{
  "id": "1307.7377",
  "version": "v3",
  "published": "2013-07-28T16:09:22.000Z",
  "updated": "2014-09-16T10:40:55.000Z",
  "title": "Basic nets in the projective plane",
  "authors": [
    "S. Yu. Orevkov"
  ],
  "comment": "14 pages, 15 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The notion of basic net (called also basic polyhedron) on $S^2$ plays a central role in Conway's approach to enumeration of knots and links in $S^3$. Drobotukhina applied this approach for links in $RP^3$ using basic nets on $RP^2$. By a result of Nakamoto, all basic nets on $S^2$ can be obtained from a very explicit family of minimal basic nets (the nets $(2\\times n)^*$, $n\\ge3$, in Conway's notation) by two local transformations. We prove a similar result for basic nets in $RP^2$. We prove also that a graph on $RP^2$ is uniquely determined by its pull-back on $S^3$ (the proof is based on Lefschetz fix point theorem).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-08-29T22:50:44.000Z",
      "abstract": "The notion of basic net (called also basic polyhedron) on $S^2$ plays a central role in Conway's approach to enumeration of knots and links in $S^3$. Drobotukhina applied this approach for links in $\\RP^3$ using basic nets on $\\RP^2$. By a result of Nakamoto, all basic nets on $S^2$ can be obtained from a very explicit family of minimal basic nets (the nets $(2\\times n)^*$, $n\\ge3$, in Conway's notation) by two local transformations. We prove a similar result for basic nets in $\\RP^2$. We prove also that a graph on $\\RP^2$ is uniquely determined by its pull-back on $S^3$ (the proof is based on Lefschetz fix point theorem).",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-09-16T10:40:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10"
    ],
    "keywords": [
      "projective plane",
      "lefschetz fix point theorem",
      "minimal basic nets",
      "basic polyhedron",
      "local transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.7377O"
    }
  }
}