Reverse implications in the Lagrangian slice–ribbon hierarchy

Determine whether regularly Lagrangian slice knots are necessarily decomposably Lagrangian slice and whether Lagrangian slice knots are necessarily regularly Lagrangian slice, including the corresponding versions for prescribed disks, prescribed Legendrian knots, and topological knot types.

Background

The paper introduces decomposable, regular, and general Lagrangian sliceness as symplectic analogues of ribbonness, handle-ribbonness, and sliceness. Decomposable disks are known to be regular, but the converse questions remain unresolved at several levels of specification.

References

Decomposable Lagrangian disks are regular , but we do not know if the converse holds. Likewise, we do not know if every Lagrangian disk is regular . In fact, under any interpretation (for prescribed disks, for prescribed Legendrian knots, or simply for topological knot types) each reverse implication in the "Lagrangian slice-ribbon conjecture" eq:SSR is open.

eq:SSR:

{decomposably slice}⇒{regularly slice}⇒{Lagrangian slice}.\{\text{decomposably slice}\} \Rightarrow \{\text{regularly slice}\} \Rightarrow \{\text{Lagrangian slice}\}.

— Derivative links in contact topology  (2609.30182 - Breen et al., 24 Sep 2026) in Section 1, Context, equation (1.2) discussion

Specifically, while the max-tb unknot is decomposably slice, we do not know if contact-$(+1)$ handleslides preserve decomposable sliceness.

— Derivative links in contact topology  (2609.30182 - Breen et al., 24 Sep 2026) in Remark 2.14, subsection “Legendrian R-links”