| In search of the "optimal embedding" of a knot | p. 1 |
| Introduction | p. 3 |
| Motivational problem | p. 3 |
| Notations and remarks | p. 8 |
| [alpha]-energy functional E[superscript (alpha)] | p. 9 |
| Renormalizations of electrostatic energy of charged knots | p. 9 |
| Renormalizations of r[superscript -alpha]-modified electrostatic energy, E[superscript (alpha)] | p. 15 |
| Asymptotic behavior of r[superscript -alpha] energy of polygonal knots | p. 22 |
| The self-repulsiveness of E[superscript (alpha)] | p. 26 |
| On E[superscript (2)] | p. 31 |
| Continuity | p. 31 |
| Behavior of E[superscript (2)] under "pull-tight" | p. 34 |
| Mobius invariance | p. 34 |
| The cosine formula for E[superscript (2)] | p. 35 |
| Existence of E[superscript (2)] minimizers | p. 42 |
| Average crossing number and finiteness of knot types | p. 44 |
| Gradient, regularity of E[superscript (2)] minimizers, and criterion of criticality | p. 45 |
| Unstable E[superscript (2)]-critical torus knots | p. 54 |
| Energy associated to a diagram | p. 57 |
| General framework | p. 58 |
| "X-energy" | p. 59 |
| Normal projection energies | p. 60 |
| Generalization to higher dimensions | p. 60 |
| L[superscript p] norm energy with higher index | p. 63 |
| Definition of ([alpha], p)-energy functional for knots e[superscript alpha,p] | p. 63 |
| Control of knots by E[superscript alpha,p] (e[superscript alpha,p]) | p. 65 |
| Complete system of admissible solid tori and finiteness of knot types | p. 75 |
| Existence of E[superscript alpha,p] minimizers | p. 77 |
| The circles minimize E[superscript alpha,p] | p. 78 |
| Definition of [alpha]-energy polynomial for knots | p. 79 |
| Brylinski's beta function for knots | p. 79 |
| Other L[superscript p]-norm energies | p. 81 |
| Numerical experiments | p. 83 |
| Numerical experiments on E[superscript (2)] | p. 84 |
| [alpha] [greater than sign] 2 cases. The limit as n to [infinity] when [alpha] [greater than or equal] 3 | p. 84 |
| Table of approximate minimum energies | p. 86 |
| Stereo pictures of E[superscript (2)] minimizers | p. 91 |
| Energy of knots in a Riemannian manifold | p. 101 |
| Definition of the unit density ([alpha],p)-energy E[superscript alpha,p subscript M] | p. 101 |
| Control of knots by E[superscript alpha,p subscript M] | p. 102 |
| Existence of energy minimizers | p. 105 |
| Examples: Energy of knots in S[superscript 3] and H[superscript 3] | p. 107 |
| Energy of circles in S[superscript 3] | p. 107 |
| Energy of trefoils on Clifford tori in S[superscript 3] | p. 109 |
| Existence of E[superscript (2) subscript S superscript 3] minimizers | p. 109 |
| Energy of knots in H[superscript 3] | p. 110 |
| Other definitions | p. 114 |
| The existence of energy minimizers | p. 115 |
| Physical knot energies | p. 117 |
| Thickness and ropelength | p. 117 |
| Four thirds law | p. 118 |
| Osculating circles and osculating spheres | p. 119 |
| Global radius of curvature | p. 121 |
| Self distance type energies defined via the distance function | p. 127 |
| Relation between these geometric quantities and e[superscript alpha,p] | p. 131 |
| Numerical computations and applications | p. 132 |
| Energy of knots from a conformal geometric viewpoint | p. 133 |
| Preparation from conformal geometry | p. 135 |
| The Lorentzian metric on Minkowski space | p. 135 |
| The Lorentzian exterior product | p. 143 |
| The space of spheres | p. 144 |
| The 4-tuple map and the cross ratio of 4 points | p. 147 |
| Pencils of spheres | p. 151 |
| Modulus of an annulus | p. 156 |
| Cross-separating annuli and the modulus of four points | p. 158 |
| The measure on the space of spheres [Lambda] | p. 165 |
| Orientations of 2-spheres | p. 168 |
| The space of non-trivial spheres of a knot | p. 175 |
| Non-trivial spheres of a knot | p. 175 |
| The 4-tuple map for a knot | p. 177 |
| Generalization of the 4-tuple map to the diagonal | p. 179 |
| Twice tangent spheres | p. 179 |
| Tangent spheres | p. 183 |
| Osculating spheres | p. 184 |
| Lower semi-continuity of the radii of non-trivial spheres | p. 186 |
| The infinitesimal cross ratio | p. 189 |
| The infinitesimal cross ratio of the complex plane | p. 189 |
| The real part as the canonical symplectic form of T* S[superscript 2] | p. 191 |
| The infinitesimal cross ratio for a knot | p. 201 |
| From the cosine formula to the original definition of E[superscript (2)] | p. 205 |
| E[superscript (2) subscript o]-energy for links | p. 207 |
| The conformal sin energy E[subscript sin theta] | p. 213 |
| The projection of the inverted open knot | p. 213 |
| The geometric meaning of E[subscript sin theta] | p. 215 |
| Self-repulsiveness of E[subscript sin theta] | p. 216 |
| E[subscript sin theta] and the average crossing number | p. 221 |
| E[subscript sin theta] for links | p. 221 |
| Measure of non-trivial spheres | p. 223 |
| Non-trivial spheres, tangent spheres and twice tangent spheres | p. 223 |
| The volume of the set of the non-trivial spheres | p. 224 |
| The measure of non-trivial spheres in terms of the infinitesimal cross ratio | p. 226 |
| Non-trivial annuli and the modulus of a knot | p. 231 |
| Self-repulsiveness of the measure of non-trivial spheres | p. 235 |
| The measure of non-trivial spheres for non-trivial knots | p. 237 |
| Measure of non-trivial spheres for links | p. 241 |
| Generalization of the Gauss formula for the linking number | p. 243 |
| The Gauss formula for the linking number | p. 243 |
| The writhe and the self-linking number | p. 245 |
| The total twist | p. 247 |
| Average crossing number | p. 249 |
| The conformal angle and the Gauss integral | p. 250 |
| Mobius invariance of the writhe | p. 252 |
| The circular Gauss map and the inverted open knots | p. 253 |
| The 3-tuple map to the set of circles in S[superscript 3] | p. 257 |
| The set of unoriented circles in S[superscript 3] | p. 257 |
| The 3-tuple map to the set of circles | p. 257 |
| Conformal moduli of a solid torus | p. 259 |
| Modulus of a 2-dimensional annulus revisited | p. 259 |
| Conformal moduli of a solid torus | p. 260 |
| Kirchhoff elastica | p. 263 |
| Open problems and dreams | p. 265 |
| Bibliography | p. 271 |
| Index | p. 285 |
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