Open problemQuestion: Literature-sourcedProof: Not applicablePipeline-generated‡ sources checked 2026-10-22‡ unresolved

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.

This problem is unresolved. Its sources and status may be reviewed, but no solution is accepted here. It cannot be used as a proof premise.

The slice-ribbon conjecture

Open problem

For every smooth oriented knot $K \subset S^{3}$ that bounds a smooth properly embedded disk in $B^{4}$ , must there also exist a smooth immersion $f : D^{2} \to S^{3}$ whose boundary is $K$ and whose only self-intersections are finitely many transverse double arcs, each with one properly embedded preimage arc and one preimage arc contained in the interior of $D^{2}$ ?

Open problem references

The ribbon-immersion formulation and banded-link movie convention

Ray, Slice Knots and Knot Concordance, Conjecture 1.7

Hom and Park, Ribbon knots and iterated cables of fibered knots, §1

Hom and Park, author-hosted full-text version, arXiv:2507.20455v1

Livingston, A Survey of Classical Knot Concordance, §10

Attached PDF to Zenodo record 21752847, A Reduction of the Slice–Ribbon Conjecture

Zenodo record 21752847 metadata

Status

Status reviewed 2026-10-21. Hom and Park's version of record, published 18 June 2026, still identifies the general converse from smooth sliceness to ribbonness as open in §1; it proves restricted ribbon classifications and gives a conditional dichotomy, not a resolution. Ray's lecture notes record positive results for 2-bridge knots and most 3-strand pretzel knots. The 71-page PDF currently attached to Zenodo record 21752847 is titled A Reduction of the Slice–Ribbon Conjecture. Its abstract states only $H_{Q} \Rightarrow S R C$ and explicitly says no unconditional proof of $H_{Q}$ is claimed. Assumption 10.5 (PDF p.61) leaves $H_{n c}$ as a package-level hypothesis; Theorem 10.26 (p.67) assumes $H_{Q}$ , and Corollary 10.27 (p.67) lists $H_{n c}$ , $H_{r e d}$ , and $H_{a p p}$ as remaining conditions. By contrast, the current Zenodo API metadata title is A Conditional Proof of the Slice–Ribbon Conjecture and its description claims an unconditional proof; it credits Jianming Wang, while the attached PDF title page credits Kianming.W. This metadata/PDF discrepancy does not supply the missing proof. The assessed article and inspected attachment leave the general conjecture open.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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

Arunima Ray, Slice Knots and Knot Concordance, Winter Braids Lecture Notes 8 (standard reference, not scraped)
Jennifer Hom and JungHwan Park, Ribbon knots and iterated cables of fibered knots, arXiv:2507.20455v1; published in Mathematische Zeitschrift 313 (2026) (standard reference, not scraped)
Charles Livingston, A Survey of Classical Knot Concordance, Handbook of Knot Theory, Chapter 7 (standard reference, not scraped)
A Reduction of the Slice–Ribbon Conjecture: Certified Quenching and a Single Completion Condition, Zenodo record 21752847, attached PDF (standard reference, not scraped)
Current metadata for Zenodo record 21752847 (standard reference, not scraped)