The Milnor invariants of clover links

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 $μ_c(I)$ for a sequence $I$ is well-defined up to the greatest common devisor of $μ_{c}(J)'s$, 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 $μ_c(I)$ is given explicitly. As an application, we give an edge-homotopy classification of $4$-clover links.

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