{
  "id": "2007.03721",
  "version": "v1",
  "published": "2020-07-07T18:22:42.000Z",
  "updated": "2020-07-07T18:22:42.000Z",
  "title": "Property G and the $4$--genus",
  "authors": [
    "Yi Ni"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "We say a null-homologous knot $K$ in a $3$--manifold $Y$ has Property G, if the properties about the Thurston norm and fiberedness of the complement of $K$ is preserved under the zero surgery on $K$. In this paper, we will show that, if the smooth $4$--genus of $K\\times\\{0\\}$ (in a certain homology class) in $(Y\\times[0,1])\\#N\\overline{\\mathbb CP^2}$, where $Y$ is a rational homology sphere, is smaller than the Seifert genus of $K$, then $K$ has Property G. When the smooth $4$--genus is $0$, $Y$ can be taken to be any closed, oriented $3$--manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-07T18:22:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational homology sphere",
      "zero surgery",
      "seifert genus",
      "homology class",
      "thurston norm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}