This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This project has two goals, both concerning graphs embedded in 3-space. The topological symmetry group of a graph embedded in 3-space is defined as the group of automorphisms of the graph which are induced by homeomorphisms of 3-space. Our first goal is to characterize which groups can occur as the topological symmetry group of a particular embedded graph or family of graphs. The second goal of our project concerns intrinsic properties of a graph -- properties of an embedded graph which do not depend on the particular embedding. For example, a graph which has the property that every embedding of it in 3-space contains a nontrivial link is said to be intrinsically linked, while one which has the property that every embedding of it contains a nontrivial knot is said to be intrinsically knotted. We have previously shown that for any natural number n there is a graph such that every embedding of it contains a link with n components such that every pair of components has linking number at least n, and every component is a knot with minimal crossing number at least n. However, there cannot exist a graph which has the property that every embedding of it contains at least one composite knot. We would now like to study for which measures of complexity of knots and links there are graphs which are arbitrarily intrinsically complex by that measure.
A molecule can be represented as a graph in 3-dimensional space. Most molecules are rigid, and the geometry of their graphs determines many of their properties. However, some molecules can rotate around particular bonds, and others are large enough to be somewhat flexible. For these non-rigid molecules, their topology is important in predicting their behavior. The study of the topology of graphs embedded in 3-dimensional space is used in determining the symmetries of non-rigid molecules. In particular, while the group of rigid symmetries of a molecule (known as the point group) is useful for analyzing the symmetries of rigid molecules, a molecular structure which is not rigid may have symmetries which are not included in the point group. The topological symmetry group was created in order to classify the symmetries of non-rigid molecules. The main goal of our project is to characterize all topological symmetry groups. Understanding molecular symmetries has many important application in chemistry. Symmetry is used in interpreting results in crystallography, spectroscopy, and quantum chemistry, as well as in analyzing the electron structure of a molecule. Symmetry is also used in pharmacology and in designing new molecules and new types of reactions.
