Computing Reeb dynamics on 4d convex polytopes

TL;DR

This paper introduces a method to analyze Reeb dynamics on 4D convex polytopes by linking combinatorial and smooth Reeb orbits, enabling computational exploration of their properties and testing conjectures.

Contribution

It establishes a correspondence between combinatorial and smooth Reeb orbits on 4D convex polytopes, facilitating computational analysis and new insights into systolic ratios.

Findings

Identified new examples of polytopes with systolic ratio 1

Developed a computational approach to enumerate Reeb orbits

Provided experimental evidence related to Viterbo's conjecture

Abstract

We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between "combinatorial Reeb orbits" for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action and Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbo's conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $1$.