On the symmetric molecular conjectures

Abstract

A molecular linkage consists of a set of rigid bodies pairwise connected by revolute hinges, so that all hinge lines of each body are concurrent in a single point. It is an important problem in biochemistry, as well as in robotics, to efficiently analyze the motions and configuration spaces of such linkages. The well-developed mathematical theory of generic rigidity of body-bar frameworks permits a rigidity and flexibility analysis of molecular linkages via fast combinatorial algorithms. However, recent work in rigidity theory has shown that symmetry (a common feature of many molecular and mechanical linkages) can lead to additional motions in body-bar and molecular frameworks. These motions typically maintain the original symmetry of the structure throughout the path, and they can often be detected via simple combinatorial counts. In this paper, we outline how these symmetry-based mathematical counts and methods can be used to efficiently predict the motions of symmetric molecular linkages, and we provide numerical companion configuration spaces supporting the symmetric Molecular Conjectures in rigidity theory.