Symmetries of Möbius ladders
This paper considers the relationship between the automorphisms of a graph and the symmetries of that graph when it is embedded in S 3 or \({\mathbb{R}}^ 3.\) More precisely, the following questions are addressed: Suppose \(\Phi\) is a homeomorphism of a graph G, is there an embedding of G in S 3 or \({\mathbb{R}}^ 3 \)such that \(\Phi\) is induced by some diffeomorphism h of S 3 or \({\mathbb{R}}\) 3? If h is also required to be orientation reversing, is there still such an embedding? Finally if, in addition, \(\Phi\) is of finite order, will there be an h of the same finite order? At the beginning of the paper an example is presented of a graph G with a homeomorphism \(\Phi\) such that for any embedding of G in S 3 there is no diffeomorphism of S 3 which induces \(\Phi\) on G. After this example, the above questions are answered in detail for a class of graphs which has arisen in chemistry, the Möbius ladders. A Möbius ladder \(M_ n\) is a simple closed curve K together with n segments joining n pairs of antipodal points of K. In particular, it is shown that for n odd there is no embedding of \(M_ n\) in S 3 which is invariant under an orientation reversing diffeomorphism of S 3, whereas for n even there is always such an embedding. The paper concludes with some observations about the symmetries of a slightly larger class of graphs which includes the Möbius ladders.
scientific article; zbMATH DE number 4046707
scientific article