{
  "id": "1807.01500",
  "version": "v1",
  "published": "2018-07-04T09:42:46.000Z",
  "updated": "2018-07-04T09:42:46.000Z",
  "title": "On the conjugacy problem in braid groups: Garside theory and subsurfaces",
  "authors": [
    "Saul Schleimer",
    "Bert Wiest"
  ],
  "comment": "4 figures",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "abstract": "Garside-theoretical solutions to the conjugacy problem in braid groups depend on the determination of a characteristic subset of the conjugacy class of any given braid, e.g. the sliding circuit set. It is conjectured that, among rigid braids with a fixed number of strands, the size of this set is bounded by a polynomial in the length of the braids. In this paper we suggest a more precise bound: for rigid braids with $N$ strands and of length $L$, the sliding circuit set should have at most $C\\cdot L^{N-2}$ elements, for some constant $C$. We construct a family of braids which realise this potential worst case. Our example braids suggest that having a large sliding circuit set is a geometric property of braids, as our examples have multiple subsurfaces with large subsurface projection; thus they are \"almost reducible\" in multiple ways, and act on the curve graph with small translation distance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-04T09:42:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F36",
      "20F10"
    ],
    "keywords": [
      "conjugacy problem",
      "braid groups",
      "garside theory",
      "rigid braids",
      "potential worst case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}