Energy of Knots and Conformal Geometry

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World Scientific, 2003 - Mathematics - 306 pages
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Energy of knots is a theory that was introduced to create a OC canonical configurationOCO of a knot OCo a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a OC canonical configurationOCO of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments. Contents: In Search of the OC Optimal EmbeddingOCO of a Knot: -Energy Functional E (); On E (2); L p Norm Energy with Higher Index; Numerical Experiments; Stereo Pictures of E (2) Minimizers; Energy of Knots in a Riemannian Manifold; Physical Knot Energies; Energy of Knots from a Conformal Geometric Viewpoint: Preparation from Conformal Geometry; The Space of Non-Trivial Spheres of a Knot; The Infinitesimal Cross Ratio; The Conformal Sin Energy E sin (c) Measure of Non-Trivial Spheres; Appendices: Generalization of the Gauss Formula for the Linking Number; The 3-Tuple Map to the Set of Circles in S 3; Conformal Moduli of a Solid Torus; Kirchhoff Elastica; Open Problems and Dreams. Readership: Graduate students and researchers in geometry & topology and numerical & computational mathematics."
 

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Contents

Introduction
3
On E2
31
L norm energy with higher index
63
Numerical experiments
83
Stereo pictures of E2 minimi2ers
91
Energy of knots in a Riemannian manifold
101
Physical knot energies
117
Preparation from conformal geometry
135
The conformal sin energy Ens
213
Measure of nontrivial spheres
223
Appendix A Generali2ation of the Gauss formula for the link
243
Average crossing number
249
Appendix B The 3tuple map to the set of circles in S2
257
Kirchhoff elastica
263
Bibliography
271
Index
285

The space of nontrivial spheres of a knot
175
The infinitesimal cross ratio
189

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