Semi-equivelar maps on the torus and the Klein bottle are Archimedean |
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Authors: | Basudeb Datta Dipendu Maity |
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Affiliation: | Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India |
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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 or on the Klein bottle 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. |
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Keywords: | Polyhedral maps on torus and Klein bottle Vertex-transitive map Equivelar map Archimedean tiling |
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