LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The right trefoil is prime

Statement

Assume full AC. The standard right-handed trefoil is a prime oriented knot.

Statement references

Definition of a prime knot

Exact genus of the right trefoil

Facts & Assumptions

Given: Full AC and the standard right-handed trefoil $T$ .

[F1]

$g (T) = 1$ . (The right trefoil has genus one)

[F2]

Genus is additive under connected sum, and genus zero characterizes the unknot, under AC $_{ω}$ . (Knot genus is additive under connected sum, A knot has genus zero exactly when it is the unknot)

[F3]

Full AC implies AC $_{ω}$ . (The Axiom of Choice, AC implies DC implies countable choice)

Proof

Given: Full AC and $T$ .

1.1givenF1F2F3

By [F1] and [F2], $T$ is nontrivial. If $T = K # J$ with both factors nontrivial, [F2] and [F3] make their genera positive integers, so $g (K) + g (J) \geq 2$ .

2.1F1F2F3step 1.1∎

Additivity gives $1 = g (T) = g (K) + g (J) \geq 2$ , a contradiction. Thus $T$ is nontrivial and has no connected-sum presentation with both factors nontrivial, exactly the definition of prime.

Sources

Oriented connected sum and prime knot

Depends on

Used by

The Alexander polynomial does not determine the prime knot summands Counterexample

Dependency tree · two levels

30 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

Burde and Zieschang, Knots, 2nd revised and extended edition (standard reference, not scraped)
The right-trefoil genus-one proof in this batch (standard reference, not scraped)