{
  "id": "2501.15642",
  "version": "v2",
  "published": "2025-01-26T19:02:04.000Z",
  "updated": "2025-05-06T00:09:39.000Z",
  "title": "On winding numbers of almost embeddings of $K_4$ in the plane",
  "authors": [
    "Emil Alkin",
    "Alexander Miroshnikov"
  ],
  "comment": "8 pages, 4 figures",
  "categories": [
    "math.GT",
    "cs.CG",
    "math.CO"
  ],
  "abstract": "Let $K_4$ be the complete graph on four vertices. Let $f$ be a continuous map of $K_4$ to the plane such that $f$-images of non-adjacent edges are disjoint. For any vertex $v \\in K_4$ take the winding number of the $f$-image of the cycle $K_4 - v$ around $f(v)$. It is known that the sum of these four integers is odd. We construct examples showing that this is the only relation between these four numbers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-05-06T00:09:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M15",
      "55M25",
      "05C10"
    ],
    "keywords": [
      "winding number",
      "embeddings",
      "complete graph",
      "non-adjacent edges",
      "continuous map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}