Energy Functions for Knots

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9407132 Simon The investigator and his colleague undertake an extensive computational and analytical study of the energy surfaces given by the energy functions on finite-dimensional conformation spaces of knots. The focus is on two conformation spaces: polygonal knots, and harmonic knots, where harmonic knots are those given by finite Fourier series parametrizations. Recent theorems relate the invariants given by these energy functions to classical invariants such as crossing number. So this project should lead to a better understanding of knots realizable with relatively low degrees of freedom. Knot theory is the mathematical study of the everyday objects called knots. Once thought the purview of theoretical mathematicians, knot theory has recently found a wide range of applications, in fields as diverse as molecular biology and cosmology. For example, DNA molecules are usually knotted, and in the replication process must somehow become untangled. In this project the investigators apply sophisticated computing techniques to the study of knots. In particular, they continue the development of a new approach to knots: physical knot theory. The mathematical knot is given physical properties, and knot types are distinguished by computations involving these properties. For example, knot conformations can be assigned an energy, and then knot types can be distinguished by considering minimum energy conformations. The conceptual basis of the approach is familiar to researchers in biology, chemistry, and physics as well as mathematics. The project produces both theoretical results and software.