Mathematics Colloquia and Seminars

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Geometric knot theory is the study of geometric properties of space curves that derive from their topological knottedness. Perhaps the most famous result is the Fáry/Milnor theorem relating total curvature to bridge number, but the field can be dated back to work of Pannwitz on quadrisecant lines for knots. There has been a surge of interest in geometric knot theory over the past 20 years, partly due to biophysical applications to the shapes of knotted polymers like DNA molecules. One interesting problem with some surprising answers asks for the shapes of knots and links tied tight in rope of fixed thickness. We will survey recent results on this so-called ropelength problem, as well as some new strengthened versions of the theorems of Pannwitz and Fary/Milnor. We will also consider a question of Gromov on the distortion of knots.