Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)
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Noncoprime torus slopes can be links

Statement refuted

Assume the full Axiom of Choice. The claim that every nonzero integer pair (p,q) determines a single torus knot and satisfies the coprime torus-knot genus formula is false: (p,q)=(2,2) determines a two-component link.

Facts & Assumptions

Given: Full AC and the nonzero parameter pair (2,2).

[A1]

Full AC is the choice-function axiom used by the knot and formula statements (The Axiom of Choice).

[L1]

Full AC implies Countable Choice, the hypothesis of the torus-knot definition (AC implies DC implies countable choice, The Axiom of Countable Choice (ACω)).

[F1]

Coprime torus knot specifies that when d=gcd⁡(∣p∣,∣q∣)>1, the corresponding torus link is the union of d disjoint parallel copies of the primitive slope (p/d,q/d).

[F2]

Torus-knot Alexander polynomial and genus formulas assumes coprime nonzero parameters and defines genus for a single knot.

Counterexample

1.1A1F2givenalgebra

The greatest common divisor of ∣2∣ and ∣2∣ is d=2>1, so the pair (2,2) is outside the coprime hypothesis in [F2].

2.1A1L1F1step 1.1construct

Full AC [A1] supplies the countable-choice hypothesis of [F1] by [L1]. The associated torus link then consists of d=2 disjoint parallel copies of the primitive slope (2/2,2/2)=(1,1). The two copies are separate embedded circles, so the result has two components and is not a knot.

3.1F2step 1.1step 2.1algebra∎

The knot genus formula in [F2] has no application to this pair: its coprimality hypothesis fails and its knot-genus input is not a single-component knot. Thus this example refutes extending that knot formula to all nonzero integer pairs.

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Sources