The Milnor invariants of clover links

TL;DR

This paper studies the properties of Milnor invariants for clover links, establishing conditions for their well-definedness, and applies these results to classify 4-clover links up to edge-homotopy.

Contribution

It provides new conditions under which Milnor invariants are well-defined for clover links and offers an explicit classification of 4-clover links based on these invariants.

Findings

Milnor numbers of certain lengths are well-defined under specific conditions.

The range of Milnor numbers for non-repeated sequences is explicitly determined.

An edge-homotopy classification of 4-clover links is achieved.

Abstract

J.P. Levine introduced a clover link to investigate the indeterminacy of the Milnor invariants of a link. It is shown that for a clover link, the Milnor numbers of length at most $2k+1$ are well-defined if those of length at most $k$ vanish, and that the Milnor numbers of length at least $2k+2$ are not well-defined if those of length $k+1$ survive. For a clover link $c$ with the Milnor numbers of length at most $k$ vanishing, we show that the Milnor number $\mu_c(I)$ for a sequence $I$ is well-defined up to the greatest common devisor of $\mu_{c}(J)'s$, where $J$ is a subsequence of $I$ obtained by removing at least $k+1$ indices. Moreover, if $I$ is a non-repeated sequence with length $2k+2$, the possible range of $\mu_c(I)$ is given explicitly. As an application, we give an edge-homotopy classification of $4$-clover links.