Knotted Legendrian surfaces with few Reeb chords
Georgios Dimitroglou Rizell

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 , we construct Legendrian embeddings of a surface of genus into which lie in pairwise distinct Legendrian isotopy classes and which all have transverse Reeb chords ( is the conjecturally minimal number of chords). Furthermore, for of the embeddings the Legendrian contact homology DGA does not admit any augmentation over , 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 from a similar perspective.
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