{
  "schema_version": 1,
  "problem_number": "OWR-17471-010",
  "title": "Zero Surgery Obstructs Persistent Free Subgroups of the Figure-Eight Knot Group",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "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.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "keywords": [
    "figure-eight knot",
    "Dehn filling",
    "persistent subgroup",
    "zero surgery",
    "metabelian group",
    "OWR-17471-010"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-17471-010/",
  "pdf_url": "https://eulersolve.org/papers/owr-17471-010/paper.pdf?v=981b590f6e6b",
  "doi": "10.5281/zenodo.23511715",
  "zenodo_record_url": "https://zenodo.org/records/23511715",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the universal frozen OWR-17471-010 formulation. This is not a nonexistence theorem for all hyperbolic knots and does not settle whether some other hyperbolic knot admits a persistent rank-two free subgroup, or what happens when zero surgery is excluded. Classical inputs are attributed. The written general proof is supplemented, not replaced, by exact diagnostics. AI-assisted, self-audited and unrefereed; no independent review, formal verification, certified novelty or absolute priority.",
  "files": {
    "paper.pdf": {
      "sha256": "981b590f6e6b642e3b9125d05ae1054b619c9cbbab821287a288424749fa2ea7"
    },
    "source.zip": {
      "sha256": "2febc54a2f98152e22aa3b3af4bba104e3b865babc6366fd2006598ae4747e1d"
    },
    "verification_report.md": {
      "sha256": "e77743298c63e8c486b7a3240062ed61d46dab77a980a9c22fda875ab792a29a"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}