Semi-equivelar maps on the surface of Euler genus 3

2020

If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types then the map is said to be a Semi-equivelar map. In this article we classify all semi-equivelar and vertex transitive maps on the surface of Euler genus 3, i.e., on the surface of Euler characteristic -1.

Indian Journal of Pure and Applied Mathematics

Semi Equivelar maps are generalizations of Archimedean solids to surfaces other than 2-sphere. Semi Equivelar Maps were introduced by Upadhyay et. al. in 2014. They also studied semi equivelar maps on the surface of Euler characteristics v ¼ À1. In this article we classify all the semi equivelar maps on this surface with upto 12 vertices. We show that there are exactly four such maps. We also prove that there are at least 10 semi equivelar maps on this surface. We compute their Automorphism groups and show that none of these maps are vertex transitive. Keywords Vertex transitive maps Á Archimedean solids Á Semi-Equivelar Maps Á Automorphism group AMS classification 52B70 Á 52C20 Á 57M20 Á 57N05 1 Introduction Semi equivelar maps are generalizations of the maps on surfaces of Archemedian solids to the maps on surfaces other than the 2-sphere. It is observed that not much study has been made of maps on this surface except for the study of weakly neighbourly polyhedral maps (see [9]) and some semi equivelar maps (see [1] and [2]). In this article we attempt to give a classification of semi equivelar maps on this surface with at most 12 vertices and study their automorphism group. Recall that a surface is a closed 2-dimensional manifold without boundary. A Polyhedral complex is a finite collection X of convex polytopes, in Euclidean space E n such that the following two conditions are satisfied for X : first, if r 2 X and s is a face of r, then s 2 X and second, if r 1 and r 2 2 X then r 1 \ r 2 is a face of r 1 and r 2. The dimension d of X is the maximum of the dimensions of the elements in X. We also call X a polyhedral d-complex. In this article the objects of our study are polyhedral 2-complexes. For a 2dimensional polyhedral complex X, the elements of dimension 0, 1 an 2 are called vertex, edges and faces. Let v be a vertex of X. Then star of v, denoted by st(v), is a polyhedral complex fr 2 X : fvg rg and the link of a vertex v in X, denoted by lk(v) is the polyhedral complex fr 2 ClðstðvÞÞ : r \ fvg ¼ /g, where Cl(A) denote closure of A. If the link of each vertex in X is topologically a circle then X is also called a combinatorial 2-manifold. The set Aut(X) for a polyhedral complex X is a collection of all the automorphism

Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. In earlier work a complete classification of semi-equivelar map of type $(3^5, 4)$ on the surface of Euler characteristic -1 was given. In the meantime Karabas an Nedela classified vertex transitive semi-equivelar maps on the double torus. In this article we study the types of semi-equivelar maps on double torus that are also available on the surface of Euler characteristic -1. We classify them and show that none of them are vertex transitive.

2019

We enumerate and classify all the semi equivelar maps on the surface of $ \chi=-2 $ with up to 12 vertices. We also determine which of these are vertex-transitive and which are not.

2011

Semi-Equivelar maps are generalizations of Archimedean Solids (as are equivelar maps of the Platonic solids) to the surfaces other than 2−Sphere. We classify some semi equivelar maps on surface of Euler characteristic −1 and show that none of these are vertex transitive. We establish existence of 12-covered triangulations for this surface. We further construct double cover of these maps to show existence of semi-equivelar maps on the surface of double torus. We also construct several semi-equivelar maps on the surfaces of Euler characteristics −8 and −10 and on non-orientable surface of Euler characteristics −2.

2021

If the face-cycles at all the vertices in a map are of the same type, then the map is said to be a semi-equivelar map. Automorphism (symmetry) of a map can be thought of as a permutation of the vertices which preserves the vertex-edge-face incidences in the embedding. The set of all symmetries forms the symmetry group. In this article, we discuss the maps’ symmetric groups on higher genus surfaces. In particular, we show that there are at least 39 types of the semi-equivelar maps on the surface with Euler char. −2m,m ≥ 2 and the symmetry groups of the maps are isomorphic to the dihedral group or cyclic group. Further, we prove that these 39 types of semi-equivelar maps are the only types on the surface with Euler char. −2. Moreover, we know the complete list of semi-equivelar maps (up to isomorphism) for a few types. We extend this list to one more type and can classify others similarly. We skip this part in this article. MSC 2010 : 52C20, 52B70, 51M20, 57M60.

arXiv (Cornell University), 2020

A vertex v in a map M has the face-sequence (p n1 1 . . . . .p n k k ), if there are consecutive n i numbers of p i -gons incident at v in the given cyclic order, for 1 ≤ i ≤ k. A map M is called a semi-equivelar map if each of its vertex has same face-sequence. Doubly semi-equivelar maps are a generalization of semi-equivelar maps which have precisely 2 distinct face-sequences. In this article, we enumerate the types of doubly semi-equivelar maps on the plane and torus which have combinatorial curvature 0. Further, we present classification of doubly semi-equivelar maps on the torus and illustrate this classification for those doubly semi-equivelar maps which comprise of face-sequence pairs {(3 6 ), (3 3 .4 2 )} and {(3 3 .4 2 ), (4 4 )}.

Journal of Combinatorial Theory, Series B, 2012

In an earlier paper by A. Breda, R. Nedela and J. Širáň, a classification was given of all regular maps on surfaces of negative prime Euler characteristic. In this article we extend the classification to surfaces with Euler characteristic −3p (equivalently, to non-orientable surfaces of genus 3p + 2) for all odd primes p.

Journal of Combinatorial Theory, Series B, 2012

In an earlier paper by A. Breda, R. Nedela and J. Širáň, a classification was given of all regular maps on surfaces of negative prime Euler characteristic. In this article we extend the classification to surfaces with Euler characteristic −3p (equivalently, to non-orientable surfaces of genus 3p + 2) for all odd primes p.

We give a classification of all regular maps on nonorientable surfaces with a negative odd prime Euler characteristic (equivalently, on nonorientable surfaces of genus p + 2 where p is an odd prime). A consequence of our classification is that there are no regular maps on nonorientable surfaces of genus p + 2 where p is a prime such that p ≡ 1 (mod 12) and p = 13.