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 $C = \frac{1}{20} \sqrt{3} \cdot L( \frac{\pi}{3} ) \zeta^{-1}(4) \zeta(Eis, 2) \approx 0.2087432125056015...$, where $L$ is the Lobachevsky function and $\zeta(Eis,2) =\sum\limits_{(a,b)\in\mathbb Z^2\setminus 0}{\frac{1}{|a+b\omega^2|^4}}$, and $\omega^6=1$.