Energy of knots and the infinitesimal cross ratio

Abstract

This is a survey article on two topics. The Energy E of knots can be obtained by generalizing an electrostatic energy of charged knots in order to produce optimal knots. It turns out to be invariant under Moebius transformations. We show that it can be expressed in terms of the infinitesimal cross ratio, which is a conformal invariant of a pair of 1-jets, and give two kinds of interpretations of the real part of the infinitesimal cross ratio.