Planar Legendrian graphs

TL;DR

This paper classifies trivial Legendrian embeddings of planar graphs in the standard contact 3-sphere, establishing invariants and conditions for Legendrian simplicity, especially for 3-connected graphs or those containing K4 as a minor.

Contribution

It introduces a complete set of invariants for trivial Legendrian embeddings of planar graphs and characterizes when such embeddings are Legendrian simple.

Findings

Oriented Legendrian ribbon and rotation invariant form a complete invariant set.

Unique trivial embedding is Legendrian simple for 3-connected graphs or graphs with K4 minor.

Abstract

We prove two results on the classification of trivial Legendrian embeddings $g: G \rightarrow (S^3,\xi_{std})$ of planar graphs. First, the oriented Legendrian ribbon $R_g$ and rotation invariant $\text{rot}_g$ are a complete set of invariants. Second, if $G$ is 3-connected or contains $K_4$ as a minor, then the unique trivial embedding of $G$ is Legendrian simple.