Non-Leaving-Face property for marked surfaces

TL;DR

This paper proves that the non-leaving-face property, previously known for associahedra, holds for all unpunctured marked surfaces, advancing understanding of the geometric structure of these polytopes.

Contribution

It extends the non-leaving-face property from associahedra to all unpunctured marked surfaces, generalizing a key geometric feature.

Findings

Non-leaving-face property holds for all unpunctured marked surfaces.

Supports geometric understanding of flip graphs in surface triangulations.

Generalizes previous results from associahedra to broader classes of polytopes.

Abstract

We consider the polytope arising from a marked surface by flips of triangulations. Sleator, Tarjan and Thurston studied in 1988 the diameter of the associahedron, which is the polytope arising from a marked disc by flips of triangulations. They showed that every shortest path between two vertices in a face does not leave that face. We establish that same non-leaving-face property for all unpunctured marked surfaces.