{
  "id": "2202.01755",
  "version": "v1",
  "published": "2022-02-03T18:21:18.000Z",
  "updated": "2022-02-03T18:21:18.000Z",
  "title": "Classification of doubly periodic untwisted (p,q)-weaves by their crossing number",
  "authors": [
    "Mizuki Fukuda",
    "Motoko Kotani",
    "Sonia Mahmoudi"
  ],
  "comment": "22 pages, 12 figures. arXiv admin note: text overlap with arXiv:2108.09464",
  "categories": [
    "math.GT"
  ],
  "abstract": "A weave is the lift to the Euclidean thickened plane of a set of infinitely many planar crossed geodesics, that can be characterized by a number of sets of threads describing the organization of the non-intersecting curves, together with a set of crossing sequences representing the entanglements. In this paper, the classification of a specific class of doubly periodic weaves, called untwisted (p,q)-weaves, is done by their crossing number, which is the minimum number of crossings that can possibly be found in a unit cell of its infinite weaving diagrams. Such a diagram can be considered as a particular type of quadrivalent periodic planar graph with an over or under information at each vertex, whose unit cell corresponds to a link diagram in a thickened torus. Moreover, considering that a weave is not uniquely defined by its sets of threads and its crossing sequences, we also specify the notion of equivalence classes by introducing a new parameter, called crossing matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-03T18:21:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K10",
      "57K12",
      "57M15",
      "05A05"
    ],
    "keywords": [
      "crossing number",
      "doubly periodic",
      "classification",
      "quadrivalent periodic planar graph",
      "unit cell corresponds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}