纽结与共形几何的能量ENERGY OF KNOTS AND CONFORMAL GEOMETRY

2003-12  

World Scientific Pub Co Inc  

O'Hara, Jun  

288  

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Energy of knots is a theory that was introduced to create a "canonical configuration" of a knot — 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 "canonical configuration" 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.

Part 1 In search of the "optimal embedding" of a knot Chapter 1 Introduction　　1.1 Motivational problem　　1.2 Notations and remarks　Chapter 2 a-energy functional E(a)　　2.1 Renormalizations of electrostatic energy of charged knots . . .　　2.2 Renormalizations of r-a-modified electrostatic energy, E(~) . .　　2.3 Asymptotic behavior of r-a energy of polygonal knots　　2.4 The self-repulsiveness of E(a)　Chapter 3 On E(2)　　3.1 Continuity　　3.2 Behavior of E(2) under "pull-tight" .　　3.3 M5bius invariance　　3.4 The cosine formula for E(2)　　3.5 Existence of E(2) minimizers　　3.6 Average crossing number and finiteness of knot types　　3.7 Gradient, regularity of E(2) minimizers, and criterion of criticality　　3.8 Unstable E(2)-critical torus knots　　3.9 Energy associated to a diagram 3.9.1 General framework 3.9.2 "X-energy" .　　3.10 Normal projection energies　　3.11 Generalization to higher dimensions　Chapter 4 LPnorm energy with higher index　　4.1 Definition of (a,p)-energy functional for knots ea'r'　　4.2 Control of knots by Ea'p (e(a'p)　　4.3 Complete system of admissible solid tori and finiteness of knot types　　4.4 Existence of Ea'P minimizers　　4.5 The circles minimize Ea'p　　4.6 Definition of a-energy polynomial for knots　　4.7 Brylinski's beta function for knots　　4.8 Other LV-norm energies　Chapter 5 Numerical experiments　　5.1 Numerical experiments on E(2)　　5.2 a > 2 cases. The limit as n-when a> 3　　5.3 Table of approximate minimum energies　Chapter 6 Stereo pictures of E(2) minimizers　Chapter 7 Energy of knots in a Riemannian manifold　　7.1 Definition of the unit density (a,p)-energy Eap　　7.2 Control of knots by Eap　　7.3 Existence of energy minimizers　　7.4 Examples : Energy of knots in Sa and Ha 7.4.1 Energy of circles in Sa 7.4.2 Energy of trefoils on Clifford tori in Sa 7.4.3 Existence of E('s22 minimizers 7.4.4 Energy of knots in Ha　　7.5 Other definitions　　7.6 The existence of energy minimizers　Chapter 8 Physical knot energies　　8.1 Thickness and ropelength　　8.2 Four thirds law　　8.3 Osculating circles and osculating spheres　　……Part 2 Energy of knots from a conformal geometric viewpointAppendix A Generalization of the Gauss formula for the linking numberAppendix B The 3-tulple map to the set of circles in S3Appendix C Conformal moduli of a solid torus Appendix D Kirchhoff elastica Appendix E Open problems and dreams Bibliography Index

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