ExampleConstruction: Literature-sourcedVerification: 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.

Connected-sum genus and ribbon examples

Example

Assume full AC. Let $T$ be the standard right-handed trefoil, let $T^{*}$ be its image under a fixed orientation-reversing reflection of $S^{3}$ , and let $- T^{*}$ denote reversal of that image's knot orientation. Then $g (T # T) = 2$ and $g (T # (- T^{*})) = 2$ . The knot $T # (- T^{*})$ bounds a smooth ribbon disk in $B^{4}$ , meaning a smooth properly embedded disk whose radial-height function has no interior local maxima; in particular it bounds a smooth properly embedded disk and is smoothly slice.

Example references

Connected sum and knot orientation conventions

Facts & Assumptions

Given: Full AC, the standard right-handed trefoil, its reverse mirror, and the stated connected-sum operations.

[F1]

The trefoil genus is exactly $1$ . (The right trefoil has genus one)

[F2]

Genus is additive under connected sum. (Knot genus is additive under connected sum)

[F3]

Mirroring and reversing orientation preserve genus. (Knot genus is invariant under mirroring and reversal)

[F4]

The trefoil sum with its reverse mirror bounds a smooth ribbon disk. (The trefoil sum with its reverse mirror is ribbon)

[A1]

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

[A2]

Full AC implies AC $_{ω}$ . (AC implies DC implies countable choice)

[A3]

The ribbon convention excludes interior radial maxima; index-zero minima and index-one saddles are allowed.

Verification

Given: Full AC and the two connected sums in the statement.

1.1givenF1F2F3A1A2algebra

By [F1], $g (T) = 1$ , and by [F3], $g (- T^{*}) = 1$ . Full AC implies the AC $_{ω}$ hypothesis of [F2] by [A2], so additivity gives $g (T # T) = 1 + 1 = 2$ and $g (T # (- T^{*})) = 1 + 1 = 2$ .

2.1givenF4A3∎

By [F4], $T # (- T^{*})$ bounds a smooth ribbon disk. A ribbon disk is in particular a smooth properly embedded disk, so the knot is smoothly slice. The explicit doubled-arc disk has no interior radial maxima, as required by the stated convention.

Sources

The right trefoil has genus one

Knot genus is additive under connected sum

Knot genus is invariant under mirroring and reversal

The trefoil sum with its reverse mirror is ribbon

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

31 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

Charles Livingston, A Survey of Classical Knot Concordance, arXiv:math/0307077v4 (standard reference, not scraped)
Burde and Zieschang, Knots, 2nd revised and extended edition (standard reference, not scraped)
Brendan Owens and Frank Swenton, An Algorithm to Find Ribbon Disks for Alternating Knots (standard reference, not scraped)