Séminaire de Mathématique

Normal Forms of Knots and Plane Curves

by Prof. Alexey SOSSINSKY (Independent University of Moscow & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

The talk summarizes my work on the classification of knots and closed plane curves, most of which was done jointly with O.Karpenkov and S.Avvakumov. The main idea is to supply the knot (or the closed plane curve) with an energy functional and to classify these objects via their {\it normal forms}, which we define as the shapes of these objects that minimize the functional.


For any closed $\mathcal C^2$ curve $\gamma$, the functional that we choose is the {\it Euler functional} $E(\gamma)$ equal to the integral along $\gamma$ of the squared curvature of $\gamma$. We prove (using some fairly sophisticated methods of the calculus of variations) that:


(1) the critical points of $E$ are given by circles passed once or several times and by $\infty$-shaped curves passed once or several times.


(2) the minima of $E$ are given by $\infty$-shaped curves passed once and by circles passed once or several times. }


This solves a (long forgotten) problem set by Euler in 1774. The same result was obtained at about the same time (2012) by Yu.Sachkov by completely different methods. The result also gives a new proof of the famous Whitney--Graustein theorem on the classification of plane curves. It will be illustrated in the talk by an animation that shows in real time how a plane curve is homotoped to its normal form (by gradient descent along the functional).


For knots $k:S^1 \to R^3$, we use a functional $F$ equal to the sum of the Euler functional $E(k)$ and a simple {\it repulsive functional} $R(k)$ (the latter prevents self-intersections of $k$). We construct an algorithm (implemented in an animation that will be shown in the talk) which yields the isotopy of a given knot to its normal form (corresponding to the minimum of $F$) via a discretized version of gradient descent along $F$.


We then discuss to what extent this algorithm gives a practical solution of the knot classification problem and compare our theoretical normal forms with those obtained by physical experiments (which will be shown during the talk) with models of knots made out of flexible wire.