Doubly Semiequivelar Maps on Torus and Klein Bottle

Anand K Tiwari,Amit Tripathi,Punam Gupta,Yogendra Singh

doi:10.1155/2020/5674172

Anand K Tiwari, Amit Tripathi + Show 2 more

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https://doi.org/10.1155/2020/5674172

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Abstract

A tiling of the Euclidean plane, by regular polygons, is called 2-uniform tiling if it has two orbits of vertices under the action of its symmetry group. There are 20 distinct 2-uniform tilings of the plane. Plane being the universal cover of torus and Klein bottle, it is natural to ask about the exploration of maps on these two surfaces corresponding to the 2-uniform tilings. We call such maps as doubly semiequivelar maps. In the present study, we compute and classify (up to isomorphism) doubly semiequivelar maps on torus and Klein bottle. This classification of semiequivelar maps is useful in classifying a category of symmetrical maps which have two orbits of vertices, named as 2-uniform maps.

Highlights

Equivelar and semiequivelar maps are generalizations of the maps on the surfaces of well-known Platonic solids and Archimedean solids to the closed surfaces other than the 2sphere, respectively
Analogues to the Archimedean tilings, here we initiate the theory of maps on torus and Klein bottle corresponding to the 2-uniform tilings
We present infinite series of doubly semiequivelar map (DSEM) of type [33, 42: 32, 4, 3, 4], we do not know whether this DSEM can be obtained from any semiequivelar map by the above elementary map operations. is observation leads to the following question

Summary

Introduction

Equivelar and semiequivelar maps are generalizations of the maps on the surfaces of well-known Platonic solids and Archimedean solids to the closed surfaces other than the 2sphere, respectively. E 3 regular tilings provide equivelar maps of types [36], [44], and [63] and 8 semiregular tilings provide semiequivelar maps of types [34, 6], [33, 42], [32, 4, 3, 4], [3, 4, 6, 4], [3, 6, 3, 6], [3, 122], [4, 6, 12], and [4, 82] on torus and Klein bottle. Maity and Upadhyay [17] have presented a way to classify the eight types of semiequivelar maps on the torus for arbitrary number of vertices

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