Exact Lagrangians in four-dimensional symplectisations

TL;DR

This paper constructs explicit examples of exact Lagrangian embeddings in four-dimensional symplectisations, revealing new phenomena about their displaceability and linking properties of Legendrians in overtwisted contact manifolds.

Contribution

It provides explicit constructions of exact Lagrangians in symplectisations and explores their displaceability and Legendrian linking properties.

Findings

Exact Lagrangian embeddings of tori and Klein bottles are constructed.

Any closed exact Lagrangian in the symplectisation is displaceable.

Examples of topologically linked Legendrians that are dynamically non-interlinked are presented.

Abstract

In this note we provide explicit constructions of exact Lagrangian embeddings of tori and Klein bottles inside the symplectisation of an overtwisted contact three-manifold. Note that any closed exact Lagrangian in the symplectisation is displaceable by a Hamiltonian isotopy. We also use positive loops to exhibit elementary examples of topologically linked Legendrians that are dynamically non-interlinked in the sense of Entov-Polterovich.