Unknotting orientable surfaces
Abstract
It is shown that every locally flatly embedded genus g ∈ {1,2} surface in the $4$-sphere with knot group ℤ is unknotted. The same proof establishes that any two genus g ∈ {1,2} surfaces in D4 with knot group 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 ℤ[t± 1], which was solved in genus g ≥ 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 ℤ, this shows that a locally flatly embedded orientable surface in S4 is unknotted if and only if its knot group is ℤ.
Related
- The unknotting problem for surfaces in the 4-sphere
- Complete non-orientable minimal surfaces of finite total curvature: Approximation and interpolation
- The proper geometric dimension of the mapping class groups of non-orientable surfaces with punctures
- Unknotting number is not additive under connected sum
- Ascending, Bridge, and Unknotting Numbers of Knotted Graphs
BibTeX
@misc{conway2026unknotting,
title = {{Unknotting orientable surfaces}},
author = {Anthony Conway},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06544},
eprint = {2609.06544},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06544},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Unknotting orientable surfaces
AU - Anthony Conway
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06544
UR - https://arxiv.org/abs/2609.06544
AB - It is shown that every locally flatly embedded genus g ∈ {1,2} surface in the $4$-sphere with knot group ℤ is unknotted. The same proof establishes that any two genus g ∈ {1,2} surfaces in D^4 with knot group 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 ℤ[t^± 1], which was solved in genus g ≥ 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 ℤ, this shows that a locally flatly embedded orientable surface in S^4 is unknotted if and only if its knot group is ℤ.
ER -