{
  "id": "2501.11912",
  "version": "v2",
  "published": "2025-01-21T06:23:51.000Z",
  "updated": "2025-01-22T21:10:55.000Z",
  "title": "Detecting Free Products in the Mapping Class Group of Punctured Disks via Dynnikov Coordinates",
  "authors": [
    "Elif Medetoğulları",
    "Elif Dalyan",
    "S. Öykü Yurttaş"
  ],
  "comment": "Mistake in the proof",
  "categories": [
    "math.GT"
  ],
  "abstract": "We prove that Dehn twists about certain $k$ curves on an $n$-punctured disk generate either a free group of rank $k$ or a free product of Abelian groups, depending on the configuration of the curves. Our approach refines previously known criteria, providing a different method for detecting freeness in the mapping class group of the punctured disk. Using Dynnikov coordinates, we further present an algorithm that checks whether Dehn twists about these $k$ curves generate a free group or a free product of Abelian groups.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2025-01-22T21:10:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50",
      "57M60"
    ],
    "keywords": [
      "mapping class group",
      "detecting free products",
      "dynnikov coordinates",
      "dehn twists",
      "abelian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}