Knots are self-entangled filaments. In mathematics, they have been most commonly studied as purely topological objects. But physical knots are knots made of real physical stuff, from rope to DNA or other large flexible molecules; or purely mathematical knots endowed with physical-like properties such as energy or thickness. The goals of physical knot theory are to mathematically model and help understand real physical systems, and to use physically inspired measures of knot complexity to develop novel methods for knot recognition/classification and gain deeper understanding of configuration spaces of knots. The investigator also casts a wider net, looking for connections between physical knot theory and polymer theory, as well as analogies between physical knot theory and other important optimization and configuration problems such as protein folding.
Modern laboratory technology lets scientists see the tiniest structures of life. Modern computers let researchers visualize and simulate how the structures interact. This combined technology has helped make geometry and topology an essential tool in medicine, polymers, and other areas of chemistry, physics, and biology. Knotting, or other tangling of filaments, is one of the fundamental ways that matter behaves, and is a key phenomenon in this scientific interaction. Knotting and tangling happen at every scale studied by science, from microscopic DNA loops to everyday rope to tangled magnetic field loops in the solar corona. The investigator, collaborators, and students study fundamental problems that arise in all these physical systems: How are knots and tangles created? What properties of the various systems cause essentially different kinds of knots and tangles? Can the structure be simplified or completely untangled? If so, how? How do the mathematical properties of different kinds of knots or tangles influence physical behavior? The project contributes to the effort of finding good data structures and good manipulation and visualization tools for topological and geometric objects -- essential tools for work in the realm of the very small. These efforts are at the interface of information technology, nanotechnology, and biotechnology. In materials and manufacturing, as well as in biotechnology, the work could have considerable impact by providing effective models of the topological and geometric behavior of polymers in general, and DNA in the specific.
