Papers by Debashis Bhowmik
arXiv (Cornell University), Feb 15, 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.
A New Class of Quantum Codes Associate with a Class of Maps
Proceedings of the Seventh International Conference on Mathematics and Computing, 2022
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.
Some New Classes of Homological Quantum Codes Associated with Surface Maps
National Academy Science Letters, 2021
Error-correcting quantum codes and related combinatorial constructs play an important role in sev... more Error-correcting quantum codes and related combinatorial constructs play an important role in several results in computational theory. In this article, we introduce two new classes of codes. These codes are associated with a class of combinatorial structures of surfaces. The encoding rates of these classes of codes are 1.
Indian Journal of Pure and Applied Mathematics, 2021
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.
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 $.
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.
arXiv: Geometric Topology, 2021
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).
Surface codes and color codes associated with non-orientable surfaces
Quantum Information and Computation, 2021
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, 2020
Semi-equivelar maps are generalizations of Archimedean solids. We classify all the semi-equivelar... more Semi-equivelar maps are generalizations of Archimedean solids. We classify all the semi-equivelar maps on the surface of Euler Characteristics-2 with vertices up to 12. We calculate their automorphism groups and study their vertex-transitivity.
arXiv: Geometric Topology, 2019
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.
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Papers by Debashis Bhowmik