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

Yasuhara, Akira

doi:10.1090/S0002-9947-09-04840-5

Self delta-equivalence for links whose Milnor’s isotopy invariants vanish
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Trans. Amer. Math. Soc. 361 (2009), 4721-4749 Request permission

Abstract:

For an $n$-component link, Milnor’s isotopy invariants are defined for each multi-index $I=i_1i_2...i_m~(i_j\in \{1,...,n\})$. 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 $S^3$ is link-homotopic 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 $r\leq 2$ are self $\Delta$-equivalence invariants. In this paper, 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 all Milnor invariants of the link with $r\leq 2$ vanish. This is a geometric characterization for links whose Milnor invariants with $r\leq 2$ 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.

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