ProofAtlas

Low-dimensional topology and knot concordance

Slice–Ribbon Conjecture

Collaboration beta

Does every knot bounding a smooth disk in the four-ball also bound one with no interior maxima?

Kslice⇒Kribbon
Known results and sources
A knot above the boundary of a four-ball cross-section with slice and ribbon disk silhouettes separated by an open converse question.
Every ribbon knot is slice; the open question asks whether every slice knot is ribbon.

Research problem

Exact mathematical statement

A smooth knot K⊂S3K\subset S^3 is slice if it bounds a smoothly and properly embedded disk D⊂B4D\subset B^4. It is ribbon if it bounds such a disk for which the radial Morse function has no interior local maxima, equivalently a disk built only from minima and saddles. The Slice–Ribbon Conjecture is

Kslice⇒Kribbon.K\text{ slice}\Longrightarrow K\text{ ribbon}.

Problem infographic

Problem at a glance

A four-panel problem-first explainer defines slice and ribbon knots, checks a valid ribbon-disk example with two minima, one saddle, and no maxima, shows the one-way known implication, and states the open converse.
Slice–Ribbon asks whether every smooth slice knot admits some disk with no interior local maxima; the displayed ribbon example has Euler characteristic two minus one equals one.

Current mathematical picture

Where work on Slice–Ribbon Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: A Laurent-chain unit or a group-ring simplification automatically gives a smooth handle cancellation. Turn algebraic or Floer information into a boundary-level ribbon conclusion or an explicit geometric cancellation, while respecting the distinction between one disk and all disks for the same knot.

Route status · Narrowed route
Main reductionCurrent reduction

Modulo the stated ambient movie audit, Slice–Ribbon is equivalent to the assertion that every knot with a one-maximum slice disk is ribbon, or to elementary ribbon descent.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeWrite the ambient band-movie interchange that completes the elementary ribbon-concordance factorization rel boundary.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Slice–Ribbon Conjecture in numbers

1.5kretained lines of mathematical investigation1,496 in the current working snapshot
Argument development
1,258 · 84%
Explored or eliminated routes
37 · 2%
Computational analysis
3 · 0%
Open obligations
76 · 5%
Definitions and setup
122 · 8%
8selected mapped statements1routes investigated4open questions4contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byRalph FoxProposed1962Status checkedAug 14, 2026
Source recordedAug 14, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Slice–Ribbon ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Does every slice knot admit a ribbon disk? — Depends on missing premiseDoes every slice knot admita ribbon disk?Current reduction — Depends on missing premiseCurrent reductionFirst unresolved Morse profile — Depends on missing premiseFirst unresolved MorseprofileOne-maximum reduction — Depends on missing premiseOne-maximum reductionChosen-disk versus knot boundary — Depends on missing premiseChosen-disk versus knotboundaryClosing target — Depends on missing premiseClosing targetExact conjecture — Depends on missing premiseExact conjectureSource-reported Floer descent branch — Depends on missing premiseSource-reported Floerdescent branchSource-reported limitation — stoppedSource-reported limitationWrite the ambient band-movie interchange that completes the elementary ribbon-concordance factorization rel boundary. — OpenWrite the ambient band-movieinterchange that completesthe…Resolve the one-maximum (2,2,1) profile by boundary-level ribbonness or a genuine knot obstruction. — OpenResolve the one-maximum(2,2,1) profile byboundary-level…Convert surviving algebraic data into an embedded sphere–belt pivot or another valid smooth geometric move. — OpenConvert surviving algebraicdata into an embeddedsphere–belt…No proof or counterexample in source — OpenNo proof or counterexamplein source
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded
Narrowed routeSource-reported limitation

The following claim is rejected or insufficient in the recorded route: A Laurent-chain unit or a group-ring simplification automatically gives a smooth handle cancellation. Turn algebraic or Floer information into a boundary-level ribbon conclusion or an explicit geometric cancellation, while respecting the distinction between one disk and all disks for the same knot.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Write the ambient band-movie interchange that completes the elementary ribbon-concordance factorization rel boundary.Suggested move: Give an explicit isotopy moving a leaf birth–saddle pair past the other surface handles without changing the boundary knots.
Ready to work on
02
Resolve the one-maximum (2,2,1) profile by boundary-level ribbonness or a genuine knot obstruction.Suggested move: Apply the current work's exact group-ring-before-abelianization protocol to concrete marked movies and separate fixed-disk defects from boundary-knot conclusions.
Ready to work on
03
No proof or counterexample in source

