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CONFORMALLY INVARIANT ENERGIES OF KNOTS

Published online by Cambridge University Press:  08 March 2005

R. Langevin  and
J. O’Hara
R. Langevin
Affiliation:
Institut de Mathématiques de Bourgogne, CNRS-UMR 5584, Université de Bourgogne, France (remi.langevin@u-bourgogne.fr)
J. O’Hara
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Tokyo, Japan (ohara@comp.metro-u.ac.jp)
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Abstract

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Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of knots and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form, which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.

AMS 2000 Mathematics subject classification: Primary 57M25; 53A30. Secondary 57R17

Keywords

knotenergyconformalcross-ratioMöbius transformations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 4 , Issue 2 , April 2005 , pp. 219 - 280
Copyright
2005 Cambridge University Press