Ideal Knots

CONTENTS......Page 10
Preface......Page 6
CHAPTER 1
IDEAL KNOTS AND THEIR RELATION TO THE PHYSICS OF REAL KNOTS......Page 12
1 Ideal knots......Page 13
2 Ideal trajectories as knot invariants......Page 14
3 Length/diameter ratio of ideal knots as a scale independent measure of the complexity of knots......Page 15
5 Relation between length/diameter ratio and the average crossing number of ideal representations of knots......Page 18
6 Writhe of ideal representations of knots......Page 21
7 Ideal composite knots......Page 23
8 Relation between ideal and real trajectories of knots......Page 24
9 Gel electrophoresis as a physical method to test the relationship between ideal and real knots......Page 27
10 Conclusions......Page 28
References......Page 29
1 Introduction......Page 31
2 The knot inflation process......Page 32
3.2 FindNeighbours(FN) and RemoveOverlaps (RO)procedures......Page 33
3.4 ReduceNodeNumber (RNN), DoubleNodeNumber (DNN) and NormalizeNodeNumber (NNN) procedures......Page 35
3.5 The core of the SONO algorithm......Page 36
4.1 Untagling an entangled unknot......Page 37
4.2 The Moffat test......Page 38
4.3 The Perko Pair test......Page 39
5 In search of ideal prime knots......Page 40
5.1 Prime knots up to 9 crossings......Page 42
5.2 Tightening of the torus knots T2,M......Page 44
5.3 Towards the ground state conformation of the T2,m knots......Page 46
Acknowledgements......Page 51
References......Page 52
2 General Method......Page 53
3.2 Vertex to Edge distance......Page 54
3.3 Edge to Edge distance......Page 55
4 What Does "Local" Mean......Page 56
5 Avoiding Change of Type......Page 57
7.1 Writhe and Average Crossing Number......Page 58
7.2 Smoothness......Page 60
9 Future Directions......Page 61
11 References......Page 62
1. INTRODUCTION......Page 63
2. DEFINITIONS AND PROPERTIES OF ENERGY FUNCTIONS......Page 65
3. ENERGIES DEFINED BY KNOT THICKNESSES......Page 66
4. OPEN THICKNESS ENERGY......Page 69
5. ENERGIES OF POLYGONAL KNOTS......Page 76
References......Page 79
1 Introduction......Page 81
2 Computing the writhe of lattice polygons......Page 83
3 Width of the writhe distribution......Page 85
4 Numerical methods......Page 90
5 Writhe as a function of knot type......Page 92
6 Writhe as a function of link type......Page 95
7 Discussion......Page 96
Acknowledgements......Page 97
1. Introduction......Page 99
2. Knot Complexity and the Minimal Number......Page 101
3. Numerical Results......Page 107
3.1 Estimating the Minimal Number......Page 108
3.2 Minimal Curvature......Page 111
3.3 Bounds on the minimal edge index and the minimal curvature index......Page 112
4. Discussion......Page 113
References......Page 115
1. Introduction, vocabulary, and history of geometric knots......Page 118
1.1. Minimal stick number......Page 119
1.2. The space of geometric knots......Page 123
1.3. Equilateral polygons......Page 126
2.1. Random knot generation......Page 129
3. Results of search......Page 130
4. Conclusions......Page 136
References......Page 137
1 Introductory remarks......Page 140
2 Basic definitions......Page 141
3.1 A reminder: probability of a trivial knot for an N-link ring decays exponentially with N......Page 143
3.3 Applying scaling arguments: confinement of an ideal trivial knot......Page 145
4 The central idea of this work......Page 148
5 Discussion......Page 149
6 Additional Note......Page 151
References......Page 152
1 Introduction......Page 154
4 Polygonal Thickness......Page 155
5 Continuity......Page 159
Acknowledgments......Page 160
References......Page 161
Ideal Knots......Page 162
Gel velocity......Page 164
References......Page 165
Energy functions for knots : beginning to predict physical behavior......Page 167
1 Gel velocity, random knot frequency, and topological ground state energy of a knot......Page 168
2 Knots......Page 176
3.1 Motivation and vertex-energy......Page 180
3.2 Energies for smooth curves......Page 182
3.3 Energies for polygons......Page 184
References......Page 190
