Zero Surgery Obstructs Persistent Free Subgroups of the Figure-Eight Knot Group
Manuscript 20 October 2026 · Online 20 October 2026
Abstract
A question in Oberwolfach Report 8/2020 asks whether every hyperbolic knot without finite surgery has a rank-two free subgroup that injects under every rational Dehn filling. The figure-eight knot gives a negative answer. Its zero-filling kernel is exactly the second derived subgroup of the knot group. The zero-surgery quotient is metabelian, so zero filling kills a nontrivial element of every rank-two free subgroup. An explicit length-eighteen word witnesses the obstruction. The note combines classical surgery facts with elementary group theory; no absolute priority claim is made. Existence for some other hyperbolic knot, or a variant excluding zero surgery, remains unaddressed. AI-assisted, self-audited and unrefereed; not independently reviewed or formally verified.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete counterexample
- Categories
- math.GT · math.GR
- Manuscript
- 20 October 2026
- Online release
- 20 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Zero Surgery Obstructs Persistent Free Subgroups of the Figure-Eight Knot Group,” EulerSolve Research Papers, OWR-17471-010, 2026. https://doi.org/10.5281/zenodo.23511715.
BibTeX
@misc{ferudun2026figureeight,
author = {Ferudun, Alper},
title = {Zero Surgery Obstructs Persistent Free Subgroups of the Figure-Eight Knot Group},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-17471-010/},
doi = {10.5281/zenodo.23511715},
note = {OWR-17471-010; unrefereed preprint}
}