Existence of compressing cycles for general exact Lagrangian fillings

Determine whether an arbitrary exact Lagrangian filling of a Legendrian link admits an $L$-compressing cycle, and develop methods to identify such a cycle explicitly when it exists.

Background

LL-compressing cycles are homology cycles in a filling that bound embedded Lagrangian disks in the complement. They are central to the paper because their intersection data supplies the quivers and cluster coordinates associated with fillings.

For fillings arising from Legendrian weaves, the paper gives explicit combinatorial descriptions of several classes of such cycles. The general existence and identification problem remains unresolved outside the weave setting.

References

For a general filling, determining the existence of an $L$-compressing cycle -- let alone explicitly identifying one -- remains an open problem.

— Cluster structures from Legendrian double twist knots: a comparative approach  (2610.02086 - Hughes et al., 1 Oct 2026) in Section 2, subsection “Homology of weaves”