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A generalization of 2-uniform maps on the torus

Speaker: Yogendra Singh (RSS2018503, PhD Student)

Abstract: A covering of the Euclidean plane, without gaps or overlap, using polygons is called a tiling of the plane. A tiling T is called k-uniform if it is edge to edge and has exactly k-orbits of vertices under its automorphism group. The k-uniform tilings of the plane are completely classified for k ∈ {1, 2, . . . , 7}. A map associated with such tilings is called k-uniform map. By the fact that the plane is the universal cover of the torus, one can explore the same type maps (locally) on the torus that are associated with these tilings. In this regard, the eleven 1-uniform tilings (also known as Archimedean tilings) and twenty 2-uniform tilings of the plane provide semi-equivelar and doubly semi-equivelar maps on the torus, respectively. Here, we give a construction to classify and enumerate doubly semi-equivelar maps on the torus. For this we describe a planar representation of a doubly semi-equivelar map. We also determine distinct classes of such representations by defining, if any, an isomorphism between them. This gives the exact number of maps up to isomorphism.

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