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EBOOK
Author O'Hara, Jun.
Title Energy of knots and conformal geometry / Jun O'Hara.
Imprint River Edge, NJ : World Scientific, [2003]
©2003

Series K & E series on knots and everything ; v. 33
K & E series on knots and everything ; v. 33.
Subject Knot theory.
Conformal geometry.
Series K & E series on knots and everything ; v. 33
K & E series on knots and everything ; v. 33.
Subject Knot theory.
Conformal geometry.
Description 1 online resource (xiv, 288 pages) : illustrations.
Bibliography Note Includes bibliographical references (pages 271-284) and index.
Contents pt. 1. In search of the "optimal embedding" of a knot. ch. 1. Introduction. 1.1. Motivational problem. 1.2. Notations and remarks -- ch. 2. [symbol]-energy functional E([symbol]). 2.1. Renormalizations of electrostatic energy of charged knots. 2.2. Renormalizations of r[symbol] -modified electrostatic energy, E([symbol]). 2.3. Asymptotic behavior of r-[symbol] energy of polygonal knots. 2.4. The self-repulsiveness of E([symbol]) -- ch. 3. On E(2). 3.1. Continuity. 3.2 Behavior of E(2) under "pull-tight" -- 3.3. Möbius 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. 3.10. Normal projection energies. 3.11. Generalization to higher dimensions -- ch. 4. L[symbol] norm energy with higher index. 4.1. Definition of ([symbol], p)-energy functional for knots e[symbol, p]. 4.2. Control of knots by E[symbol, p] (e[symbol, p]). 4.3. Complete system of admissible solid tori and finiteness of knot types. 4.4. Existence of E[symbol, p] minimizers. 4.5. The circles minimize E[symbol, p]. 4.6. Definition of [symbol]-energy polynomial for knots. 4.7. Brylinski's beta function for knots. 4.8. Other L[symbol]-norm energies -- ch. 5. Numerical experiments. 5.1. Numerical experiments on E(2). 5.2. [symbol]> 2 cases. The limit as n [symbol][symbol] when [symbol][symbol] 3. 5.3. Table of approximate minimum energies -- ch. 6. Stereo pictures of E(2) minimizers -- ch. 7. Energy of knots in a Riemannian manifold. 7.1. Definition of the unit density ([symbol], p)-energy E[symbol][symbol]. 7.2. Control of knots by E[symbol][symbol]. 7.3. Existence of energy minimizers. 7.4. Examples : energy of knots in S3 and H3. 7.5. Other definitions. 7.6. The existence of energy minimizers -- ch. 8. Physical knot energies. 8.1. Thickness and ropelength. 8.2. Four thirds law. 8.3. Osculating circles and osculating spheres. 8.4. Global radius of curvature. 8.5. Self distance type energies defined via the distance function. 8.6. Relation between these geometric quantities and e[symbol][symbol]. 8.7. Numerical computations and applications.
pt. 2. Energy of knots from a conformal geometric viewpoint. ch. 9. Preparation from conformal geometry. 9.1. The Lorentzian metric on Minkowski space. 9.2. The Lorentzian exterior product. 9.3. The space of spheres. 9.4. The 4-tuple map and the cross ratio of 4 points. 9.5. Pencils of spheres. 9.6. Modulus of an annulus. 9.7. Cross-separating annuli and the modulus of four points. 9.8. The measure on the space of spheres A. 9.9. Orientations of 2-spheres -- ch. 10. The space of non-trivial spheres of a knot. 10.1. Non-trivial spheres of a knot. 10.2. The 4-tuple map for a knot. 10.3. Generalization of the 4-tuple map to the diagonal. 10.4. Lower semi-continuity of the radii of non-trivial spheres -- ch. 11. The infinitesimal cross ratio. 11.1. The infinitesimal cross ratio of the complex plane. 11.2. The real part as the canonical symplectic form of T*S2. 11.3. The infinitesimal cross ratio for a knot. 11.4. From the cosine formula to the original definition of E(2). 11.5.E[symbol]-energy for links -- ch. 12. The conformal sin energy E[symbol][symbol]. 12.1. The projection of the inverted open knot. 12.2. The geometric meaning of E[symbol][symbol]. 12.3. Self-repulsiveness of E[symbol][symbol]. 12.4. E[symbol][symbol] and the average crossing number. 12.5. E[symbol][symbol] for links -- ch. 13. Measure of non-trivial spheres. 13.1. Non-trivial spheres, tangent spheres and twice tangent spheres. 13.2. The volume of the set of the non-trivial spheres. 13.3. The measure of non-trivial spheres in terms of the infinitesimal cross ratio. 13.4. Non-trivial annuli and the modulus of a knot. 13.5. Self-repulsiveness of the measure of non-trivial spheres. 13.6. The measure of non-trivial spheres for non-trivial knots. 13.7. Measure of non-trivial spheres for links.
Summary 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 problem 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.
Note Print version record.
ISBN 9789812795304 (electronic bk.)
9812795308 (electronic bk.)
9812383166 (alk. paper)
OCLC # 268676330
Additional Format Print version: O'Hara, Jun. Energy of knots and conformal geometry. River Edge, NJ : World Scientific, ©2003 9812383166 9789812383167 (DLC) 2003041104 (OCoLC)51478044