#### Volume 12, issue 3 (2012)

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On Legendrian graphs

### Danielle O’Donnol and Elena Pavelescu

Algebraic & Geometric Topology 12 (2012) 1273–1299
##### Abstract

We investigate Legendrian graphs in $\left({ℝ}^{3},{\xi }_{std}\right)$. 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 $\left(tb,rot\right)$ 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 $\left(tb,rot\right)$ determines two Legendrian isotopy classes of the lollipop graph and a pair $\left(tb,rot\right)$ 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