LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Connected alternating diagrams represent nonsplit links

Statement

If a regular link diagram on $S^{2}$ is connected and alternating, then the represented link is nonsplit.

Statement references

Greene, Alternating links and definite surfaces, Corollary 3.2 and Theorem 5.1

Kindred, A geometric proof of the flyping theorem, Fact 2.4

Facts & Assumptions

Given: A connected alternating regular diagram $D$ and its represented link $L$ .

[F1]

Connectedness and alternation are properties of the projection and its crossing visits; a nonempty connected crossingless diagram consists of one circle. (Reduced alternating link diagram)

[F2]

For a nonempty connected alternating diagram, the checkerboard construction gives a connected embedded surface $F$ with boundary $L$ , a finite disk-and-band presentation, and a definite quadratic form on $H_{1} (F; Q)$ satisfying $q ([c]) = lk (c, τ c)$ for all oriented rational multicurves. This also includes the zero-dimensional form of a crossingless spanning disk. (The checkerboard normal pairing of a connected alternating diagram is definite)

[F3]

A connected spanning surface with that definite normal-pairing property has nonsplit boundary. (A definite normal pairing forces a nonsplit boundary)

[F4]

A link is split if a sphere disjoint from it has link points on both complementary sides. (Split and nonsplit links)

Proof

1.1givenF1F2F4construct

If $D$ is empty, the represented link $L$ is empty, so no sphere has link points on both sides; $L$ is nonsplit by [F4]. Henceforth assume $D$ is nonempty. Apply [F2] to the supplied connected alternating diagram. The resulting surface $F$ is connected, has the represented link as its entire boundary, and carries the stated definite normal pairing. If the diagram has no crossings, [F1] identifies it as a single circle and [F2] uses a spanning disk, so the same supplied-surface conclusions hold.

2.1F3F4step 1.1∎

The surface from step 1.1 satisfies every hypothesis of [F3], which therefore says that no sphere of the kind in [F4] splits its boundary $L$ . Thus $L$ is nonsplit, with no reduction or choice assumption required.

Depends on

Used by

Reduced alternating diagrams minimize crossing number Theorem
Tait flyping theorem Theorem

Dependency tree · two levels

12 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

Joshua E. Greene, Alternating links and definite surfaces, §§2–3 and §5, especially Corollary 3.2 and Theorem 5.1 (standard reference, not scraped)
Thomas Kindred, A geometric proof of the flyping theorem, §2.1, Fact 2.4 (standard reference, not scraped)