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.
Worked cases: identity and cable operators
Example
Consider the identity pattern I given by the core of the solid torus, the (p,1)-cable pattern for a positive integer p, and the local-knot pattern P_J obtained by tying a fixed knot J into a small arc of the core. Their winding numbers are respectively 1, p and 1. The identity pattern fixes every knot. The closure of the (p,1)-cable pattern is the unknot, so it satisfies the closure sliceness hypothesis for the localized weak-injectivity corollary. The local-knot pattern has closure J, which may be nontrivial, yet its solid-torus meridian normally generates the closure-knot group.
Example references
Facts & Assumptions
Given: The identity, (p,1)-cable and local-knot patterns with the stated orientations.
Winding number is algebraic intersection with a meridional disk (Satellite operators on concordance classes).
Strong winding is normal generation by the solid-torus meridian in the closure complement (Satellite operators on concordance classes).
Verification
The identity core meets a meridional disk once positively. A (p,1)-cable represents p times the core class, and the local knot changes the core by a knot tied into a ball, not its homology class. Thus the listed winding numbers are 1,p,1.
The (p,1) torus knot is the unknot, so its closure bounds the standard disk in B⁴ and is Z[1/|p|]-slice. The identity map is K↦K.
For the local-knot pattern the closure complement is the exterior of J with the axis meridian identifying to a meridian of J. Meridional Dehn filling gives S³; van Kampen therefore says the normal closure of this meridian is the whole knot group. J can be nontrivial, so knotted closure does not preclude strong winding.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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)
- Arunima Ray, “Slice Knots and Knot Concordance” (standard reference, not scraped)