DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 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.

Smooth knot concordance

Definition

Let $K_{0}, K_{1}$ be smooth oriented knots in the standard oriented three-sphere $S^{3}$ , and let $I = [0, 1]$ . A smooth concordance from $K_{0}$ to $K_{1}$ is the image of a smooth proper embedding $f : S^{1} \times I \to S^{3} \times I$ whose lower and upper boundary circles are $K_{0} \times {0}$ and $K_{1} \times {1}$ . The embedding has product collars at both ends: in a neighborhood of each boundary circle it is the corresponding knot embedding crossed with the interval coordinate. Orient the annulus so that its boundary, with the boundary-orientation convention, is $- K_{0} \times {0} ⊔ K_{1} \times {1}$ . The knots are smoothly concordant when such an annulus exists.

Definition references

Smooth oriented knots and their orientations

Properness, local boundary charts, and product collars

Depends on

Used by

An abstract annulus need not be a smooth concordance Counterexample
Equal zero-traces do not force smooth concordance Counterexample
Exotic and localized homology concordance classes Definition
Locally flat topological knot concordance Definition
Metabolic Seifert forms and algebraic knot concordance Definition
Ribbon disks and ribbon knots Definition
Satellite operators on concordance classes Definition
A knot and its reverse mirror sum to a slice knot Example
A genus-g knot cobordism bounds the tau difference Lemma
A slice disk induces a Seifert metabolizer Lemma
A smooth slice knot has a knot Floer complex stably equivalent to the unknot Lemma
Epsilon is stable-equivalence invariant but not a homomorphism Lemma
Lee cobordism maps give the genus filtration bound Lemma
Local resolution of ribbon singularities Lemma
Odd Khovanov and Bar–Natan LEO foundations Lemma
Smooth and locally flat topological concordance are equivalence relations Lemma
Smooth and topological concordance respect connected sum Lemma
Smooth sliceness is equivalent to concordance to the unknot Lemma
The slice-ribbon conjecture Open problem
Epsilon and Upsilon concordance data Theorem
Every ribbon knot is smoothly slice Theorem
LEO local-equivalence group and concordance map Theorem
Smooth and topological knot concordance classes form abelian groups Theorem
Smooth concordance implies stable filtered equivalence of knot Floer complexes Theorem
Tau is an additive smooth concordance invariant and bounds four-ball genus Theorem
The algebraic concordance map is a surjective homomorphism Theorem

Dependency tree · two levels

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