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PreprintCC BY-NC-SA 4.0arXiv

The unknotting problem for surfaces in the 4-sphere

  • Cameron Gordon
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Abstract

We show that there is no algorithm to decide whether or not a closed, connected, orientable, locally flat PL surface in S4 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 S4 of genus 18 whose group has unsolvable word problem, and an orientable surface in S4 of genus 40 whose group contains an isomorphic copy of every recursively presented group. Similar results hold for non-orientable surfaces.

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Cite

BibTeXRIS
BibTeX
@misc{gordon2026unknotting,
  title = {{The unknotting problem for surfaces in the 4-sphere}},
  author = {Cameron Gordon},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36331},
  eprint = {2610.36331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36331},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - The unknotting problem for surfaces in the 4-sphere
AU  - Cameron Gordon
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36331
UR  - https://arxiv.org/abs/2610.36331
AB  - 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.
ER  -