Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)
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Descending diagrams are unknotted and unknotting number is finite

Statement

Assume the Axiom of Countable Choice. Let K be a smooth oriented knot and let D be an admissible finite planar code for a regular diagram of a representative of K, based at an interior point of an edge. After moving the basepoint slightly forward within the same edge to avoid bends if necessary, the descending-diagram procedure changes at most c(D) crossings, where c(D) is the number of crossing records in the code, and its output diagram represents the standard unknot. Consequently the set of unknotting-sequence lengths for K is a nonempty subset of N, so the unknotting number u(K) is finite and satisfies u(K)≤c(D).

Facts & Assumptions

Given: ACω, a smooth oriented knot K, a finite regular diagram code D, and a basepoint in the interior of an edge.

[A1]

The Axiom of Countable Choice, written ACω. The Axiom of Countable Choice (ACω)

[F1]

After moving an edge-interior bend basepoint slightly forward within the same edge if necessary, the algorithm terminates, changes at most one crossing per crossing record and returns an admissible updated code; the adjustment does not change crossing encounter order or the output. Descending-diagram crossing-change sequence, The descending-diagram procedure terminates on admissible finite codes

[F2]

A code whose first encounter at every crossing is an overpass represents the standard unknot. A descending based knot diagram represents the unknot

[F3]

The unknotting number is the minimum length of a crossing-change sequence from some regular diagram of a representative to the standard unknot diagram. Bridge number and unknotting number of an oriented knot

Proof

Given: ACω, a knot K, and an admissible finite code D for a regular diagram of a representative of K, with a basepoint in an edge interior.

1.1A1F1givenalgebra

Make the basepoint adjustment in [F1] if necessary, then apply the procedure to D. Since D has c(D) crossing records, the algorithm terminates after that finite scan and returns at most c(D) crossing changes.

2.1F2F3step 1.1algebra∎

After those changes every crossing is first encountered as an overpass, so [F2] says the updated code represents the standard unknot. It follows from [F3] that the returned list witnesses u(K)≤c(D). In particular the set defining u(K) is nonempty and its minimum is finite.

Depends on

Used by

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Sources