Legendrian theta-graphs

TL;DR

This paper characterizes Legendrian theta-graphs by their Thurston-Bennequin and rotation numbers, introduces the transverse push-off concept, and relates its topological type to pretzel links, revealing limitations of classical invariants.

Contribution

It provides necessary and sufficient conditions for Legendrian theta-graphs and introduces the transverse push-off, linking it to pretzel links for topologically planar cases.

Findings

Thurston-Bennequin and rotation numbers do not fully classify Legendrian theta-graphs.

Defined the transverse push-off and computed its self-linking number.

Topological type of the transverse push-off is a pretzel link for planar theta-graphs.

Abstract

In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar theta-graph. We define the transverse push-off of a Legendrian graph and we determine its self linking number for Legendrian theta-graphs. In the case of topologically planar theta-graphs, we prove that the topological type of the transverse push-off is that of a pretzel link.