[Paper Review] Color or cover

Ivan Izmestiev|arXiv (Cornell University)|Mar 2, 2015

Geometric Analysis and Curvature Flows8 references3 citations

TL;DR

This paper proves that on a triangulated sphere, if all but two vertices have degrees divisible by $k$, those two exceptional vertices cannot be adjacent, using coloring monodromy for $k=2$ and vertex colorings of Platonic solids for $k=3,4,5$. It generalizes monodromy to branched covers between triangulated surfaces and connects the construction to Belyi surfaces and constant curvature cone-metrics.

ABSTRACT

If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can associate a branched cover. This generalizes to a space of germs between two triangulated surfaces. We also discuss relations with Belyi surfaces and with cone-metrics of constant curvature.

Motivation & Objective

• To establish a topological constraint on triangulated spheres where all but two vertices have degrees divisible by $k$.
• To investigate the geometric and combinatorial implications of such degree conditions using coloring monodromy.
• To generalize the monodromy construction from $k=2$ to higher $k$, including $k=3,4,5$, via Platonic solid colorings.
• To connect the resulting branched covers to Belyi surfaces and cone-metrics of constant curvature.
• To extend the monodromy framework to germs between triangulated surfaces, broadening its applicability.

Proposed method

• Uses coloring monodromy as a tool to analyze vertex degree conditions on triangulated spheres.
• Applies vertex colorings of Platonic solids to handle cases $k=3,4,5$, leveraging their symmetric structures.
• Constructs a branched cover from the monodromy data associated with vertex colorings.
• Extends the monodromy construction to a space of germs between triangulated surfaces, generalizing the framework.
• Relates the resulting structures to Belyi surfaces through the monodromy representation.
• Connects the geometric data to cone-metrics of constant curvature via the monodromy and surface structure.

Experimental results

Research questions

• RQ1 Under what conditions can two exceptional vertices of non-divisible degree coexist on a triangulated sphere?
• RQ2 How does coloring monodromy facilitate the proof of non-adjacency for $k=2$?
• RQ3 What role do Platonic solids play in extending the monodromy method to $k=3,4,5$?
• RQ4 How can monodromy be generalized from individual surfaces to germs between triangulated surfaces?
• RQ5 What is the relationship between the constructed branched covers and Belyi surfaces or constant curvature cone-metrics?

Key findings

• For $k=2$, the non-adjacency of the two exceptional vertices is proven using coloring monodromy.
• For $k=3,4,5$, the non-adjacency result is established via vertex colorings of Platonic solids.
• A branched cover can be associated with a coloring monodromy, generalizing the construction beyond $k=2$.
• The monodromy framework extends to a space of germs between triangulated surfaces, enabling broader geometric applications.
• The construction yields connections to Belyi surfaces through the monodromy representation.
• The method provides a pathway to studying constant curvature cone-metrics via combinatorial and monodromic data.