Mathematics Colloquia and Seminars

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Experiments on knotted DNA molecules suggest that certain physical properties of DNA knots can be predicted from a corresponding ideal shape. Intuitively, when a given knot in a piece of string is pulled tight, it always achieves roughly the same geometrical configuration, with a minimum length of string within the knot. Such a configuration is called an ideal shape for the knot, and approximations of ideal shapes in this sense have been found via a series of computer experiments. These shapes have intriguing physical features and have been shown to capture average properties of knotted polymers. But when does a shape satisfy the intuitive geometrical definition for ideality? In this talk I show that ideal shapes can be understood using only elementary (but new!) mathematics. In particular, I show that global curvature, a very natural and simple generalization of the classic concept of local curvature, leads to a simple characterization of an ideal shape and to a necessary condition for ideality. Another application of global curvature can be found in characterizing the equilibria of knotted curves or rods, which may exhibit self-contact after sufficient twisting. Here global curvature provides a simple way to formulate the constraint that prevents a rod from passing through itself.