A zonotope and a product of two simplices with disconnected flip graphs

TL;DR

This paper presents a three-dimensional zonotope with disconnected tilings and demonstrates that triangulations of certain products of simplices are also disconnected for large dimensions, using probabilistic methods.

Contribution

It provides the first example of a disconnected flip graph in zonotopal tilings and extends this to triangulations of products of simplices, employing probabilistic construction techniques.

Findings

A 3D zonotope with disconnected tight tilings

Disconnection of triangulations of ^4 imes ^n for large n

Use of probabilistic methods in geometric combinatorics

Abstract

We give an example of a three-dimensional zonotope whose set of tight zonotopal tilings is not connected by flips. Using this, we show that the set of triangulations of $\Delta^4 \times \Delta^n$ is not connected by flips for large $n$. Our proof makes use of a non-explicit probabilistic construction.