## Oliver Club

Chemists have defined the point group of a molecule as the group of rigid

symmetries of its molecular graph in $ {\mathbb R}^3 $. While this group is useful for

analyzing the symmetries of rigid molecules, it does not include all of the

symmetries of molecules which are flexible or can rotate around one or more

bonds. To study the symmetries of such molecules, we define the topological

symmetry group of a graph embedded in $ {\mathbb R}^3 $ to be the subgroup of the auto-

morphism group of the abstract graph that is induced by homeomorphisms

of $ {\mathbb R}^3 $. This group gives us a way to understand not only the symmetries

of non-rigid molecular graphs, but the symmetries of any graph embedded

in $ {\mathbb R}^3 $. The study of such symmetries is a natural extension of the study of

symmetries of knots. In this talk we will present a survey of results about

the topological symmetry group and how it can play a role in analyzing the

symmetries of non-rigid molecules.