Geometric Knot Theory

Abstract

Geometric knot theory studies relations between geometric properties of a space curve and the knot type it represents. As examples, knotted curves have quadrisecant lines, and have more distortion and more total curvature than (some) unknotted curves. Geometric energies for space curves – like the Möbius energy, ropelength and various regularizations – can be minimized within a given knot type to give an optimal shape for the knot. Increasing interest in this area over the past decade is partly due to various applications, for instance to random knots and polymers, to topological fluid dynamics and to the molecular biology of DNA. This workshop focused on the mathematics behind these applications, drawing on techniques from algebraic topology, differential geometry, integral geometry, geometric measure theory, calculus of variations, nonlinear optimization and harmonic analysis.