Semi-equivelar maps on the surface of Euler characteristic -1

TL;DR

This paper classifies semi-equivelar maps on the double torus with Euler characteristic -1, showing none are vertex transitive, extending previous classifications of such maps on different surfaces.

Contribution

It provides a classification of semi-equivelar maps on the double torus with Euler characteristic -1 and demonstrates their lack of vertex transitivity.

Findings

No semi-equivelar maps on the double torus with Euler characteristic -1 are vertex transitive.

The classification extends previous work on semi-equivelar maps on other surfaces.

The study identifies specific types of semi-equivelar maps present on the double torus.

Abstract

Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. In earlier work a complete classification of semi-equivelar map of type $(3^5, 4)$ on the surface of Euler characteristic -1 was given. In the meantime Karabas an Nedela classified vertex transitive semi-equivelar maps on the double torus. In this article we study the types of semi-equivelar maps on double torus that are also available on the surface of Euler characteristic -1. We classify them and show that none of them are vertex transitive.