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.

A knot and its reverse mirror sum to a slice knot

Example

Assume the Axiom of Countable Choice. For every smooth oriented knot $K \subset S^{3}$ , the connected sum $K # K^{†}$ bounds a smooth disk in $B^{4}$ with only radial minima and saddles, is therefore also bounded by a locally flat topological disk, and satisfies $g_{4}^{s m} (K # K^{†}) = g_{4}^{t o p} (K # K^{†}) = 0$ . Its reverse-mirror summand represents the inverse of $[K]$ in each concordance group.

Example references

The reverse-mirror knot

The two slice-genus minima

The concordance group operations and inverse convention

Facts & Assumptions

Given: ${A C}_{ω}$ and a smooth oriented knot $K \subset S^{3}$ .

[F1]

The reverse mirror is formed by applying the fixed orientation-reversing reflection and then reversing the knot orientation. (Reverse mirror of an oriented knot)

[F2]

A smooth proper disk is locally flat as a topological proper surface, and its genus is $0$ because $χ (D^{2}) = 1$ . (Locally flat proper surfaces and concordance collars, Smooth and topological slice genus)

[F3]

The smooth and topological slice genera are the least genera in the respective nonempty surface classes. (Smooth and topological slice genus, Nonemptiness and minima for the smooth and locally flat slice genera)

[F4]

The general reverse-mirror disk construction applies to every smooth oriented knot. (A knot summed with its reverse mirror bounds a ribbon disk)

[F5]

In the smooth concordance group, the reverse-mirror class is the additive inverse; the current theorem's smooth branch proves this using the disk in [F4]. (Smooth and topological knot concordance classes form abelian groups)

[F6]

The full TOP group interface likewise gives the reverse-mirror class as the inverse for locally flat concordance, with the same countable-choice hypothesis. Its TOP connected-sum input includes the compact real-code/ $L [r]$ specialization in the compatibility supplier at exactly this standard-product annulus scope; the unqualified literature assertion alone is not treated as an AC_omega theorem. No new local normal-neighborhood proof is asserted here. (Smooth and topological knot concordance classes form abelian groups)

Verification

technique · direct

1.1F1F2F4givenconstruct

By [F1], the notation $K^{†}$ refers to the fixed reverse-mirror operation. Apply [F4] to obtain a smooth disk $D$ bounded by $K # K^{†}$ with radial critical indices $0$ or $1$ . Since the disk is smooth, [F2] makes it an admissible locally flat surface as well. Its Euler characteristic gives $g (D) = 0$ .

2.1F2F3step 1.1algebra

Every admissible surface genus is in $N_{0}$ , while $D$ contributes genus $0$ to both genus sets. Thus their least elements are $0$ by [F3].

3.1F5F6step 1.1∎

By [F5], the class of $K^{†}$ is the inverse of $[K]$ in the smooth group; [F6] gives the same conclusion in the locally flat topological group. The common disk from step1.1 is admissible in both categories, and all input choice hypotheses are ${A C}_{ω}$ . This proves the entire example, including its group-class assertion.

Sources

A knot summed with its reverse mirror bounds a ribbon disk

Smooth and topological knot concordance classes form abelian groups

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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)