Energy of knots and the infinitesimal cross ratio

TL;DR

This survey explores the energy of knots, its invariance under Moebius transformations, and its expression via the infinitesimal cross ratio, linking knot theory with conformal geometry.

Contribution

It provides a comprehensive overview of the energy of knots, its invariance properties, and its relation to the infinitesimal cross ratio, offering new geometric interpretations.

Findings

Energy of knots is invariant under Moebius transformations

Energy can be expressed using the infinitesimal cross ratio

Provides interpretations of the real part of 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.