On Legendrian Graphs

TL;DR

This paper extends classical invariants to Legendrian graphs in three-dimensional space, characterizes when graphs can be realized with specific Legendrian properties, and explores how these invariants distinguish different Legendrian classes.

Contribution

It introduces extensions of Thurston-Bennequin and rotation numbers to Legendrian graphs and characterizes their effectiveness in classifying Legendrian realizations.

Findings

A graph can be Legendrian realized with all cycles as unknots with tb=-1 and rot=0 iff it lacks a K_4 minor.

The pair (tb, rot) does not fully classify Legendrian graphs with cut edges or vertices.

For specific graphs, (tb, rot) determines multiple Legendrian classes.

Abstract

We investigate Legendrian graphs in $(\R^3, \xi_{std})$. We extend the classical invariants, Thurston-Bennequin number and 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 $(tb, rot)$ does not characterize a Legendrian graph up to Legendrian isotopy if the graph contains a cut edge or a cut vertex. For the lollipop graph the pair $(tb,rot)$ determines two Legendrian classes and for the handcuff graph it determines four Legendrian classes.