Milnor invariants of clover links

Levine introduced clover links to investigate the indeterminacy of Milnor invariants of links. He proved that for a clover link, Milnor numbers of length up to 2k + 1 are well-defined if those of length ≤ k vanish, and that 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 vanishing Milnor numbers of length ≤ k, we show that the Milnor number μ_c(I) for a sequence I is well-defined by taking modulo the greatest common divisor of the μ_c(J)′s, where J is any proper subsequence of I obtained by removing at least k + 1 indices. Moreover, if I is a non-repeated sequence of 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.