DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

How 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.

Abstract oriented cobordism of knot circles

Definition

An oriented circle is a smooth one-dimensional manifold diffeomorphic to the standard oriented circle. An abstract oriented cobordism from an oriented circle $C_{0}$ to an oriented circle $C_{1}$ is a compact oriented smooth surface $F$ together with orientation-preserving identifications of its incoming and outgoing boundary components such that $\partial F = - C_{0} ⊔ C_{1}$ . Here $- C_{0}$ denotes the same circle with its orientation reversed. The cobordism is abstract: no embedding of $F$ , $C_{0}$ , or $C_{1}$ into a common three- or four-manifold is included in the data.

Definition references

Oriented knot circles in the standard three-sphere

Depends on

Oriented links in the three-sphere and ambient isotopy

Used by

An abstract annulus need not be a smooth concordance Counterexample
Lee cobordism maps give the genus filtration bound Lemma
Odd Khovanov and Bar–Natan LEO foundations Lemma

Dependency tree · two levels

6 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

Charles Livingston, A Survey of Classical Knot Concordance, Handbook of Knot Theory, Chapter 7 (standard reference, not scraped)
Arunima Ray, Slice Knots and Knot Concordance, Winter Braids Lecture Notes 8 (standard reference, not scraped)