Loops of Legendrians in contact 3-manifolds

TL;DR

This paper investigates the topology of Legendrian loops in contact 3-manifolds, revealing non-trivial homotopy classes and the relationship between Legendrian and formal Legendrian loops, with implications for contact topology.

Contribution

It establishes a homotopy injection from the contactomorphism group into Legendrian spaces and provides examples of Legendrian loops with non-trivial Legendrian properties but trivial smooth knot types.

Findings

Homotopy injection of contactomorphism group into Legendrian space.

Existence of Legendrian loops non-trivial as Legendrians but trivial as smooth knots.

Examples of formal Legendrian loops that are non-trivial.

Abstract

We study homotopically non-trivial spheres of Legendrians in the standard contact R3 and S3. We prove that there is a homotopy injection of the contactomorphism group of S3 into some connected components of the space of Legendrians induced by the natural action. We also provide examples of loops of Legendrians that are non-trivial in the space of formal Legendrians, and thus non-trivial as loops of Legendrians, but which are trivial as loops of smooth embeddings for all the smooth knot types.