Planar Legendrian $\Theta$-graphs

TL;DR

This paper classifies topologically trivial Legendrian $ heta$-graphs, identifies all nondestabilizeable realizations, and introduces moves that alter Legendrian types within isotopy classes, revealing an infinite family of such graphs.

Contribution

It provides the first classification of Legendrian $ heta$-graphs and introduces new moves, expanding understanding beyond Legendrian knots.

Findings

Infinite family of nondestabilizeable Legendrian $ heta$-graphs.

Any planar graph with a $ heta$-subdivision has infinitely many Legendrian embeddings.

Introduces vertex stabilization and vertex twist moves.

Abstract

We classify topologically trivial Legendrian $\Theta$-graphs and identify the complete family of nondestabilizeable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of a $\Theta$-graph or $S^1 \vee S^1$ as a subgraph will have an infinite number of distinct, topologically trivial nondestabilizeable Legendrian embeddings. Additionally, we introduce two moves, vertex stabilization and vertex twist, that change the Legendrian type within a smooth isotopy class.