The governing source explicitly reports no full proof or counterexample.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Convert surviving algebraic data into an embedded sphere–belt pivot or another valid smooth geometric move.Suggested move: Track whiskers, Peiffer data, and the chosen disk throughout any proposed handle trade instead of inferring cancellation from abelian algebra.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The smooth Slice–Ribbon Conjecture remains open. Recent work proves family-specific non-ribbon results and conditional dichotomies but neither supplies a slice non-ribbon knot nor proves every slice knot ribbon.

[1][2]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Peer reviewedHom and Park proved that either distinct iterated cables in their family are linearly independent in concordance or Slice–Ribbon fails; this dichotomy does not resolve which alternative holds.[2]
  2. Peer reviewedSchultens's survey records the conjecture as still open and discusses band sums and ribbon concordance context.[1]
  3. Historical sourceModern literature attributes the converse from slice to ribbon to Fox's 1962 problem list.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusSlice–Ribbon Conjecture
Logical consequenceRibbon implies slice

Every ribbon knot is slice; the conjecture asks for the converse implication.

[1][2]
Related problemConcordance independence of iterated cables

For iterated cables of tight fibered knots, a linear-independence alternative is tied to the truth or failure of Slice–Ribbon.

[2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetFormal smooth knot, properly embedded disk, and radial Morse-function definitions in dimensions three and four.
  • Formalization targetFormal support for ribbon singularities and the equivalence with no-maxima disk movies.
  • Formalization targetA theorem-alignment review separating boundary-knot existence from properties of one chosen disk.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role8 selected mapped statements
  • theorem candidate1 of 81
  • reduction2 of 82
  • lemma3 of 83
  • equivalence1 of 81
  • negative result1 of 81
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.21 displayed rows · 1 route included
  • retained route statementDoes every slice knot admit a ribbon disk?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact conjectureintermediate
  • retained route statementOne-maximum reductionintermediate
  • retained route statementFirst unresolved Morse profileintermediate
  • retained route statementSource-reported Floer descent branchintermediate
  • retained route statementChosen-disk versus knot boundaryintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureSource-reported limitationreported failure
  • Research targetWrite the ambient band-movie interchange that completes the elementary ribbon-concordance factorization rel boundary.open
  • Research targetResolve the one-maximum (2,2,1) profile by boundary-level ribbonness or a genuine knot obstruction.open
  • Research targetConvert surviving algebraic data into an embedded sphere–belt pivot or another valid smooth geometric move.open
  • Research targetNo proof or counterexample in sourceopen
  • Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: A Laurent-chain unit or a group-ring simplification automatically gives a smooth handle cancellation. Turn algebraic or Floer information into a boundary-level ribbon conclusion or an explicit geometric cancellation, while respecting the distinction between one disk and all disks for the same knot.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeWrite the ambient band-movie interchange that completes the elementary ribbon-concordance factorization rel boundary.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointWrite the ambient band-movie interchange that completes the elementary ribbon-concordance factorization rel boundary.

Slice–Ribbon Conjecture · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Does every knot bounding a smooth disk in the four-ball also bound one with no interior maxima?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
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Sources and references2 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Band sums and concordance of knotspeer reviewed result · Jennifer Schultens · Boletín de la Sociedad Matemática Mexicana / Springer Nature · 2025-03-10 · DOI 10.1007/s40590-025-00722-y · accessed Aug 14, 2026
  2. 2
    Ribbon knots and iterated cables of fibered knotspeer reviewed result · Jennifer Hom, JungHwan Park · Mathematische Zeitschrift / Springer Nature · 2026-06-18 · DOI 10.1007/s00209-026-04050-3 · accessed Aug 14, 2026

Important qualifications

  • This bounded pass uses two recent peer-reviewed publisher pages and does not decide the status of every proposed counterexample or variant.
  • The original Fox problem is represented indirectly through modern sources; the 1962 proceedings were not independently opened.
  • No incoming-packet URL was fetched and packet claims were not used as external evidence.
  • No formalization or candidate-counterexample computation was independently reproduced.

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