DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Locally flat topological knot concordance

Definition

Let $K_{0}, K_{1}$ be smooth oriented knots in the standard oriented three-sphere $S^{3}$ , and let $I = [0, 1]$ . They are locally flat topologically concordant if there is a topological proper embedding $f : S^{1} \times I \to S^{3} \times I$ whose image is a locally flat annulus, whose boundary circles are $K_{0} \times {0}$ and $K_{1} \times {1}$ , and whose annulus orientation induces the boundary $- K_{0} \times {0} ⊔ K_{1} \times {1}$ . The annulus must have product collars at both ends. No differentiability of the annulus is assumed.

Definition references

Smooth oriented endpoint knots

The same endpoint and orientation convention in the smooth category

Properness, local-flatness charts, and collars

Depends on

Used by

Exotic and localized homology concordance classes 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 slice disk induces a Seifert metabolizer Lemma
Local resolution of ribbon singularities Lemma
Smooth and locally flat topological concordance are equivalence relations Lemma
Smooth and topological concordance respect connected sum Lemma
The slice-ribbon conjecture Open problem
Reported refined s-invariant computations Remark
Every ribbon knot is smoothly slice Theorem
Smooth and topological knot concordance classes form abelian groups Theorem
The algebraic concordance map is a surjective homomorphism Theorem

Dependency tree · two levels

8 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)