Self Delta-equivalence for Links Whose Milnor’s Isotopy Invariants Vanish

For an n-component link, Milnor’s isotopy invariants are defined for each multi-index I = i1i2...im (ij ∈ {1, ..., n}). Here m is called the length. Let r(I) denote the maximum number of times that any index appears in I. It is known that Milnor invariants with r = 1, i.e., Milnor invariants for all multi-indices I with r(I) = 1, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with r = 1 coincide. This gives us that a link in S3 is linkhomotopic to a trivial link if and only if all Milnor invariants 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 Milnor invariants with r ≤ 2 are self Δ-equivalence invariants. In this paper, we give a self Δ-equivalence classification of the set of n-component links in S3 whose Milnor invariants with length ≤ 2n − 1 and r ≤ 2 vanish. As a corollary, we have that a link is self Δ-equivalent to a trivial link if and only if all Milnor invariants of the link with r ≤ 2 vanish. This is a geometric characterization for links whose Milnor invariants with r ≤ 2 vanish. The chief ingredient in our proof is Habiro’s clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory.