{
  "id": "2311.16776",
  "version": "v1",
  "published": "2023-11-28T13:38:20.000Z",
  "updated": "2023-11-28T13:38:20.000Z",
  "title": "Thick braids and other non-trivial homotopy in configuration spaces of hard discs",
  "authors": [
    "Patrick Ramsey"
  ],
  "comment": "20 pages, 6 figures",
  "categories": [
    "math.GT",
    "math.AT",
    "math.MG"
  ],
  "abstract": "We study the homotopy groups of the configuration space of discs inside a unit disc just beyond the first critical radius. We find a non-trivial `thick braid' (loop in the configuration space), which is trivial in the ordered configuration space of points when replacing each disc by its centre, in the case $n=4$. We find a non-contractible $(n-3)$-sphere for $n\\ge 5$, where $r$ is between the first two critical radii, and explore the persistence of this homotopy class when the ambient unit disc is deformed. For sufficiently large $n$, we demonstrate the existence of non-trivial $\\pi_k$-classes with $k<n-3$ beyond higher critical radii.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-28T13:38:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R80",
      "51F99"
    ],
    "keywords": [
      "configuration space",
      "thick braid",
      "non-trivial homotopy",
      "hard discs",
      "ambient unit disc"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}