Papers by debashis bhowmik
Proceedings of the Seventh International Conference on Mathematics and Computing
A 2-uniform tiling is an edge-to-edge tiling by regular polygons having 2 distinct transitivity c... more A 2-uniform tiling is an edge-to-edge tiling by regular polygons having 2 distinct transitivity classes of vertices. There are 20 distinct 2-uniform tilings (these are of 14 different types) on the plane, and since the plane is the universal cover of the torus, it is natural to explore maps on the torus that correspond to the 2-uniform tilings. In this article, we discuss that if a map is the quotient of a plane's 2-uniform lattice then what would be the bounds of the number of vertex orbits. A 3-uniform tiling is an edge-to-edge tiling by regular polygons having 3 distinct transitivity classes of vertices. There are 61 distinct 3-uniform tilings on the plane. In this article, we discuss that if a map is the quotient of a plane's 3-uniform lattice then what would be the bounds of the number of vertex orbits.
If the facecycles at all the vertices in a map are of the same type, then the map is said to be a... more If the facecycles 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 vertexedgeface 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.
We enumerate and classify all the semi equivelar maps on the surface of χ=-2 with up to 12 vertic... more We enumerate and classify all the semi equivelar maps on the surface of χ=-2 with up to 12 vertices. We also determine which of these are vertex-transitive and which are not.
If the cyclic sequences of face typesat all vertices in a map are the same, then the map is said ... more If the cyclic sequences of face typesat all vertices in a map are the same, then the map is said to be a semi-equivelar map. In particular, a semi-equivelar map is equivelar if the faces are the same type. Homological quantum codes represent a subclass of topological quantum codes. In this article, we introduce thirteen new classes of quantum codes. These codes are associated with the following: (i) equivelar maps of type [k^k], (ii) equivelar maps on the double torus along with the covering of the maps, and (iii) semi-equivelar maps on the surface of -1, along with their covering maps. The encoding rate of the class of codes associated with the maps in (i) is such that k/n→ 1 as n→∞, and for the remaining classes of codes, the encoding rate is k/n→α as n→∞ with α< 1.
If the cyclic sequence of faces for all the vertices in a polyhedral map are of the same types th... more 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. Se... more 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
ArXiv, 2020
If the cyclic sequences of {face types} {at} all vertices in a map are the same, then the map is ... more If the cyclic sequences of {face types} {at} all vertices in a map are the same, then the map is said to be a semi-equivelar map. In particular, a semi-equivelar map is equivelar if the faces are the same type. Homological quantum codes represent a subclass of topological quantum codes. In this article, we introduce {thirteen} new classes of quantum codes. These codes are associated with the following: (i) equivelar maps of type $ [k^k]$, (ii) equivelar maps on the double torus along with the covering of the maps, and (iii) semi-equivelar maps on the surface of \Echar{-1}, along with {their} covering maps. The encoding rate of the class of codes associated with the maps in (i) is such that $ \frac{k}{n}\rightarrow 1 $ as $ n\rightarrow\infty $, and for the remaining classes of codes, the encoding rate is $ \frac{k}{n}\rightarrow \alpha $ as $ n\rightarrow \infty $ with $ \alpha< 1 $.
arXiv: Combinatorics, 2020
If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is s... more If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is said to be a semi-equivelar map. In this article, we classify all the types of semi-equivelar maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$. That is, we present {a complete map types of} semi-equivelar maps (if exist) on the surface of Euler char. $-1$. We know the complete list of semi-equivelar maps (upto isomorphism) for some types. Here, we also present a complete list of semi-equivelar maps for one type and for other types, similar steps can be followed.
We enumerate and classify all the semi equivelar maps on the surface of $ \chi=-2 $ with up to 12... more 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.
If the face-cycles at all the vertices in a map are of the same type, then the map is said to be ... more 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.
A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Au... more A vertex-transitive map $X$ is a map on a closed surface on which the automorphism group ${\rm Aut}(X)$ acts transitively on the set of vertices. If the face-cycles at all the vertices in a map are of two types, then the map is said to be a $2$-semiequivelar map. A 2-uniform tiling is an edge-to-edge tiling of regular polygons having $2$ distinct transitivity classes of vertices. Clearly, a $2$-uniform map is $2$-semiequivelar. The converse of this is not true in general. We know that there are infinitely many tilings of $2$-semiequivelar type. In this article, we study $2$-semiequivelar maps on the torus that are the quotient of the plane's $2$-unform latices. We have found bounds of the number of orbits of vertices of the quotient $2$-semiequivelar maps on the torus under the automorphism groups (symmetric groups).
Quantum Information and Computation
Silva et al. produced quantum codes related to topology and coloring, which are associated with t... more Silva et al. produced quantum codes related to topology and coloring, which are associated with tessellations on the orientable surfaces of genus $\ge 1$ and the non-orientable surfaces of the genus 1. Current work presents an approach to build quantum surface and color codes} on non-orientable surfaces of genus $ \geq 2n+1 $ for $n\geq 1$. We also present several tables of new surface and color codes related to non-orientable surfaces. These codes have the ratios $k/n$ and $d/n$ better than the codes obtained from orientable surfaces.
National Academy Science Letters
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Papers by debashis bhowmik