On Legendrian graphs

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.