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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Finite-order characters of a cyclic branched cover
Definition
Assume AC. Let be a smooth oriented knot, let , and let be its -fold cyclic branched cover. A character is a homomorphism . Its order is its order in ; it is finite-order when this order is finite and has prime-power order when its order is for a prime and . It vanishes on a subgroup when for every .
Definition references
Depends on
Used by
- Torsion linking form and Casson–Gordon bordism data Definition
- The square knot has a nontrivial character vanishing on a linking metabolizer Example
- Characters vanishing on the disk metabolizer extend over the branched cover Lemma
- Prime-power characters admit a multiple bounding bordism Lemma
- Casson–Gordon signature-defect obstruction to sliceness Theorem
Dependency tree · two levels
16 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
- Patrick Gilmer, Slice Knots in S3 (standard reference, not scraped)