Contents -- Preface -- 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 Ea -- 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 Mobius 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 Lp norm energy with higher index -- 4.1 Definition of (a p)-energy functional for knots eap -- 4.2 Control of knots by Eap (eap) -- 4.3 Complete system of admissible solid tori and finiteness of knot types -- 4.4 Existence of Eap minimizers -- 4.5 The circles minimize Eap -- 4.6 Definition of a-energy polynomial for knots -- 4.7 Brylinski's beta function for knots -- 4.8 Other Lp-norm energies -- Chapter 5 Numerical experiments -- 5.1 Numerical experiments on E(2) -- 5.2 a > 2 cases. The limit as n -> oo 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 EapM -- 7.2 Control of knots by EapM -- 7.3 Existence of energy minimizers -- 7.4 Examples : Energy of knots in S3 and H3 -- 7.4.1 Energy of circles in S3.

7.4.2 Energy of trefoils on Clifford tori in S3 -- 7.4.3 Existence of E(2)S(3) minimizers -- 7.4.4 Energy of knots in H3 -- 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 -- 8.4 Global radius of curvature -- 8.5 Self distance type energies denned via the distance function -- 8.6 Relation between these geometric quantities and eap -- 8.7 Numerical computations and applications -- Part 2 Energy of knots from a conformal geometric viewpoint -- Chapter 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 -- Chapter 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 -- 10.3.1 Twice tangent spheres -- 10.3.2 Tangent spheres -- 10.3.3 Osculating spheres -- 10.4 Lower semi-continuity of the radii of non-trivial spheres -- Chapter 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 Eo(2)-energy for links -- Chapter 12 The conformal sin energy Esin0 -- 12.1 The projection of the inverted open knot -- 12.2 The geometric meaning of Esin0 -- 12.3 Self-repulsiveness of Esin0.

12.4 Esin0 and the average crossing number -- 12.5 Esin0 for links -- Chapter 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 infinitesimalcross 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 -- Appendix A Generalization of the Gauss formula for the linking number -- A.l The Gauss formula for the linking number -- A.2 The writhe and the self-linking number -- A.3 The total twist -- A.4 Average crossing number -- A.5 The conformal angle and the Gauss integral -- A.6 Mobius invariance of the writhe -- A.7 The circular Gauss map and the inverted open knots -- Appendix B The 3-tuple map to the set of circles in S3 -- B.l The set of unoriented circles in S3 -- B.2 The 3-tuple map to the set of circles -- Appendix C Conformal moduli of a solid torus -- C.l Modulus of a 2-dimensional annulus revisited -- C.2 Conformal moduli of a solid torus -- Appendix D Kirchhoff elastica -- Appendix E Open problems and dreams -- Bibliography -- Index.