants of length 1 vanish.