Chirality
by
Marston Conder
University of Auckland

Chirality, or handedness, is an interesting property in many branches of science and medicine. Roughly speaking, an object is chiral if it is non-isomorphic to its mirror image. For example, the left and right trefoils are chiral knots (with the same Alexander polynomial but different Jones polynomials). Remarkably, when a discrete object is assumed to have a large degree of rotational symmetry, it often happens that it possesses also reflectional symmetry, so that chirality is not the norm. Instances occur in the study of compact Riemann surfaces with large automorphism groups, regular maps on surfaces (generalising the platonic solids), and higher-dimensional polytopes. This talk will look at the notion of chirality in contexts like these, and describe some recent results (including the discovery of the first known finite chiral 5-polytopes with maximum rotational symmetry, and an infinite sequence of gaps in the spectrum of orientably-regular but chiral maps) obtained in joint work with a number of co-authors.

Date received: October 14, 2007