The Milnor invariants of clover links

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 ${\mu }_c(J{)}^{\prime }s$, 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.

arXiv admin note: substantial text overlap with arXiv:1506.03922

References in corpus (1)

• Bottom tangles and universal invariants