Math Conjectures

Slice–ribbon conjecture

OPENMajorConjectureProposed 1962 · Canonical special case

Canonical statement

If a smooth knot K:S1↪S3K:S^{1}\hookrightarrow S^{3} bounds a smoothly and properly embedded disk D↪D4D\hookrightarrow D^{4}, then KK bounds a smooth immersion f:D2↬S3f:D^{2}\looparrowright S^{3} whose only self-intersections are ribbon singularities: transverse double arcs for which one preimage arc lies in int⁡D2\operatorname{int}D^{2} and the other has both endpoints on ∂D2\partial D^{2}.
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If a smooth knot \(K:S^{1}\hookrightarrow S^{3}\) bounds a smoothly and properly embedded disk \(D\hookrightarrow D^{4}\), then \(K\) bounds a smooth immersion \(f:D^{2}\looparrowright S^{3}\) whose only self-intersections are ribbon singularities: transverse double arcs for which one preimage arc lies in \(\operatorname{int}D^{2}\) and the other has both endpoints on \(\partial D^{2}\).
GEO-TOP-016:[Fox1962KnotProblems][Livingston2005KnotConcordanceSurvey][K3ProblemList2026][DunfieldGong2025RibbonExperiments]
Geometry and topology

Notes

A knot K⊂S3K\subset S^3 is smoothly slice if it bounds a smooth, properly embedded disk in the four-ball, and ribbon if it bounds an immersed disk in S3S^3 whose only self-intersections are ribbon singularities: transverse double arcs with one preimage arc interior and the other joining two boundary points. A ribbon disk can be pushed into D4D^4 to give an embedded slice disk, so every ribbon knot is slice. Fox asked in his 1962 problem list whether the converse holds — is every smoothly slice knot ribbon? [Fox1962KnotProblems]

The conjecture has resisted both proof and refutation. The concordance invariants surveyed by Livingston [Livingston2005KnotConcordanceSurvey], together with extensive computation, rule out many candidate counterexamples, and large-scale experiments with ribbon concordances and slice obstructions continue this program without finding a slice knot that fails to be ribbon [DunfieldGong2025RibbonExperiments]. The question retains a central place on current problem lists in low-dimensional topology [K3ProblemList2026].

It remains open in both directions: there is no proof of the conjecture, and no smoothly slice knot has been shown not to be ribbon.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.