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TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
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Bridge number one characterizes the unknot and signature bounds unknotting number

Statement

Assume full AC. For every smooth oriented knot K, b(K)=1 if and only if K is ambient isotopic through orientation-preserving diffeomorphisms to the fixed oriented coordinate circle, and the classical signature satisfies ∣σ−1(K)∣≤2u(K), where u(K) is the minimum number of crossing changes needed to reach that circle.

Facts & Assumptions

Given: Full AC and a smooth oriented knot K.

[A1]

Full AC implies the countable-choice hypothesis of the bridge-number-one theorem. The Axiom of Choice, AC implies DC implies countable choice

[F1]

Under countable choice, b(K)=1 if and only if K is ambient isotopic through orientation-preserving diffeomorphisms to the fixed oriented coordinate circle. Bridge number one characterizes the unknot

[F2]

Under full AC, the classical signature is well-defined and satisfies ∣σ−1(K)∣≤2u(K). The classical signature bounds unknotting number

Proof

1.1A1F1given

By [A1], the hypotheses of [F1] hold, so it supplies both implications of the asserted bridge-number characterization, with the specified oriented standard unknot.

2.1F2step 1.1∎

Apply [F2] to the same knot. It gives the asserted signature inequality with the same unknotting-number convention. Combining it with step 1.1 proves the conjunction without changing either interface.

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