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CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)
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Jones span detects crossing number for reduced alternating knots

Statement

Assume full AC. If D is a connected reduced alternating diagram of a knot K, then the library Jones polynomial has t-span exactly c(K)=c(D), in the convention t=A−4. The span of a nonzero Laurent polynomial is its largest exponent minus its smallest exponent.

Facts & Assumptions

Given: Full AC, a connected reduced alternating knot diagram D, and the crossing number of its underlying knot type.

[F1]

Full AC implies countable choice (The Axiom of Countable Choice (ACω)).

[F2]

The crossing number is the least crossing count among regular diagrams of the same underlying unoriented link type (Crossing number of a link).

[F3]

Under full AC, a reduced alternating knot diagram with c(D) crossings has Jones span c(D), while every competing c′-crossing knot diagram has Jones span at most c′ (Adequacy and Jones span of reduced alternating diagrams).

[F4]

A reduced alternating diagram is connected and alternating by definition (Reduced alternating link diagram).

[A1]

Full AC, the choice-function principle for arbitrary families of nonempty sets (The Axiom of Choice).

Proof

1.1A1F1F2F3F4given

By [A1] and [F1], the crossing-number definition [F2] applies. Since D is a connected reduced alternating diagram by [F4], [F3] gives span⁡tVK(t)=c(D).

2.1F2F3step 1.1choosealgebra∎

For every regular diagram E of the same underlying knot type, choose its single-component orientation to match K. By [F3], c(D)≤c(E) because the common Jones span is c(D) and is at most c(E). The diagram D itself is one of the competitors in [F2], so c(K)≤c(D); the preceding inequality for every E gives c(D)≤c(K). Thus c(K)=c(D), and step 1.1 gives the asserted Jones span.

Depends on

Used by

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Sources