Energetic reduction of elastic knots to normal form: Gradient descent along the Euler functional
Alexey Sossinsky, Independent University of Moscow, Laboratoire Poncelet
Simple physical experiments show that a deformed knotted resilient elastic wire always returns to its equilibrium shape (which we call the normal form of the wire knot). The aim of this research (joint work with S.Avvakumov and O.Karpenkov) is to construct a mathematical model of such physical knots. This is done by supplying the space of knots with an energy functional and performing gradient descent w.r.t. the functional. The energy functional that we use is the sum of the Euler functional (the integral along the curve defining the knot of the square of the curvature) and a simple repulsive functional (that forbids crossing changes). Using a discretized version of gradient descent along that functional, we construct an algorithm that brings the knot to the shape that minimizes its energy. The algorithm is implemented in a computer animation that shows the isotopy bringing the knot to normal form.
We will show some of these animations and demonstrate a few of the physical experiments with wire knots, compare the results of physical and mathematical experiments, and outline how our algorithm can be used in practice to recognize knots and to unravel trivial knots.