{
  "id": "1908.10113",
  "version": "v1",
  "published": "2019-08-27T10:01:42.000Z",
  "updated": "2019-08-27T10:01:42.000Z",
  "title": "A characteristics for a surface sum of two handlebodies along an annulus or a once-punctured torus to be a handlebody",
  "authors": [
    "Fengchun Lei",
    "He Liu",
    "Fengling Li",
    "Andrei Vesnin"
  ],
  "comment": "11 pGES, 3 FIGURES",
  "categories": [
    "math.GT"
  ],
  "abstract": "The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum $H=H_1\\cup_A H_2$ of two handlebodies $H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a longitude for either $H_1$ or $H_2$. 2. Let $H=H_1\\cup_T H_2$ be a surface sum of two handlebodies $H_1$ and $H_2$ along a once-punctured torus $T$. Suppose that $T$ is incompressible in both $H_1$ and $H_2$. Then $H$ is a handlebody if and only if the there exists a collection $\\{\\delta, \\sigma\\}$ of simple closed curves on $T$ such that either $\\{\\delta, \\sigma\\}$ is primitive in $H_1$ or $H_2$, or $\\{\\delta\\}$ is primitive in $H_1$ and $\\{\\sigma\\}$ is primitive in $H_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-27T10:01:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10"
    ],
    "keywords": [
      "handlebody",
      "surface sum",
      "characteristics",
      "annulus sum",
      "main results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}