{
  "abstract": "It is shown that every locally flatly embedded genus $g \\in \\{1,2\\}$ surface in the $4$-sphere with knot group $\\mathbb{Z}$ is unknotted. The same proof establishes that any two genus $g \\in \\{1,2\\}$ surfaces in $D^4$ with knot group $\\mathbb{Z}$ and common boundary an Alexander polynomial one knot are isotopic rel. boundary. In previous work, the author and Powell reduced such unknotting problems to a question concerning the cancellation of $(-t)$-quadratic forms over $\\mathbb{Z}[t^{\\pm 1}]$, which was solved in genus $g \\geq 3$ using work of Bass. In genus $g=2$, we observe that the same proof goes through using further work of Bass. In genus $g=1,$ the result instead follows from a statement in commutative algebra which was proved with the assistance of AI. Combined with earlier work of Freedman on locally flat spheres with knot group $\\mathbb{Z}$, this shows that a locally flatly embedded orientable surface in $S^4$ is unknotted if and only if its knot group is $\\mathbb{Z}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06544",
  "author_total": 1,
  "authors": [
    {
      "name": "Anthony Conway"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b93f1egf3ycxk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06544",
  "fulltext_url": null,
  "id": "b93f1egf3ycxk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06544",
  "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": "W7211991054",
  "pdf_url": "https://arxiv.org/pdf/2609.06544",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b93f1egf3ycxk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06544",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06544"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Unknotting orientable surfaces",
  "title_text": "Unknotting orientable surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/b93f1egf3ycxk",
  "venue": "arXiv",
  "versions": []
}