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Marbarisha M. Kharkongor

Publications and source records attributed to Marbarisha M. Kharkongor.

4 recordsLinked to original sources
arXiv2022

A class of maps on the torus and their vertex orbits

A tiling (edge-to-edge) of the plane is a family of tiles that cover the plane without gaps or overlaps. Vertex figure of a vertex in a tiling to be the union of all edges incident to that vertex. A tiling is $k$-vertex-homogeneous if any two vertices with congruent vertex figures are symmetric with each other and the vertices form precisely $k$ transitivity classes with respect to the group of all symmetries of the tiling. In this article, we discuss that if a map is the quotient of a plane's $k$-vertex-homogeneous lattice ($k \ge 4$) then what would be the sharp bounds of the number of vertex orbits.

math.CO↗
arXiv2021

Quotient maps of 2,3-uniform tilings of the plane on the torus

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.

math.GT↗
arXiv2021

Dihedral and cyclic symmetric maps on surfaces

If the face\mbox{-}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\mbox{-}edge\mbox{-}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. $-2m, m \ge 2$ 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.

math.CO↗