Random Knots, Random Groups and DNA

This talk will explore two possible connections between random topology and circular molecules such as plasmids of DNA. One connection is through the knot which the molecule forms in space. For the knot there are two basic types of randomness to consider. If crossings are unimportant there are random walk type models yielding distributions on knots. If crossings are difficult there are models for configurations within a knot type or random walks on the set of knot types. The second connection is through the space of possible chemical species. A circular plasmid can be viewed as a map from a circle to a tiling space of short linear sequences. If a plasmid is modified chemically by homologous recombination with some collection of short circular plasmids then the short plasmids give 2-cells that can be added to the tiling space and the modifications amount to homotopies between the associated maps of circles. Thus the process is a random walk on representatives of a fixed homotopy class in the fundamental group of the linear sequence tiling space with the short plasmid 2-cells. It might be interesting to apply similar analysis to other geometries of large molecules.

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