{
  "abstract": "We show that there is no algorithm to decide whether or not a closed, connected, orientable, locally flat PL surface in $S^4$ of genus at least 18 (resp. 19) is TOP (resp. PL) unknotted. We prove analogous results for non-orientable surfaces, and for surfaces in any compact PL 4-manifold. We also show that there is an orientable surface in $S^4$ of genus 18 whose group has unsolvable word problem, and an orientable surface in $S^4$ of genus 40 whose group contains an isomorphic copy of every recursively presented group. Similar results hold for non-orientable surfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36331",
  "author_total": 1,
  "authors": [
    {
      "name": "Cameron Gordon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c39x826zm41zv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36331",
  "fulltext_url": null,
  "id": "c39x826zm41zv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36331",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7224133587",
  "pdf_url": "https://arxiv.org/pdf/2610.36331",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c39x826zm41zv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36331",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36331"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The unknotting problem for surfaces in the 4-sphere",
  "title_text": "The unknotting problem for surfaces in the 4-sphere",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/c39x826zm41zv",
  "venue": "arXiv",
  "versions": []
}