{
  "id": "math/0001185",
  "version": "v1",
  "published": "2000-01-28T00:00:00.000Z",
  "updated": "2000-01-28T00:00:00.000Z",
  "title": "Claspers and finite type invariants of links",
  "authors": [
    "Kazuo Habiro"
  ],
  "comment": "83 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper1.abs.html",
  "journal": "Geom. Topol. 4 (2000), 1-83",
  "categories": [
    "math.GT",
    "math.QA"
  ],
  "abstract": "We introduce the concept of `claspers,' which are surfaces in 3-manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called `C_k-equivalence,' which is generated by surgery operations of a certain kind called `C_k-moves'. We prove that two knots in the 3-sphere are C_{k+1}-equivalent if and only if they have equal values of Vassiliev-Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev-Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3-dimensional topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2000-01-28T00:00:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "57M05",
      "18D10"
    ],
    "keywords": [
      "finite type invariants",
      "surgery operations",
      "vassiliev-goussarov invariants",
      "abelian groups",
      "equivalence relation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 83,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}