### Abstract

For an n-component link, Milnor's isotopy invariants are defined for eachmulti-index I = i_{1}i_{2}⋯i_{m} (i _{j} ∈ {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 ≤ 2 are self Δ-equivalence invariants. In this paper, we give a self Δ-equivalence classification of the set of n-component links in S^{3} whose Milnor invariants withlength ≤ 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.

Original language | English |
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Pages (from-to) | 4721-4749 |

Number of pages | 29 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2009 Sep 1 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics