Zero Filling and Persistent Free Subgroups of the Figure-Eight Knot
Year: 2026
Type: preprint
Abstract: The figure-eight knot gives a negative answer to the literal universal rank-two free-subgroup question printed on p.493 of Motegi's Oberwolfach Report 8/2020 contribution. It is hyperbolic and has no finite nonmeridional surgery, but every free rank-two subgroup has a nontrivial element killed by zero filling. The zero filling is a torus bundle, so its metabelian fundamental group kills an explicit nontrivial second-derived word. The short proof verifies the no-finite-surgery hypothesis across all rational slopes using the credited classical filling classification. Rational slopes include zero and exclude the meridian at infinity. The topology and group-law ingredients are classical; related algebra in Boden-Friedl and earlier Ito-Motegi-Teragaito text is credited. The result refutes a universal affirmative assertion and leaves the separate search for a positive hyperbolic example unresolved. Two independent primary-source priority audits found no earlier explicit full counterexample in inspected text, but priority cannot be fully cleared. The full 2024 cyclic-persistence and 2019 vanishing-elements bodies were not inspected, and some longer precursors were scanned and selectively read. No historical-first, new-topology or continued-open-status claim is made. A historical precursor's subsequently acknowledged Claim6.3 gap is disclosed; that claim is not a proof dependency. The authored-only verification package preserves 22 original inputs, independent derivations and reviews, source read scopes and hashes, exact finite controls, nested manifests and an offline replay driver. A pinned exact-inventory gate repairs a historical release verifier's unanchored manifest and constant source-exclusion flag without changing original evidence. Raw third-party sources and account data are excluded. Finite checks supplement the proof; integrity is not authenticity or priority. AI tools were used extensively in solving, research, writing, reproduction and adversarial verification. This is an unrefereed preprint with internal AI adversarial reviews; it has not received human peer review or proof-assistant certification.
Author: Kriebel, Alec
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222018076
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