Fox slice–ribbon conjecture

Determine whether every smoothly slice knot in the 3-sphere is ribbon, thereby resolving the slice–ribbon conjecture.

Background

The paper uses the classical slice–ribbon conjecture as the smooth-topological motivation for introducing analogous gradations in contact and symplectic topology. Ribbon knots are known to be slice, but the converse remains unresolved.

References

The slice--ribbon conjecture asserts that the first implication is an equivalence.

— Exceptional pretzel knots are not algebraically slice  (2609.34522 - Saurabh, 28 Sep 2026) in Section 1, Introduction

Fox's slice-ribbon conjecture asking if the converse holds has become one of the most famous open questions in knot theory; see .

— Derivative links in contact topology  (2609.30182 - Breen et al., 24 Sep 2026) in Section 1, Introduction, Context