Geometric simplicial embeddings of arc-type graphs

TL;DR

This paper studies multiarc graphs on surfaces, proving rigidity and convexity properties of simplicial maps, which deepen understanding of their geometric structure and relationships to arc and flip graphs.

Contribution

It establishes rigidity results and convexity of subsurface strata for multiarc graphs, extending known properties of arc and flip graphs.

Findings

Simplicial maps are often induced by obvious geometric operations.

Subsurface strata are convex under certain complexity conditions.

Rigidity results constrain the structure of graph embeddings.

Abstract

In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the "obvious way". We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex images.