On Legendrian graphs

O’Donnol, Danielle; Pavelescu, Elena

doi:10.2140/agt.2012.12.1273

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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 $(ℝ^{3}, ξ_{std})$ . 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 $K_{4}$ 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, $K_4$

Mathematical Subject Classification 2010

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

Volume 12, issue 3 (2012)