6 - Symmetries of Embedded Graphs

Summary

In this chapter we will discuss different types of symmetries of embedded graphs. Results about symmetries will provide an easy method of proving that certain molecular graphs are intrinsically chiral. Specifically, by a symmetry of an embedded graph we shall mean a homeomorphism of S³ or ℝ³ that takes the graph to itself. In Chapter 4, an embedded graph was defined to be rigidly achiral in S³ if there is an orientation-reversing homeomorphism h : (S³, G) → (S³, G) of finite order. By analogy with this definition, we shall define a rigid symmetry of an embedded graph G as any finite-order homeomorphism h:(S³, G) → (S³, G). It is important to distinguish the concept of a rigid symmetry from that of a physically rigid motion of space. A physically rigid graph may have rotational symmetries, planar reflections, or combinations of these two types of symmetries, but no other types of symmetries. Rigid symmetries include these three types of rigid motions as well as other finiteorder homeomorphisms that are not rigid motions. For example, Figure 6.1 illustrates an order-three homeomorphism that first deforms a trefoil to a symmetric position, then rotates it by 120°, and then deforms it back to its original shape.

The concept of a symmetry of an embedded graph should also not be confused with that of an automorphism of an abstract graph, which is a map of the graph to itself, independent of any particular embedding of the graph in space.