Classification of exact Lagrangian fillings of Legendrian links

Determine a complete classification of oriented embedded exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere (S^3, ξ_st), up to compactly supported Hamiltonian isotopy.

Background

The paper situates its main theorem within a broader program: while this work completes the classification for the standard Legendrian Hopf link, the general problem of classifying exact Lagrangian fillings for Legendrian links is far from settled. Historically, only the max-tb Legendrian unknot has a complete, non-empty classification of embedded exact Lagrangian fillings, and most subsequent advances have focused on constructing examples and distinguishing them via Floer- or sheaf-theoretic invariants.

This unresolved classification motivates the authors’ approach: they develop a pseudoholomorphic conic fibration compatible with an arbitrary exact Lagrangian filling and then show any such filling is Hamiltonian isotopic to one of two standard models for the Hopf link. The broader classification problem, however, persists for other Legendrian links.

References

Even if the classification of exact Lagrangian fillings lies at the heart of low-dimensional contact and symplectic topology, it is still largely unresolved. To wit, the only Legendrian link for which we have a complete non-empty classification of Lagrangian fillings is the max-tb Legendrian unknot.

— The Legendrian Hopf Link has exactly two Lagrangian fillings  (2506.15111 - Thomson, 18 Jun 2025) in Section 1.1 (Scientific Context)

We conjecture, in line with Conjecture 5.1, that the fillings we construct comprise an exhaustive collection.

— Cluster structures from Legendrian double twist knots: a comparative approach  (2610.02086 - Hughes et al., 1 Oct 2026) in Section 1, Main results