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

TL;DR

This paper analyzes the asymptotic growth of a special class of triangulations of the real projective plane with three conical points, providing explicit formulas involving special functions and constants.

Contribution

It establishes the quadratic growth rate of such triangulations and derives an explicit constant involving Lobachevsky and zeta functions.

Findings

Number of triangulations grows as C·n^2 + O(n^{3/2})

Explicit constant C involves Lobachevsky function and zeta functions

Provides asymptotic enumeration formula for specific triangulations

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 $\omega^6=1$.