Rigid and nonrigid achirality

Abstract

In order to completely characterize a molecule it is useful to understand the symmetries of its molecular bond graph in 3-space.For many purposes the most important type of symmetry that a molecule can exhibit is mirror image symmetry.However, the question of whether a molecular graph is equivalent to its mirror image has different interpretations depending on what assumptions are made about the rigidity of the molecular structure.If there is a deformation of 3-space taking a molecular bond graph to its mirror image then the molecule is said to be topologically achiral.If a molecular graph can be embedded in 3-space in such a way that it can be rotated to its mirror image, then the molecule is said to be rigidly achiral.We use knot theory in R 3 to produce hypothetical knotted molecular graphs which are topologically achiral but not rigidly achiral, this answers a question which was originally raised by a chemist.A molecular bond graph is a graph in R 3 which is a geometric model of the structure of a molecule, see [Walb] and [Was].We will be working primarily with molecular bond graphs which consist only of a simple closed curve K in R 3 .Since we are addressing a question raised by chemists and are working in R 3 , we choose to use the term " achiral" from the chemical literature rather than using the corresponding mathematical term "amphicheiral" which is generally used for knots in S 3 .It is not hard to show that K is topologically achiral if and only if there is an orientation reversing diffeomorphism of R 3 leaving K setwise invariant.If K is rigidly achiral then there is some embedding of K in 3-space which can be rotated to its mirror image.This embedding is said to be a symmetry presentation for K. Let h be this rotation composed with a reflection so that h(K) = K.Since K is only supposed to be a model of reality we shall make the assumption that this rotation is through a rational angle.Hence h must be of finite order.On the other hand, any finite order diffeomorphism of (R 3 , K) is conjugate to a rotation composed with a reflection.Thus K is rigidly achiral if and only if there is a finite order orientation reversing diffeomorphism of (R 3 , K).By giving K an orientation we can distinguish further between two types of topological achirality.If there is a diffeomorphism of (R 3 , K) which reverses the orientations of both R 3 and K, then K is said to be negative achiral.Whereas, if there is a diffeomorphism of R 3 and K which 57

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF