{
  "id": "2311.09924",
  "version": "v1",
  "published": "2023-11-16T14:23:33.000Z",
  "updated": "2023-11-16T14:23:33.000Z",
  "title": "Finite type invariants in low degrees and the Johnson filtration",
  "authors": [
    "Wolfgang Pitsch",
    "Ricard Riba"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "We study the behavior the Casson invariant $\\lambda$, its square, and Othsuki's second invariant $\\lambda_2$ as functions on the Johnson subgroup of the mapping class group. We show that $\\lambda$ and $d_2 = \\lambda_2 - 18 \\lambda^2$ are invariants that are morphisms on respectively the second and the third level of the Johnson filtration and as a consequence never vanish on any level of this filtration. Finally we show that the invariant $\\lambda_2-18\\lambda^2 +3\\lambda$ vanishes on the fifth level of the Johnson filtration, $\\mathcal{M}_{g,1}(5)$, and as a consequence we prove that for instance the Poincar\\'e homology sphere does not admit any Heegaard splitting with gluing map an element in $\\mathcal{M}_{g,1}(5)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-16T14:23:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M27",
      "20J05"
    ],
    "keywords": [
      "johnson filtration",
      "finite type invariants",
      "low degrees",
      "othsukis second invariant",
      "poincare homology sphere"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}