Symmetries of Möbius ladders
type: Item
instance of: scholarly article
title: Symmetries of Möbius ladders
zbMATH Open document ID: 0642.57008
DOI: 10.1007/BF01446435
author: Erica Flapan
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164508
review text: 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.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 4046707
zbMATH Keywords: automorphisms of a graph
zbMATH Keywords: Möbius ladders
zbMATH Keywords: topological chemistry
zbMATH Keywords: symmetries of embedded graphs
Wikidata QID: Q29013921
MaRDI profile type: Publication
cites work: Knots and links in spatial graphs
cites work: On the Möbius Ladders
cites work: Topology of three dimensional manifolds and the embedding problems in minimal surface theory
cites work: Q4107635
cites work: Topological chirality of certain molecules
cites work: Transformations of finite period. II
Recommended article: Topological symmetry groups of Möbius ladders
Recommended article: The Symmetry Principle for Mobius Transformations
Recommended article: Mobius Equivalence and Euclidean Symmetry
Recommended article: Symmetries of double ratios and an equation for Möbius structures
Recommended article: Inner Symmetries for Moebius Maps
Recommended article: Q3810563
Recommended article: Symmetric ladders
Recommended article: Q4027268
Recommended article: Symmetric ladders and G-biliaison
Recommended article: Counting structures in the Möbius ladder
description: scientific article; zbMATH DE number 4046707