Legendrian position of veering triangulations

TL;DR

This paper establishes a connection between veering triangulations and bicontact structures, showing how edges can be realized as Legendrian arcs in the context of Anosov flows, and introduces a steady positioning technique.

Contribution

It demonstrates that edges of veering triangulations can be realized as Legendrian arcs within bicontact structures and introduces the concept of steady position for triangulations.

Findings

Edges of veering triangulations can be realized as Legendrian arcs.

Every veering triangulation can be placed in steady position.

Horizontal surgery of veering triangulations corresponds to Goodman surgery of flows.

Abstract

We make a first step towards connecting the theory of veering triangulations and bicontact structures as tools for studying (pseudo-)Anosov flows: We show that given a veering triangulation corresponding to an Anosov flow with orientable stable and unstable foliations, the edges of the triangulation can be realized as Legendrian arcs with respect to a strongly adapted bicontact structure that supports the Anosov flow. Along the way, we show that every veering triangulation can be placed in `steady position', where each pair of edge projections that intersect in the orbit space only intersect once transversely. By a previous result of the author, this implies that horizontal surgery of veering triangulations correspond to horizontal Goodman surgery of pseudo-Anosov flows.