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

Abstract

For an n n -component link, Milnor’s isotopy invariants are defined for each multi-index I = i 1 i 2 . . . i m ( i j ∈ { 1 , . . . , n } ) I=i_1i_2...i_m~(i_j\in \{1,...,n\}) . Here m m is called the length. Let r ( I ) r(I) denote the maximum number of times that any index appears in I I . It is known that Milnor invariants with r = 1 r=1 , i.e., Milnor invariants for all multi-indices I I with r ( I ) = 1 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 r=1 coincide. This gives us that a link in S 3 S^3 is link-homotopic to a trivial link if and only if all Milnor invariants of the link with r = 1 r=1 vanish. Although Milnor invariants with r = 2 r=2 are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with r ≤ 2 r\leq 2 are self Δ \Delta -equivalence invariants. In this paper, we give a self Δ \Delta -equivalence classification of the set of n n -component links in S 3 S^3 whose Milnor invariants with length ≤ 2 n − 1 \leq 2n-1 and r ≤ 2 r\leq 2 vanish. As a corollary, we have that a link is self Δ \Delta -equivalent to a trivial link if and only if all Milnor invariants of the link with r ≤ 2 r\leq 2 vanish. This is a geometric characterization for links whose Milnor invariants with r ≤ 2 r\leq 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.