Knotted Legendrian surfaces with few Reeb chords

TL;DR

This paper constructs multiple Legendrian surface embeddings with minimal Reeb chords, explores their contact homology properties, and investigates their generating family structures, providing new insights into Legendrian topology.

Contribution

It introduces explicit constructions of Legendrian surfaces with minimal Reeb chords and analyzes their contact homology and generating families, revealing novel phenomena in Legendrian topology.

Findings

Constructed g+1 Legendrian embeddings with g+1 Reeb chords

Identified embeddings with non-augmentable contact homology DGAs

Explored generating family perspectives of these surfaces

Abstract

For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$ into $J^1(R^2)=R^5$ which lie in pairwise distinct Legendrian isotopy classes and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the Legendrian contact homology DGA does not admit any augmentation over $Z/2Z$, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in $J^1(S^2)$ from a similar perspective.