{
  "id": "2007.08474",
  "version": "v1",
  "published": "2020-07-16T17:17:16.000Z",
  "updated": "2020-07-16T17:17:16.000Z",
  "title": "Domino tilings and flips in dimensions 4 and higher",
  "authors": [
    "Caroline Klivans",
    "Nicolau C. Saldanha"
  ],
  "comment": "28 pages, 14 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper we consider domino tilings of bounded regions in dimension $n \\geq 4$. We define the twist of such a tiling, an elements of ${\\mathbb{Z}}/(2)$, and prove it is invariant under flips, a simple local move in the space of tilings. We investigate which regions $D$ are regular, i.e. whenever two tilings $t_0$ and $t_1$ of $D \\times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We prove that all boxes are regular except $D = [0,2]^3$. Furthermore, given a regular region $D$, we show that there exists a value $M$ (depending only on $D$) such that if $t_0$ and $t_1$ are tilings of equal twist of $D \\times [0,N]$ then the corresponding tilings can be joined by a finite sequence of flips in $D \\times [0,N+M]$. As a corollary we deduce that, for regular $D$ and large $N$, the set of tilings of $D \\times [0,N]$ has two twin giant components under flips, one for each value of the twist.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-16T17:17:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B45",
      "52C20",
      "52C22",
      "05C70"
    ],
    "keywords": [
      "domino tilings",
      "twin giant components",
      "simple local move",
      "extra vertical space",
      "regular region"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}