{
  "id": "math/0703347",
  "version": "v1",
  "published": "2007-03-12T15:59:03.000Z",
  "updated": "2007-03-12T15:59:03.000Z",
  "title": "Surgery formulae for finite type invariants of rational homology 3--spheres",
  "authors": [
    "Christine Lescop"
  ],
  "comment": "51 pages, uses pstricks",
  "categories": [
    "math.GT"
  ],
  "abstract": "We first present three graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres. Each of these three formulae determines an alternate sum of the form $$\\sum_{I \\subset N} (-1)^{\\sharp I}Z_n(M_I)$$ where $N$ is the set of components of a framed algebraically split link $L$ in a rational homology sphere $M$, and $M_I$ denotes the manifold resulting from the Dehn surgeries on the components of $I$. The first formula treats the case when $L$ is a boundary link with $n$ components, while the second one is for $3n$--component algebraically split links. In the third formula, the link $L$ has $2n$ components and the Milnor triple linking numbers of its 3--component sublinks vanish. The presented formulae are then applied to the study of the variation of $Z_n$ under a $p/q$-surgery on a knot $K$. This variation is a degree $n$ polynomial in $q/p$ when the class of $q/p$ in $\\QQ/\\ZZ$ is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-03-12T15:59:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "57N10",
      "57M25",
      "55R80"
    ],
    "keywords": [
      "rational homology sphere",
      "algebraically split link",
      "kontsevich-kuperberg-thurston universal finite type invariant",
      "graphic surgery formulae",
      "milnor triple linking numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}