We investigate Legendrian graphs in . 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 and if and only if it does not contain as a minor. We show that the pair 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 determines two Legendrian isotopy classes of the lollipop graph and a pair determines four Legendrian isotopy classes of the handcuff graph.
"On Legendrian graphs." Algebr. Geom. Topol. 12 (3) 1273 - 1299, 2012. https://doi.org/10.2140/agt.2012.12.1273