Planar Legendrian graphs

Abstract

We prove two results on the classification of trivial Legendrian embeddings $g:G\to (S^3,{\xi }_{std})$ 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.

23 pages, 9 figures