{
  "id": "2208.08244",
  "version": "v1",
  "published": "2022-08-17T11:52:54.000Z",
  "updated": "2022-08-17T11:52:54.000Z",
  "title": "Minimal generating sets of moves for surfaces immersed in the four-space",
  "authors": [
    "Michal Jablonowski"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GT"
  ],
  "abstract": "For immersed surfaces in the four-space, we have a generating set of the Swenton-Hughes-Kim-Miller spatial moves that relate banded singular diagrams of ambient isotopic immersions of those surfaces. We also have Yoshikawa-Kamada-Kawauchi-Kim-Lee planar moves that relate marked graph diagrams of ambient isotopic immersions of those surfaces. One can ask if the former moves form a minimal set and if the latter moves form a generating set. In this paper, we derive a minimal generating set of spatial moves for diagrams of surfaces immersed in the four-space, which translates into a generating set of planar moves. We also show that the complements of two equivalent immersed surfaces can be transformed one another by a Kirby calculus not requiring the 1-1-handle or 2-1-handle slides. This gives a potential room for a stronger immersed surface invariant than the diffeomorphism type of its complement. We also discuss the fundamental group of the immersed surface-link complement in the four-space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-17T11:52:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minimal generating set",
      "four-space",
      "ambient isotopic immersions",
      "moves form",
      "relate marked graph diagrams"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}