Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
A knotted closure can have strong winding number one
Statement refuted
A pattern has strong winding number ±1 if and only if its closure P(U) is the unknot.
Statement references
Facts & Assumptions
Given: A nontrivial knot J and the pattern P_J obtained by tying J into a small arc of the solid-torus core.
The closure of the local-knot pattern is J and its winding number is 1 (Satellite operators on concordance classes).
Strong winding is the normal-generation property of the axis meridian in the closure complement (Satellite operators on concordance classes).
Counterexample
The pattern P_J has winding number 1 because tying a local knot does not change its homology class in the solid torus. Its closure is J, which is not the unknot by the choice of J.
The axis meridian is a meridian of J in this local-knot construction. Meridional filling of the knot exterior yields S³, whose fundamental group is trivial; van Kampen implies that the normal closure of the meridian is the entire group. Hence P_J is strong winding +1 although P_J(U) is not the unknot.
This disproves the forward implication from strong winding to an unknotted closure, while leaving the reverse implication untouched. The counterexample makes no claim that ordinary winding ±1 always implies strong winding ±1.
Depends on
Used by
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Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Tim D. Cochran, Christopher W. Davis, and Arunima Ray, “Injectivity of satellite operators in knot concordance” (standard reference, not scraped)