RUI: Characterizing Energy-Minimizing Knots

Duquesne University, Pittsburgh PA

Investigators

Abstract

Rawdon The investigator, collaborators, and students study different knot energy functions and their optimal configurations. Specific topics include: characterizing the shape of ropelength minima; finding relationships between the optima of different energy functions; studying polygonal energy optima as a function of the number of edges; determining the role of flexibility in pulling knots tight and in packing them efficiently; exploring the structure of polygonal knot space near ropelength minima; determining relationships between different notions of polygonal ropelength. From the massive knots of cosmic strings to the microscopic tangling of DNA, knotting and tangling are a part of nature. One physical attribute of a type of knot is its minimum ropelength (i.e. the amount of one-inch radius perfectly flexible rope needed to tie a configuration of the given type of knot). The idea of ropelength has been used by biologists, physicists, and mathematicians to predict the packing of helical DNA, the speed at which knotted DNA moves through gel, and the average shape of a knot within a large population of different configurations. What is so special about ropelength-minimizing conformations that they predict the behavior of real physical knots? What spatial attributes influence this behavior? Furthermore, real physical materials have inherent limitations, such as resistance to bending. How does one incorporate inflexibility into mathematical models of macromolecules? How does the flexibility of the material affect how the knot tightens and how long lengths pack into small spaces? In particular, can this knowledge be used to design knot-making materials that are both strong and resistant to slipping? The investigator, collaborators, and students study these fundamental questions about knotting and tangling in real physical systems. These studies lead to a deeper understanding of tangled structures present in nanotechnology and biotechnology and to better models of knots made with real physical materials.