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 271284) 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 selfrepulsiveness of E([symbol])  ch. 3. On E(2). 3.1. Continuity. 3.2 Behavior of E(2) under "pulltight"  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 4tuple map and the cross ratio of 4 points. 9.5. Pencils of spheres. 9.6. Modulus of an annulus. 9.7. Crossseparating annuli and the modulus of four points. 9.8. The measure on the space of spheres A. 9.9. Orientations of 2spheres  ch. 10. The space of nontrivial spheres of a knot. 10.1. Nontrivial spheres of a knot. 10.2. The 4tuple map for a knot. 10.3. Generalization of the 4tuple map to the diagonal. 10.4. Lower semicontinuity of the radii of nontrivial 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. Selfrepulsiveness 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 nontrivial spheres. 13.1. Nontrivial spheres, tangent spheres and twice tangent spheres. 13.2. The volume of the set of the nontrivial spheres. 13.3. The measure of nontrivial spheres in terms of the infinitesimal cross ratio. 13.4. Nontrivial annuli and the modulus of a knot. 13.5. Selfrepulsiveness of the measure of nontrivial spheres. 13.6. The measure of nontrivial spheres for nontrivial knots. 13.7. Measure of nontrivial 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 
