Once punctured disks, non-convex polygons, and pointihedra

TL;DR

This paper investigates the structure and properties of flip-graphs related to polygons and punctured polygons, revealing convex embeddings, diameter bounds, and connections to polytopes like type D associahedra and pointihedra.

Contribution

It introduces new insights into the geometric properties of flip-graphs of punctured polygons and their relationships with well-known polytopes, expanding understanding of their structure.

Findings

Embeddings between flip-graphs are strongly convex.

Bounds on the diameters of these flip-graphs are established.

Connections between flip-graphs and polytopes like type D associahedra and pointihedra are demonstrated.

Abstract

We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once-punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometric properties of these graphs and how they relate to one another. In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic). We also find bounds on the diameters of these graphs, sometimes using the strongly convex embeddings. Finally, we show how these graphs relate to different polytopes, namely type D associahedra and a family of secondary polytopes which we call pointihedra.