Energy Of Knots And Conformal Geometry

By: Jun O'hara

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Hardcover | 26 March 2003

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304 Pages

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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.

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.,." gives an excellent, state of the art account of many of the important results in this field."

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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
Table of Contents provided by Ingram. All Rights Reserved.

Energy Of Knots And Conformal Geometry

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