Semi-equivelar maps on the torus are Archimedean
Abstract
If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map on the torus is a quotient of an Archimedean tiling on the plane then the map is semi-equivelar. We show that each semi-equivelar map on the torus is a quotient of an Archimedean tiling on the plane. Vertex-transitive maps are semi-equivelar maps. We know that four types of semi-equivelar maps on the torus are always vertex-transitive and there are examples of other seven types of semi-equivelar maps which are not vertex-transitive. We show that the number of -orbits of vertices for any semi-equivelar map on the torus is at most six. In fact, the number of orbits is at most three except one type of semi-equivelar maps. Our bounds on the number of orbits are sharp.
Keywords
Cite
@article{arxiv.1705.05236,
title = {Semi-equivelar maps on the torus are Archimedean},
author = {Basudeb Datta and Dipendu Maity},
journal= {arXiv preprint arXiv:1705.05236},
year = {2017}
}
Comments
Corrected typos and minor errors in the proofs of Theorems 1.7 (c) and 1.8 (b). This article is a continuation of our works in [3] (arXiv:1610.01830). Naturally, some definitions here are same as in [3]. We have used three examples from [3] to prove Theorem 1.8. We have included these in Example 2.2 (and cited) for completeness