# Non-simplicial triangulations of compact surfaces with few vertices

I'm interested in triangulations with few vertices of a given orientable compact surface $S$.

By triangulation, I don't mean a "simplicial triangulation" but a "decomposition of $S$ by triangles", these being topological triangles glued by identifying edges. The only condition required is that an edge in $S$ is adjacent to two different triangles.

For instance, gluing two copies of a given euclidean triangle along the corresponding edges gives what I consider to be a triangulation of the sphere with 3 vertices (and 3 edges and 2 faces).

Question 1: the study of such "triangulations" must be classical. What are good references?

More specifically, I'm interested in the following question:

Question 2: for $S$ of low genus (say 0, 1 or 2) what are the different combinatorial types of such triangulations with few (say 2,3 or 4) vertices? And what is the method to describe them?

Thanks in advance for any help.

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This is an incomplete answer, but maybe the pointers can be of some use to you.

Question 1: It looks to me that the triangulations you describe are essentially triangulated multigraphs embedded on a surface. It is a heavily studied topic in topological graph theory, about which Graphs on Surfaces by Mohar and Thomassen is a good reference (there is also Topological Graph Theory by Gross and Tucker).

There is one small caveat in that you want edges to be adjacent to two different triangles, which is not a condition usually enforced in these references. But I think that triangles glued to themselves lead to dunce hats or dunce caps, which should not be very hard to analyze separately

Some more references stem from the neighborly concept of combinatorial maps: some bijections are known, you can check for example Gilles Schaeffer's work.

Question 2: If you are interested in the triangulations with few vertices, the first case is actually the one vertex case. This paper enumerates them all, but it is quite tricky. I do not know of similar results with more vertices.

If you restrict your attention to simplicial complexes, more is known, see this question.

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In the one vertex case the first few entries are 1 triangulation in genus 1, 9 in genus 2, 1726 in genus 3. These low genus cases, with many pictures in the genus 2 case, are discussed in my paper "A users guide to the mapping class group: once punctured surfaces". The nice thing about the one vertex case is that they can be depicted with "chord diagrams", closely related to dessins d'enfant. –  Lee Mosher Nov 22 '13 at 15:34
The first interesting case for what I have in mind is g=1 with 2 vertices... –  Lucien from IHP Nov 22 '13 at 17:02
@LeeMosher I think they all can be described with dessins d'enfant - just take the dual graph. This is the clean dessins of a semistable modular elliptic surface. For instance, in his paper Les familles stables de courbes elliptiques sur $\mathbb P^1$ admettant quatre fibres singulieres, Beauville computed all six 4-vertex triangulations of the sphere. I believe there are exactly five 2-vertex triangulations of the torus. They are uniquely determined by the degree of their lower-degree vertex, which can be between $1$ and $6$ but cannot be $5$. –  Will Sawin Nov 26 '13 at 16:42