Symmetries of Möbius Ladders
Möbius ladders, symmetry
Chemistry has recently motivated the study of graphs embedded in R³, and of their symmetries as an extension of knot theory. We are interested in the following question: Given a graph G embedded in R³ or S³ = R³ 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).
© 1989 Springer-Verlag
E. Flapan, Symmetries of Möbius Ladders, Mathematische Annalen, 283, (1989) 271-283. doi: 0.1007/BF01446435