Manifolds and cell complexes Knot theory Conjecture

Slice-ribbon conjecture

Statement

A tame knot $K \subset S^3$ is a ribbon knot if it bounds an immersed disk $D$ in $S^3$ whose only self-intersections are ribbon singularities. A ribbon singularity is an arc of double points with two preimages in $D$ . One preimage is an arc interior to $D$ , and the other is an arc meeting $\partial D$ in its two endpoints. The conjecture states that every smoothly slice knot is a ribbon knot.

Remarks

The converse holds: pushing the interior of a ribbon disk into $B^4$ removes the singularities, so every ribbon knot is slice. Equivalently, $K$ is ribbon if and only if it bounds a smoothly embedded disk in $B^4$ on which the radial distance function has no local maxima. Lisca proved the conjecture for two-bridge knots. The conjecture is open in general.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Knot concept
Slice knot concept
Slice-ribbon conjecture conjecture

Tag	`EQB39X4HGB`
URI	`m2://EQB39X4HGB`
Kind	conjecture
Writing	W0 stub
MSC	57K10, 57K40
Status	This copy of the site has no build data. Run `m2 status EQB39X4HGB`.
About	Slice knot
Uses	Knot, Slice knot
Changed	2 November 2026

Slice-ribbon conjecture

Statement

Remarks

Reading path

Further reading

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