Equilateral convex triangulations of $\mathbb{R}{P}^2$ with three conical points of equal defect

Abstract

Consider triangulations of $\mathbb{R}{P}^2$ whose all vertices have valency six except three vertices of valency $4$. In this chapter we prove that the number $f(n)$ of such triangulations with no more than $n$ triangles grows as $C\cdot n^2+O(n^{3/2})$ where $C=\frac{1}{20}\sqrt{3}\cdot L(\frac{\pi }{3}){\zeta }^{-1}(4)\zeta (Eis,2)\approx \mathrm{0.2087432125056015...}$, where $L$ is the Lobachevsky function and $\zeta (Eis,2)=\sum _{(a,b)\in {\mathbb{Z}}^2\setminus 0}\frac{1}{|a+b{\omega }^2{|}^4}$, and ${\omega }^6=1$.