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LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)
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The tau level of the standard unknot is zero

Statement

For the standard oriented unknot U and its standard genus-zero one-pointed Heegaard diagram, the normalized Alexander-filtered hat complex has total homology F2 generated at filtration level 0. Consequently the diagramwise threshold is τH,F(U)=0.

Facts & Assumptions

Given: The standard oriented unknot and its standard genus-zero one-pointed diagram.

[F1]

The Alexander filtration and tau level fixes the absolute Alexander normalization with the one-generator standard unknot diagram at filtration zero.

[F2]

The standard genus-zero hat diagram of the unknot has rank one proves that this genus-zero one-pointed diagram has one generator and zero differential, so its total homology is F2.

Proof

1.1F2givenconstruct

By [F2], the total complex has one generator and zero differential, so its only nonzero homology class is represented by that generator.

2.1F1step 1.1algebra∎

By [F1] the generator has Alexander level 0. For every s<0 its filtered subcomplex is zero, while at s=0 the inclusion is the identity on the one-dimensional homology. Thus the first level with nonzero image is 0, which is the asserted value.

Depends on

Used by

Dependency tree · two levels

7 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