1 Background......Page 194
1.1 Knot Theory......Page 195
1.2 Knot Classification Methods......Page 196
1.3 Optimization Techniques......Page 197
2 Methods......Page 199
2.1 State Description and Visualization......Page 200
2.2 Energy Functional......Page 201
2.3 Perturbation Methods......Page 202
3.1 Energy Unimodality......Page 206
3.2 SA vs. Gradient Descent......Page 207
4 Conclusions......Page 213
Acknowledgments......Page 214
COLOUR PLATES......Page 218
1 Introduction......Page 234
2 The general theory of relaxation......Page 235
3 Conservation of field helicity......Page 239
4 Relaxation of knotted fields......Page 240
5 Relaxation of strongly twisted tubes......Page 242
References......Page 244
1 Introduction......Page 245
2 Competing Interactions in Bistable Media......Page 246
3 Stochastic Model......Page 250
4 Links and Knots......Page 253
4.1 Ideal links and knots......Page 257
5 Conclusion......Page 260
Appendix: System evolution in the Boltzmann approximation......Page 261
References......Page 264
1 Kelvin's vortex atoms and the origin of topological fluid mechanics......Page 266
2.1 Topological equivalence classes for frozen fields......Page 269
2.2 Action of local flows and Reidemeister's moves......Page 270
3 Links of thin core vortex rings......Page 272
4.1 Kelvin's conjecture and vortex knot dynamics......Page 275
4.2 New results on stability of vortex knots......Page 276
5.1 Evolution of inflexional magnetic knots......Page 279
5.2 Relaxation of inflexional knots to minimal braids......Page 280
5.3 Possible consequences for energy estimates......Page 282
References......Page 283
1 Introduction......Page 285
2 A Knot Hamiltonian......Page 287
3 Aspects of Numerical Solution......Page 293
References......Page 298
1 Introduction......Page 299
2 Definition of α-energy E(α)......Page 301
3.2 Möbius invariance......Page 306
3.3 Existence of E-minimizers......Page 308
3.4 Gradient......Page 309
3.6 Unstable E-critical torus knots......Page 310
3.7 Finiteness of knot types......Page 311
4 Higher power index......Page 312
5.1 General cases......Page 315
5.2 The spherical case......Page 316
5.3 The hyperbolic case......Page 317
6 Thickness and self distance......Page 318
7.2 Other kinds of knot energy functionals......Page 320
8 Summary......Page 321
References......Page 322
1 Introduction......Page 326
2 Defining Möbius Energies......Page 328
3 The Excess-Length Picture and a Standard Choice of Regularization......Page 330
4 Other Möbius-Invariant Knot Energies......Page 332
5 Prime Decomposition......Page 333
6 Discretization and Computer Experiments......Page 335
7 Untangling Unknots......Page 338
8 Crossing Numbers and Ropelength......Page 339
9 Critical Knots and Links from Group Actions......Page 340
10 Hopf Links and Electrons on S2......Page 342
11 Surfaces and Submanifolds......Page 345
12 A Table of Knots and Links Minimizing Möbius Energy......Page 348
Acknowledgments......Page 351
Appendix......Page 352
References......Page 361
1 Introduction......Page 364
2 Harmonic knots - definition, existence, and index......Page 365
3 Superbridge index and harmonic index......Page 368
4 Torus knots and harmonic index......Page 370
5 Crossing number and harmonic index......Page 371
6 Examples of harmonic knots......Page 372
8 References......Page 373
1 Introduction......Page 375
2 Every Knot is a Fourier Knot......Page 376
3 Lissajous Knots and the Arf Invariant......Page 377
4 A Fourier Trefoil Knot......Page 379
6 A Series of Fibonacci Fourier Knots......Page 382
References......Page 384
CHAPTER 20 Symmetric knots and billiard knots......Page 385
1 Symmetric knots......Page 386
2 Polynomial invariants of links......Page 387
3 Periodic links and the Jones polynomial......Page 390
4 Periodic links and the generalized Jones polynomials......Page 398
5 rq-periodic links and Vassiliev invariants......Page 403
6 Lissajous knots and billiard knots......Page 406
7 Applications and Speculations......Page 415
References......Page 422