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Register a new userSemi-equivelar and vertex-transitive maps on the torus
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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 same type then the map is said to be a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. We show that there are eleven types of semi-equivelar maps on the torus. Three of these are equivelar maps. It is known that two of the three types of equivelar maps on the torus are always vertex-transitive. We show that this is true for the remaining one type of equivelar map and one other type of semi-equivelar maps, namely, if $X$ is a semi-equivelar map of type $[6^3]$ or $[3^3, 4^2]$ then $X$ is vertex-transitive. We also show, by presenting examples, that this result is not true for the remaining seven types of semi-equivelar maps. There are ten types of semi-equivelar maps on the Klein bottle. We present examples in each of the ten types which are not vertex-transitive.
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A map $X$ on a surface is called vertex-transitive if the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar toroidal maps which are not vertex-transitive. In this article, we show that semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps. More explicitly, we prove that each semi-equivelar toroidal map has a finite vertex-transitive cover. In 2019, Drach {em et al.} have shown that each vertex-transitive toroidal map has a minimal almost regular cover. Therefore, semi-equivelar toroidal maps are quotients of almost regular toroidal maps.
If the face-cycles at all the vertices in a map on a surface are of same type then the map is called semi-equivelar. There are eleven types of Archimedean tilings on the plane. All the Archimedean tilings are semi-equivelar maps. If a map $X$ on the torus is a quotient of an Archimedean tiling on the plane then the map $X$ is semi-equivelar. We show that each semi-equivelar map on the torus is a quotient of an Archimedean tiling on the plane. Vertex-transitive maps are semi-equivelar maps. We know that four types of semi-equivelar maps on the torus are always vertex-transitive and there are examples of other seven types of semi-equivelar maps which are not vertex-transitive. We show that the number of ${rm Aut}(Y)$-orbits of vertices for any semi-equivelar map $Y$ on the torus is at most six. In fact, the number of orbits is at most three except one type of semi-equivelar maps. Our bounds on the number of orbits are sharp.
A vertex-transitive map $X$ is a map on a surface on which the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called a semi-equivelar map. Clearly, a vertex-transitive map is semi-equivelar. Converse of this is not true in general. In particular, there are semi-equivelar maps on the torus, on the Klein bottle and on the surfaces of Euler characteristics $-1$ $&$ $-2$ which are not vertex-transitive. It is known that the boundaries of Platonic solids, Archimedean solids, regular prisms and antiprisms are vertex-transitive maps on $mathbb{S}^2$. Here we show that there is exactly one semi-equivelar map on $mathbb{S}^2$ which is not vertex-transitive. More precisely, we show that a semi-equivelar map on $mathbb{S}^2$ is the boundary of a Platonic solid, an Archimedean solid, a regular prism, an antiprism or the pseudorhombicuboctahedron. As a consequence, we show that all the semi-equivelar maps on $mathbb{RP}^2$ are vertex-transitive. Moreover, every semi-equivelar map on $mathbb{S}^2$ can be geometrized, i.e., every semi-equivelar map on $mathbb{S}^2$ is isomorphic to a semi-regular tiling of $mathbb{S}^2$. In the course of the proof of our main result, we present a combinatorial characterization in terms of an inequality of all the types of semi-equivelar maps on $mathbb{S}^2$. Here, we present self-contained combinatorial proofs of all our results.
Building on earlier results for regular maps and for orientably regular chiral maps, we classify the non-abelian finite simple groups arising as automorphism groups of maps in each of the 14 Graver-Watkins classes of edge-transitive maps.
In this paper, we investigate representations of links that are either centrally symmetric in $mathbb{R}^3$ or antipodally symmetric in $mathbb{S}^3$. By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors, we are able to present sufficient combinatorial conditions for a link $L$ to admit such representations. The latter naturally arises sufficient conditions for $L$ to be amphichiral. We also introduce another (closely related) method yielding again to sufficient conditions for $L$ to be amphichiral. We finally prove that a link $L$, associated to a map $G$, is amphichiral if the self-dual pairing of $G$ is not one of 6 specific ones among the classification of the 24 self-dual pairing $Cor(G) rhd Aut(G)$.
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