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Planar Legendrian graphs

Peter Lambert-Cole and Danielle O’Donnol

Algebraic & Geometric Topology 22 (2022) 3709–3746

We prove two results on the classification of trivial Legendrian embeddings of trivalent planar graphs, g: G (S3,ξstd). First, we show that such Legendrian graphs are completely classified by their oriented Legendrian ribbon Rg and rotation invariant  rotg. Thus, the pair (Rg,rotg) 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 K4 as a minor, then the unique trivial embedding of G is Legendrian simple.

Lambert-Cole dedicates this work to Joan Lambert.

contact topology, Legendrian graphs, convex surface theory, Legendrian simple
Mathematical Subject Classification 2010
Primary: 05C10, 53D10
Secondary: 57M15
Received: 23 July 2019
Revised: 14 June 2021
Accepted: 8 July 2021
Published: 14 March 2023
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