CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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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.
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Under full AC, the right-handed trefoil is not ribbon

Statement refuted

Assume the full Axiom of Choice. The standard right-handed trefoil is a ribbon knot.

Statement references

Full Axiom of Choice used by the nonsliceness supplier

Definition of a ribbon knot

Facts & Assumptions

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

[A1]

Full AC is the choice-function principle. (The Axiom of Choice)

[F1]

Under full AC, the standard right-handed trefoil does not bound a smooth properly embedded disk in $B^{4}$ . (The trefoil does not bound a smooth proper disk in the four-ball)

[F2]

Under ACω, every ribbon knot bounds a smooth properly embedded disk in $B^{4}$ . (Every ribbon knot is smoothly slice)

[F3]

Full AC implies ACω. (AC implies DC implies countable choice)

Counterexample

The standard right-handed trefoil $T$ has no smooth proper disk in $B^{4}$ , so it cannot admit a ribbon disk whose resolution is such a disk.

1.1givenA1F1

The given full AC is the choice-function principle in [A1]. Applying [F1] under that hypothesis, no smooth properly embedded disk in $B^{4}$ has boundary $T$ .

1.2givenF2F3construct

If $T$ were ribbon, [F3] would supply the ACω hypothesis of [F2], and [F2] would give a smooth properly embedded disk in $B^{4}$ with boundary $T$ .

2.1givenstep 1.1step 1.2contradiction∎

Step 1.2 contradicts step 1.1. Therefore the trefoil is not ribbon, refuting the statement.

Diagram

The box denotes the three-crossing positive two-strand braid whose closure is the standard right-handed trefoil. The closure arcs join equal endpoint positions outside the box. This encoded diagram fixes the standard trefoil used in the counterexample; the proof relies on the nonsliceness supplier, not on the drawing.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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

Arunima Ray, Slice Knots and Knot Concordance, Winter Braids Lecture Notes 8 (standard reference, not scraped)
Charles Livingston, A Survey of Classical Knot Concordance, Handbook of Knot Theory, Chapter 7 (standard reference, not scraped)