Planar Legendrian graphs

Lambert-Cole, Peter; O’Donnol, Danielle

doi:10.2140/agt.2022.22.3709

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Planar Legendrian graphs

Peter Lambert-Cole and Danielle O’Donnol

Algebraic & Geometric Topology 22 (2022) 3709–3746

DOI: 10.2140/agt.2022.22.3709

Abstract

We prove two results on the classification of trivial Legendrian embeddings of trivalent planar graphs, $g : G \to (S^{3}, ξ_{s t d})$ . First, we show that such Legendrian graphs are completely classified by their oriented Legendrian ribbon $R_{g}$ and rotation invariant ${r o t}_{g}$ . Thus, the pair $(R_{g}, {r o t}_{g})$ is a complete set of invariants. Second, we give two sufficient conditions for such embeddings to be Legendrian simple (ie determined by the contact framing and the rotation numbers of all of the cycles). If $G$ is planar, and is $3$ –connected or contains $K_{4}$ as a minor, then the unique trivial embedding of $G$ is Legendrian simple.

Lambert-Cole dedicates this work to Joan Lambert.

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Keywords

contact topology, Legendrian graphs, convex surface theory, Legendrian simple

Mathematical Subject Classification 2010

Primary: 05C10, 53D10

Secondary: 57M15

References

Publication

Received: 23 July 2019

Revised: 14 June 2021

Accepted: 8 July 2021

Published: 14 March 2023

Authors

Peter Lambert-Cole
Department of Mathematics University of Georgia Athens, GA United States

Danielle O’Donnol
Department of Mathematics Marymount University Arlington, VA United States

Volume 22, issue 8 (2022)