Séminaire de Géométrie Tropicale

Institut Mathématiques de Jussieu ,
Université Pierre et Marie Curie, Paris 6
UMR CNRS 7586



1er Février 2012, 11h00 salle 1525-502



Alexei Sossinski (Independent University of Moscow, Laboratoire Poncelet (CNRS-UIM))

Résumé : Knot energy and flat normal forms of knots
A new approach to knot energy, motivated by a series of physical experiments with a resilient wire contour, will be presented. Amazingly, all the experiments produced the same normal form (corresponding to the energy minimum) for any knot from a given isotopy class with seven crossings or less. Thus our little wire contour outperforms the classical knot energies (e.g. Moebius energy) studied by Fukuhara, O'Hara, Freedman and others. Some experiments will be demonstrated and many of their results (often unexpected) will be described.
Then the first steps of the mathematical modeling of the behavior of the wire contour will be sketched (this is joint work with Oleg Karpenkov). They involve some variational calculus, elliptic functions, combinatorics of knot diagrams, and the phase space of the pendulum. We will also discuss so-called flat knots (mathematical counterparts of a resilient wire contour squeezed between two parallel planes), their normal forms and their relationship with classical (3D) knots.
The talk will be accessible to mathematicians and mathematical physicists without any knowledge of knot theory: the very few basic facts from that theory needed in the exposition will be explained. A number of new problems (not involving special knowledge of knot theory, but possibly necessitating the calculus of variations, geometric combinatorics, differential geometry, and numerical simulations) will be formulated.