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

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Every smooth knot has a finite crossing-change sequence to the unknot

Statement

Assume the Axiom of Countable Choice. Every smooth oriented knot in $S^{3}$ admits a finite rational polygonal code recording a one-component diagram with only finitely many transverse double points and an overpassing choice at each, together with a finite sequence of changes that switch overpassing choices and whose final diagram represents the standard unknot.

Statement references

Every smooth knot admits a finite planar diagram code

A descending diagram represents the unknot

Facts & Assumptions

Given: AC $_{ω}$ and a smooth oriented knot $K \subset S^{3}$ .

[A1]

The Axiom of Countable Choice, written AC $_{ω}$ . The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

Under AC $_{ω}$ , every smooth oriented knot has an admissible finite rational planar code for a regular diagram. Every regular knot projection admits a finite rational planar code

[F2]

The finite traversal algorithm outputs a list of crossing changes and changes at most one crossing per finite crossing record. Descending-diagram crossing-change sequence

[F3]

If every first encounter at a crossing is an overpass, the represented diagram is ambient isotopic to the standard unknot. A descending based knot diagram represents the unknot

Proof

Given: The knot $K$ and AC $_{ω}$ .

1.1A1F1givenchoose

Apply [F1] to choose an admissible finite code for a regular diagram of a representative of $K$ , and choose a basepoint in the interior of one edge away from every crossing and geometric bend. There is such a point: a one-component code has at least one nondegenerate polygonal edge segment, including in the crossing-free case, and only finitely many crossings and bends are forbidden. Choose an interior point of such a segment outside that finite set.

2.1F2step 1.1algebra

Apply [F2] to that finite code. The algorithm terminates after the finite crossing records are scanned and produces a finite crossing-change list; after those changes, each crossing is first encountered as an overpass.

3.1F1F3step 1.1step 2.1∎

By [F3], the code after the listed changes represents the standard unknot. Thus the chosen code and its finite crossing-change list witness the asserted existence for $K$ .

Sources

Every regular knot projection admits a finite rational planar code

Descending-diagram crossing-change sequence

A descending based knot diagram represents the unknot

Depends on

Used by

Bridge number and unknotting number of an oriented knot Definition

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

Ozsváth, Stipsicz, and Szabó, Grid Homology for Knots and Links (standard reference, not scraped)
Cole Charbonneau, Computing the Wirtinger Presentation of a Knot Group from a Knot Diagram (standard reference, not scraped)