{
  "id": "math/0610492",
  "version": "v3",
  "published": "2006-10-16T15:23:33.000Z",
  "updated": "2007-04-29T16:04:56.000Z",
  "title": "Self delta-equivalence for links whose Milnor's isotopy invariants vanish",
  "authors": [
    "Akira Yasuhara"
  ],
  "comment": "25 pages,18 figures, Changed the content (main result). Solved the conjecture given in previous version. Previous main result was a partial answer to the conjecture",
  "categories": [
    "math.GT"
  ],
  "abstract": "For an $n$-component link $L$, the Milnor's isotopy invariant is defined for each 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 is known 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 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 the 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 the all Milnor invariants of the link with $r\\leq 2$ vanish. This is a geometric characterization for links whose Milnor invariants with $\\leq 2$ 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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-04-29T16:04:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M27"
    ],
    "keywords": [
      "milnor invariants",
      "milnors isotopy invariants vanish",
      "self delta-equivalence",
      "trivial link",
      "link-homotopy invariant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....10492Y"
    }
  }
}