# Symmetries of Möbius Ladders

## Document Type

Article

## Department

Mathematics (Pomona)

## Publication Date

1989

## Keywords

Möbius ladders, symmetry

## Abstract

Chemistry has recently motivated the study of graphs embedded in **R**³, and of their symmetries as an extension of knot theory. We are interested in the following question: Given a graph *G* embedded in **R**³ or *S*³ = **R**³ ᴗ ∞, what can be said about its symmetries just from the topology of the graph itself? More precisely, we shall let Sym(*G*) denote the group of homeomorphisms of *G*, up to isotopy. If *G* is embedded in a manifold *M*, then Sym(*M, G*) is the group of diffeomorphisms of *M* which leave *G* invariant, up to isotopy respecting *G*. We are interested in the general question of how an element of Sym(*G*) can be represented by an element of Sym(*S³, G*), for some embedding of *G* in *S³*. Of course, not every graph *G* can be embedded in such a way that a given element of Sym(*G*) can be represented by some element of Sym(*S³*, *G*). In Sect. 1, we will provide an example of a graph *G* and a particular element of Sym(*G*) such that, no matter what the embedding of *G* in *S³*, that element cannot be represented by an element of Sym(*S³*, *G*).

## Rights Information

© 1989 Springer-Verlag

## Terms of Use & License Information

## DOI

10.1007/BF01446435

## Recommended Citation

E. Flapan, Symmetries of Möbius Ladders, Mathematische Annalen, 283, (1989) 271-283. doi: 0.1007/BF01446435