Semi - Equivelar Maps on the Torus and the Klein Bottle with few vertices

TL;DR

This paper classifies semi-equivelar maps with few vertices on the torus and Klein bottle, extending the concept of Archimedean solid tilings to these surfaces and identifying possible map types.

Contribution

It provides a classification of semi-equivelar maps on the torus and Klein bottle with limited vertices, expanding understanding of such maps beyond the sphere.

Findings

Classified semi-equivelar maps with few vertices on the torus.

Classified semi-equivelar maps with few vertices on the Klein bottle.

Connected the types of these maps to known plane tilings.

Abstract

Semi-Equivelar maps are generalizations of maps on the surfaces of Archimedean solids to surfaces other than the $2$-sphere. The well known 11 types of normal tilings of the plane suggest the possible types of semi-equivelar maps on the torus and the Klein bottle. In this article we classify (up to isomorphism) semi-equivelar maps on the torus and the Klein bottle with few vertices.