## Algebraic & Geometric Topology

### On Legendrian graphs

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 12, Number 3 (2012), 1273-1299.

Dates
Revised: 31 January 2012
Accepted: 28 February 2012
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715398

Digital Object Identifier
doi:10.2140/agt.2012.12.1273

Mathematical Reviews number (MathSciNet)
MR2966686

Zentralblatt MATH identifier
1257.57009

#### Citation

O’Donnol, Danielle; Pavelescu, Elena. On Legendrian graphs. Algebr. Geom. Topol. 12 (2012), no. 3, 1273--1299. doi:10.2140/agt.2012.12.1273. https://projecteuclid.org/euclid.agt/1513715398

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