{
  "id": "2006.14701",
  "version": "v1",
  "published": "2020-06-25T21:06:52.000Z",
  "updated": "2020-06-25T21:06:52.000Z",
  "title": "Efficient triangulations and boundary slopes",
  "authors": [
    "Birch Bryant",
    "William Jaco",
    "J. Hyam Rubinstein"
  ],
  "comment": "21 pages, 6 figures; revised and improved version of an earlier paper arXiv:1108.2936, Annular efficient triangulations of 3-manifolds",
  "categories": [
    "math.GT"
  ],
  "abstract": "For a compact, irreducible, $\\partial$-irreducible, an-annular bounded 3-manifold $M\\ne\\mathbb{B}^3$, then any triangulation $\\mathcal{T}$ of $M$ can be modified to an ideal triangulation $\\mathcal{T}^*$ of $\\stackrel{\\circ}{M}$. We use the inverse relationship of crushing a triangulation along a normal surface and that of inflating an ideal triangulation to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. We prove that the topological conditions necessary for a compact 3-manifold $M$ admitting an annular-efficient triangulation are sufficient to modify any triangulation of $M$ to a boundary-efficient triangulation which is also annular-efficient. From the proof we have for any ideal triangulation $T^*$ and any inflation $\\mathcal{T}_{\\Lambda}$, there is a bijective correspondence between the closed normal surfaces in $\\mathcal{T}^*$ and the closed normal surfaces in $\\mathcal{T}_{\\Lambda}$ with corresponding normal surfaces being homeomorphic. It follows that for an ideal triangulation $\\mathcal{T}^*$ that is $0$-efficient, $1$-efficient, or end-efficient, then any inflation $\\mathcal{T}_{\\Lambda}$ of $\\mathcal{T}^*$ is $0$-efficient, $1$-efficient, or $\\partial$-efficient, respectively. There are algorithms to decide if a given triangulation or ideal triangulation of a $3$-manifold is one of these efficient triangulations. Finally, it is shown that for an annular-efficient triangulation, there are only a finite number of boundary slopes for normal surfaces of a bounded Euler characteristic; hence, in a compact, orientable, irreducible, $\\partial$-irreducible, and an-annular $3$-manifold, there are only finitely many boundary slopes for incompressible and $\\partial$-incompressible surfaces of a bounded Euler characteristic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-25T21:06:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10",
      "57M99",
      "57M50"
    ],
    "keywords": [
      "boundary slopes",
      "bounded euler characteristic",
      "closed normal surfaces",
      "annular-efficient triangulation",
      "study boundary-efficient triangulations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}