# Asymptotic bounds on the number of faces needed to construct a polyhedron of a certain genus

Let a polyhedron be a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices, where moreover we require that

• every edge touches exactly two faces,
• every vertex touches at least three faces and three edges,
• each face is convex, and
• the dual graph to the set of faces touching a single vertex is homotopical to a circle (i.e. a vertex cannot be a spike for two different sets of faces).

Let $F(g)$ be the minimal number of faces a polyhedron of genus $g$ must have. My question is: what can be said about $F(g)$?

What I have found is the following:

We have $F(0) = 4$ (given by the tetrahedron), and we should have $F(1) = 9$ and $F(2) = 12$.

Given $g\ge3$, we can construct a polyhedron of genus $g$ as follows: take two convex polygons with $g$ sides, and glue them together using $g$ quadrilateral faces in the obvious way. Pair vertices not sharing a face two by two, cut out a triangle around them, and link the triangles using three quadrilateral faces for each couple of vertices. This gives us the bound $$F(g)\le 2 + 4g.$$ Can this be improved?

Moreover, consider the following. Assume we can construct a polyhedron of genus $0$ with faces all having $n$ sides. Then we have $E = \frac{n}{2}F$ edges, and we can use the formula for the Euler characteristic to find that the number of vertices is $$V = 2 +\left(\frac{n}{2} - 1\right)F.$$ With the same process of pairing vertices, cutting and gluing, this allows us to construct a polyhedron of genus $g = \frac{V}{2} = 1 + \left(\frac{n}{2} - 1\right)\frac{F}2$ with $F + 3g$ faces. The asymptotic behavior is $$\lim_{n\to\infty}\frac{F+3g}{g} = 3.$$ This heuristics, together with the intuition that as every "hole" must be bounded by at least three faces (and thus $F(g)\ge3g$), naturally leads to:

Conjecture: We have $\lim_{g\to\infty}\frac{F(g)}{g} = 3$.

This however strongly relies on the existence of the polyhedron with convex faces with $n$ sides, of which I'm not sure. Can anyone prove or disprove my conjecture? And what about providing better bounds and asymptotics? Also, what happens if we remove the convexity assumption on the faces (but requiring that they are contractible, maybe star-shaped)?

Remark: The inspiration for this question comes from this question, where the OP is however a bit less restrictive on the definition of a polytope.