DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-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.

Split and nonsplit links

Definition

An oriented link $L \subset S^{3}$ is split if a smoothly embedded sphere $S \subset S^{3} ∖ L$ has link points in both components of $S^{3} ∖ S$ ; otherwise $L$ is nonsplit.

Definition references

Menasco, Closed incompressible surfaces in alternating knot and link complements (1984), Theorem 1(a)

Kindred, A geometric proof of the flyping theorem, §2.1

Facts & Assumptions

Given: A link $L \subset S^{3}$ as a finite disjoint union of embedded circles.

[F1]

A link is a finite disjoint union of smoothly embedded circles in $S^{3}$ (Oriented links in the three-sphere and ambient isotopy).

Proof

1.1givenF1construct

Forgetting the component orientations does not change the point set $L$ or the complementary regions of a sphere disjoint from it. Thus the split predicate depends only on the underlying unoriented link type, as required by [F1].

2.1givenF1step 1.1construct∎

Ambient isotopy carries any witnessing sphere and the two components of its complement to a sphere and two complementary components for the image link. The existence or nonexistence of a witness is therefore an ambient-isotopy invariant.

Depends on

Oriented links in the three-sphere and ambient isotopy

Used by

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

William W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), Theorem 1(a), pp. 37–41 (standard reference, not scraped)
Thomas Kindred, A geometric proof of the flyping theorem, §2.1, arXiv:2008.06490v3 (standard reference, not scraped)