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On Legendrian graphs

Danielle O'Donnol and Elena Pavelescu

Algebraic & Geometric Topology 12 (2012) 1273–1299

DOI: 10.2140/agt.2012.12.1273
Abstract

We investigate Legendrian graphs in (R3,ξstd). We extend the Thurston–Bennequin number and the rotation number to Legendrian graphs. We prove that a graph can be Legendrian realized with all its cycles Legendrian unknots with tb = −1 and rot = 0 if and only if it does not contain K4 as a minor. We show that the pair (tb,rot) does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. When we restrict to planar spatial graphs, a pair (tb,rot) determines two Legendrian isotopy classes of the lollipop graph and a pair (tb,rot) determines four Legendrian isotopy classes of the handcuff graph.

Keywords

Legendrian graph, Thurston–Bennequin number, rotation number, K4
Mathematical Subject Classification

Primary: 57M25, 57M50

Secondary: 05C10
References
Publication

Received: 11 October 2011
Revised: 31 January 2012
Accepted: 28 February 2012
Published: 9 June 2012
Authors
Danielle O'Donnol
Department of Mathematics and Statistics
Smith College
44 College Lane
Northampton MA 01060
USA
Elena Pavelescu
Department of Mathematics
Occidental College
1600 Campus Road
Los Angeles CA 90041-3314
USA