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For an $n$-component link $L$, the Milnor's isotopy invariant is defined foreach multi-index $I=i_1i_2...i_m (i_j\in\n)$. Here $m$ is called the length.Let $r(I)$ denote the maximam number of times that any index appears. It isknown that Milnor invariants with $r=1$ are link-homotopy invariant. N.Habegger and X. S. Lin showed that two string links are a link-homotopc if andonly if their Milnor invariants with $r=1$ coincide. This gives us that a linkin $S^3$ is link-homotopic to a trivial link if and only if the all Milnorinvariants of the link with $r=1$ vanish. Although Milnor invariants with $r=2$are not link-homotopy invariants, T. Fleming and the author showed that Milnorinvariants with $r\leq 2$ are self $\Delta$-equivalence invariants. In thispaper, we give a self $\Delta$-equivalence classification of the set of$n$-component links in $S^3$ whose Milnor invariants with length $\leq 2n-1$and $r\leq 2$ vanish. As a corollary, we have that a link is self$\Delta$-equivalent to a trivial link if and only if the all Milnor invariantsof the link with $r\leq 2$ vanish. This is a geometric characterization forlinks whose Milnor invariants with $\leq 2$ vanish. The chief ingredient in ourproof is Habiro's clasper theory. We also give an alternate proof of alink-homotopy classification of string links by using clasper theory.

Self delta-equivalence for links whose Milnor’s isotopy invariants vanish