{
  "id": "2007.07280",
  "version": "v1",
  "published": "2020-07-14T18:17:20.000Z",
  "updated": "2020-07-14T18:17:20.000Z",
  "title": "Cubic graphs induced by bridge trisections",
  "authors": [
    "Jeffrey Meier",
    "Abigail Thompson",
    "Alexander Zupan"
  ],
  "comment": "18 pages, 17 color figures",
  "categories": [
    "math.GT",
    "math.CO"
  ],
  "abstract": "Every embedded surface $\\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\\mathcal{K})$ into three simple pieces. In this case, the surface $\\mathcal{K}$ is determined by an embedded 1-complex, called the $\\textit{1-skeleton}$ of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph $\\Gamma$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $\\Gamma$ such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-14T18:17:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57Q45",
      "57M50",
      "05C15"
    ],
    "keywords": [
      "bridge trisection",
      "tri-plane diagram",
      "normal euler number",
      "unknotted surface",
      "interior reidemeister moves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}