RUI: Characterizing Energy-Minimizing Knots

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Rawdon The investigator, collaborators, and students studydifferent knot energy functions and their optimal configurations.Specific topics include: characterizing the shape of ropelengthminima; finding relationships between the optima of differentenergy functions; studying polygonal energy optima as a functionof the number of edges; determining the role of flexibility inpulling knots tight and in packing them efficiently; exploringthe structure of polygonal knot space near ropelength minima;determining relationships between different notions of polygonalropelength. From the massive knots of cosmic strings to the microscopictangling of DNA, knotting and tangling are a part of nature. Onephysical attribute of a type of knot is its minimum ropelength(i.e. the amount of one-inch radius perfectly flexible ropeneeded to tie a configuration of the given type of knot). Theidea of ropelength has been used by biologists, physicists, andmathematicians to predict the packing of helical DNA, the speedat which knotted DNA moves through gel, and the average shape ofa knot within a large population of different configurations.What is so special about ropelength-minimizing conformations thatthey predict the behavior of real physical knots? What spatialattributes influence this behavior? Furthermore, real physicalmaterials have inherent limitations, such as resistance tobending. How does one incorporate inflexibility intomathematical models of macromolecules? How does the flexibilityof the material affect how the knot tightens and how long lengthspack into small spaces? In particular, can this knowledge beused to design knot-making materials that are both strong andresistant to slipping? The investigator, collaborators, andstudents study these fundamental questions about knotting andtangling in real physical systems. These studies lead to adeeper understanding of tangled structures present innanotechnology and biotechnology and to better models of knotsmade with real physical materials.