Volume 13 (2008)

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Energy of knots and the infinitesimal cross ratio

Jun O'Hara

Geometry & Topology Monographs 13 (2008) 421–445

DOI: 10.2140/gtm.2008.13.421

arXiv: 0708.2948

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 Möbius 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.

Keywords

energy, knot, conformal geometry, cross ratio

Mathematical Subject Classification

Primary: 57M25

Secondary: 53A30

References
Publication

Received: 30 May 2006
Revised: 9 May 2007
Accepted: 16 May 2007
Published: 19 March 2008

Authors
Jun O'Hara
Department of Mathematics
Tokyo Metropolitan University
Tokyo
Japan