The Legendrian Hopf Link has exactly two Lagrangian fillings

2025/06/18 by Bryce Thomson, Thomson, Bryce · 1 citation
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2506.15111

openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

We prove that there are precisely two embedded exact Lagrangian fillings of the standard Legendrian Hopf link, up to compactly supported Hamiltonian isotopy. It was known that the standard Legendrian Hopf link admitted at least two such Lagrangian fillings: we show these are all. Specifically, we use a type of neck-stretching procedure to construct a pseudoholomorphic conic fibration that makes a given arbitrary exact Lagrangian filling fiber over a real curve, under a global pseudoholomorphic Lefschetz fibration. This then allows for an explicit Hamiltonian isotopy to be constructed from any given Lagrangian filling to one of two known standard fillings.

Citations

• Cluster structures on braid varieties
• Braid Loops with infinite monodromy on the Legendrian contact DGA
• Lagrangian Skeleta and Plane Curve Singularities
• Infinitely many Lagrangian fillings
• Legendrian knots and exact Lagrangian cobordisms
• Lagrangian spheres in S² x S²
• On the holomorphicity of genus two Lefschetz fibrations
• Introduction to Symplectic Field Theory
• Handlebody Construction of Stein Surfaces
• Symplectic manifolds and their lagrangian submanifolds

Cited by

• Regular Lagrangians in Lefschetz fibrations

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