Mathematical Approaches to Biomolecular Structure and Dynamics

Volume 82 of the series The IMA Volumes in Mathematics and its Applications pp 39-58

Energy Functions for Knots: Beginning to Predict Physical Behavior

  • Jonathan SimonAffiliated withDepartment of Mathematics, University of Iowa

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Several definitions have been proposed for the “energy” of a knot. The intuitive goal is to define a number u(K) that somehow measures how “tangled” or “crumpled” a knot K is. Typically, one starts with the idea that a small piece of the knot somehow repels other pieces, and then adds up the contributions from all the pieces. From a purely mathematical standpoint, one may hope to define new knot-type invariants, e.g by considering the minimum of u(K) as K ranges over all the knots of a given knot-type. We also are motivated by the desire to understand and predict how knot-type affects the behavior of physically real knots, in particular DNA loops in gel electrophoresis or random knotting experiments. Despite the physical naiveté of recently studied knot energies, there now is enough laboratory data on relative gel velocity, along with computer calculations of idealized knot energies, to justify the assertion that knot energies can predict relative knot behavior in physical systems. The relationships between random knot frequencies and either gel velocities or knot energies is less clear at this time.