Algebraic & Geometric Topology

On Legendrian graphs

Danielle O’Donnol and Elena Pavelescu

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We investigate Legendrian graphs in (3,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

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

Received: 11 October 2011
Revised: 31 January 2012
Accepted: 28 February 2012
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M50: Geometric structures on low-dimensional manifolds
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

Legendrian graph Thurston–Bennequin number rotation number $K_4$


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

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