Legendrian $\theta$-graphs

Danielle O’Donnol and Elena Pavelescu

Vol. 270 (2014), No. 1, 191–210

DOI: 10.2140/pjm.2014.270.191

Abstract

We give necessary and sufficient conditions for two triples of integers to be realized as the Thurston–Bennequin number and the rotation number of a Legendrian $\theta$-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar $\theta$-graph. We define the transverse push-off of a Legendrian graph, and we determine its self linking number for Legendrian $\theta$-graphs. In the case of topologically planar $\theta$-graphs, we prove that the topological type of the transverse push-off is that of a pretzel link.

Keywords

Legendrian graph, Thurston–Bennequin number, rotation number, $\theta$-graph

Mathematical Subject Classification 2010

Primary: 57M25, 57M50

Secondary: 05C10

Milestones

Received: 17 March 2013

Revised: 7 July 2013

Accepted: 12 July 2013

Published: 2 August 2014