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 [Formula: see text] are well-defined if those of length [Formula: see text] vanish, and that Milnor numbers of length at least [Formula: see text] are not well-defined if those of length [Formula: see text] survive. For a clover link [Formula: see text] with vanishing Milnor numbers of length [Formula: see text], we show that the Milnor number [Formula: see text] for a sequence [Formula: see text] is well-defined by taking modulo the greatest common divisor of the [Formula: see text], where [Formula: see text] is any proper subsequence of [Formula: see text] obtained by removing at least [Formula: see text] indices. Moreover, if [Formula: see text] is a non-repeated sequence of length [Formula: see text], the possible range of [Formula: see text] is given explicitly. As an application, we give an edge-homotopy classification of [Formula: see text]-clover links.

The jump of the Milnor number in the X 9 singularity class

The jump of the Milnor number of an isolated singularity f 0 is the minimal non-zero difference between the Milnor numbers of f 0 and one of its deformations (f s). We prove that for the singularities in the X 9 singularity class their jumps are equal to 2.