{
  "id": "0903.4614",
  "version": "v2",
  "published": "2009-03-26T15:35:32.000Z",
  "updated": "2009-04-20T14:19:31.000Z",
  "title": "Geometrically incompressible non-orientable closed surfaces in lens spaces",
  "authors": [
    "Miwa Iwakura"
  ],
  "comment": "19pages, 7figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "We consider non-orientable closed surfaces of minimum crosscap number in the $(p,q)$-lens space $L(p,q) \\cong V_1 \\cup_{\\partial} V_2$, where $V_1$ and $V_2$ are solid tori. Bredon and Wood gave a formula for calculating the minimum crosscap number. Rubinstein showed that $L(p,q)$ with $p$ even has only one isotopy class of such surfaces, and it is represented by a surface in a standard form, which is constructed from a meridian disk in $V_1$ by performing a finite number of band sum operations in $V_1$ and capping off the resulting boundary circle by a meridian disk of $V_2$. We show that the standard form corresponds to an edge-path $\\lambda$ in a certain tree graph in the closure of the hyperbolic upper half plane. Let $0=p_0/q_0, p_1/q_1, ..., p_k/q_k = p/q$ be the labels of vertices which $\\lambda$ passes. Then the slope of the boundary circle of the surface right after the $i$-th band sum is $(p_i, q_i)$. The number of edges of $\\lambda$ is equal to the minimum crosscap number. We give an easy way of calculating $p_i / q_i$ using a certain continued fraction expansion of $p/q$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-04-20T14:19:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57N10"
    ],
    "keywords": [
      "geometrically incompressible non-orientable closed surfaces",
      "lens space",
      "minimum crosscap number",
      "band sum",
      "meridian disk"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.4614I"
    }
  }
}