{
  "id": "1312.2566",
  "version": "v2",
  "published": "2013-12-09T20:21:54.000Z",
  "updated": "2015-05-22T14:56:16.000Z",
  "title": "An introduction to finite type invariants of knots and 3-manifolds defined by counting graph configurations",
  "authors": [
    "Christine Lescop"
  ],
  "comment": "68 pages. Change of title, updates and minor reorganization of notes of five lectures presented in the ICPAM-ICTP research school of Mekn\\`es in May 2012. To appear in the Proceedings of the conference \"Quantum topology\" organized by Chelyabinsk State University in July 2014 (Vestnik ChelGU)",
  "categories": [
    "math.GT"
  ],
  "abstract": "These introductory lectures show how to define finite type invariants of links and 3-manifolds by counting graph configurations in 3-manifolds, following ideas of Witten and Kontsevich. The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. As an important example, we present it as the algebraic intersection of a torus and a 4-chain called a propagator in a configuration space. In the second section, we introduce the simplest finite type 3-manifold invariant, which is the Casson invariant (or the Theta-invariant) of integer homology 3-spheres. It is defined as the algebraic intersection of three propagators in the same two-point configuration space. In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams. In Sections 4 and 5, we sketch an original construction based on configuration space integrals of universal finite type invariants for links in rational homology 3-spheres and we state open problems. Our construction generalizes the known constructions for links in the ambient space, and it makes them more flexible. In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-09T20:21:54.000Z",
      "title": "An introduction to finite type invariants of knots and 3-manifolds",
      "abstract": "These notes are the notes of five lectures presented in the ICPAM-ICTP research school of Mekn\\`es in May 2012, as an introduction to finite type invariants of links and 3-manifolds. The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first section. For an important example, we present it as the algebraic intersection of a torus and a 4-chain called a propagator in a configuration space. In the second section, we introduce the simplest finite type 3-manifold invariant that is the Casson invariant of integral homology spheres. It is defined as the algebraic intersection of three propagators in a two-point configuration space. In the third section, we explain the general notion of finite type invariants and introduce relevant spaces of Feynman Jacobi diagrams. In Sections 4 and 5, we sketch a construction based on configuration space integrals of universal finite type invariants for links in rational homology spheres and we state open problems. In Section 6, we present the needed properties of parallelizations of 3-manifolds and associated Pontrjagin classes, in details.",
      "comment": "64 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-05-22T14:56:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "introduction",
      "simplest finite type invariant",
      "universal finite type invariants",
      "algebraic intersection",
      "rational homology spheres"
    ],
    "tags": [
      "conference paper",
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 68,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1268416,
      "adsabs": "2013arXiv1312.2566L"
    }
  }
}