2-semi-equivelar maps on the torus and Klein bottle with few vertices

TL;DR

This paper classifies all 2-semi-equivelar maps with up to 12 vertices on the torus and Klein bottle, expanding understanding of these maps on non-spherical surfaces.

Contribution

It provides an exhaustive classification of 2-semi-equivelar maps of curvature 0 on the torus and Klein bottle, including all with up to 12 vertices.

Findings

Complete classification of 2-semi-equivelar maps on the torus and Klein bottle.

Identification of all such maps with up to 12 vertices.

Contribution to the understanding of maps on non-spherical surfaces.

Abstract

The $k$-semi equivelar maps, for $k \geq 2$, are generalizations of maps on the surfaces of Johnson solids to closed surfaces other than the 2-sphere. In the present study, we determine 2-semi equivelar maps of curvature 0 exhaustively on the torus and the Klein bottle. Furthermore, we classify (up to isomorphism) all these 2-semi equivelar maps on the surfaces with up to 12 vertices.