The physical properties of knotted and linked configurations in space have long been of interest to mathematicians. More recently, these properties have become significant to biologists, physicists, and engineers among others. Their depth of importance and breadth of application are now widely appreciated and valuable progress continues to be made each year. This volume presents several contributions from researchers using computers to study problems that would otherwise be intractable. While computations have long been used to analyze problems, formulate conjectures, and search for special structures in knot theory, increased computational power has made them a staple in many facets of the field. The volume also includes contributions concentrating on models researchers use to understand knotting, linking, and entanglement in physical and biological systems. Topics include properties of knot invariants, knot tabulation, studies of hyperbolic structures, knot energies, the exploration of spaces of knots, knotted umbilical cords, studies of knots in DNA and proteins, and the structure of tight knots. Together, the chapters explore four major themes: physical knot theory, knot theory in the life sciences, computational knot theory, and geometric knot theory.
Author(s): Calvo , Millet and Rawdon
Series: Series on Knots and Everything 38
Year: 2005
Language: English
Pages: 628
CONTENTS......Page 18
Preface......Page 8
1. Problem Statement and Results......Page 22
2. A Lemma about Polygonal Knots......Page 26
3. Solid Knots Made of Congruent Components......Page 28
4. Solid Knots of Uniform Thickness d......Page 37
References......Page 39
1. Introduction to Knot Energies......Page 40
2. Examples of Knot Energies......Page 42
3. Properties of Knot Energies: Polygonal Knots......Page 51
4. Thickness Energies......Page 57
References......Page 61
1. Introduction......Page 66
2. Review of previous physical results on tight knots and links......Page 67
3. Exact calculations......Page 68
4.1. Magnetic relaxation......Page 69
4.2. Abelian helicity......Page 70
4.4. "Freeze-in” condition......Page 71
5.1. QCD......Page 72
5.2. Knot energies......Page 74
5.3. Model......Page 75
6. Discussion and conclusions......Page 80
References......Page 82
1. Introduction......Page 86
2. Our Experimental System......Page 87
2.1. Spontaneous Knots......Page 88
3.2. Open Knot Length in Our Chain......Page 89
4. Untying Dynamics: Dependence on Knot Type......Page 92
Acknowledgments......Page 94
References......Page 95
1. Introduction......Page 96
2. Criteria for the assessment of closeness to ideality......Page 99
3. Why compute with Biarcs?......Page 102
4. Simulated annealing with biarcs......Page 105
5 . Results for the 3.1-knot......Page 106
5.2. The shape......Page 107
5.3. The contact sets......Page 110
6.2. The shape......Page 116
6.3. The contact sets......Page 120
7. Discussion......Page 124
References......Page 128
2. Description......Page 130
3. Knots......Page 131
4. History......Page 133
5. Knotting frequency......Page 134
6. Contributing factors......Page 135
7. A simple model......Page 136
8. Clinical significance......Page 139
9. Complex and multiple knots......Page 140
10. Handedness and perversion......Page 141
References......Page 143
1. Introduction......Page 148
2. The model......Page 150
3. Persistence length & Stretching curves......Page 153
4. Loading curves & DNA packaging......Page 157
5. Perspectives and Conclusions......Page 165
References......Page 166
1. Introduction......Page 170
2. Methods......Page 174
3. Results......Page 177
References......Page 179
1. Introduction......Page 182
2. Methods......Page 183
3. Characterization of the conformation of a DNA molecule bound to a surface......Page 184
4. Images of knotted DNA......Page 186
References......Page 190
1. Introduction......Page 192
2.1.1. Chain smoothing......Page 194
2.1.2. A self-avoiding chain......Page 195
2.1.3. Removing redundant points......Page 196
2.1.4. A simple knot nomenclature......Page 197
2.2. Knots in proteins......Page 199
2.2.1. A protein trefoil knot......Page 200
2.2.2. A protein figure-of-eight knot......Page 201
2.3. Protein Pseudo-knots......Page 202
2.3.1. A pseudo-knot an a SET domain......Page 203
2.3.2. Generalased protein knots......Page 204
2.3.3. Folding covalent and pseudo-knots......Page 205
3.1. Topological indices......Page 206
3.1.1. “Tornado” plots......Page 208
3.2. Topological accessibility......Page 210
4. Random proteins......Page 213
4.1.2. Secondary structure lattice folds......Page 214
4.1.3. Off-lattice folds......Page 216
4.2.2. Off-lattice folds......Page 217
4.2.3. Secondary structure lattice folds......Page 218
5. Conclusions......Page 220
Acknowledgements......Page 221
References......Page 222
1. Introduction......Page 224
2 . Random Walks and Knotting......Page 226
3. Visualization of the Knotting Spectrum......Page 232
4. Applications to the Identification of Knotting in Proteins......Page 235
References......Page 237
1. Introduction......Page 240
2. Simulation methods......Page 242
3.1. (ACN) scaling in linear and closed random walks......Page 243
3.2. ( A C N ) scaling in the individual knot types......Page 244
3.3. The equilibrium length of a knot......Page 247
3.4. Scaling of (ACN) in natural protein structures......Page 248
4. Conclusions and Outlook......Page 249
References......Page 251
1. Introduction......Page 254
2. Simulation method and shape characterization......Page 257
3. Scaling behaviour as a function of the bond-length ratio......Page 258
4. Conclusions......Page 263
References......Page 264
1. Introduction......Page 268
2. Models of Knotting......Page 270
3. Generation and Analysis of Knot Probability Data......Page 271
4. Knot Probabilities and Associated Functional Models......Page 272
4.1. The Exponential Decay Model (ED)......Page 276
4.2. The Deguchi-Tsurusaki Model (DT)......Page 277
4.3. The Dobay et al. Model (DSDS)......Page 278
4.4. The Quadratic Variation (QV)......Page 280
4.5. The Full Variation Model (FV)......Page 281
4.6. Applications to Unknot Data......Page 283
5. Analysis of Functional Models of Non-trivial Knot Probability......Page 285
5.1. The Trefoil Knot......Page 287
5.2. The Figure-Eight Knot......Page 288
6. Conclusions and Speculations......Page 290
References......Page 293
1. Introduction......Page 296
2. The Gaussian Random Walks and Polygons......Page 298
3. The Main Results and their Proofs......Page 300
References......Page 312
1. Introduction......Page 324
2.1. Ropelength of knots tied on the perfect rope......Page 326
2.2. Ropelength of polygonal knots......Page 329
3. Interpretation of simulations performed with the SONO algorithm......Page 336
3.1. Basic procedures of SONO......Page 337
3.2. Physical sense of the SONO algorithm and practical details of simulations......Page 338
3.3. The problem offinding the right ropelength, an experimental approach......Page 339
4. Ropelength of SONO knots......Page 340
4.1. The problem of finding the right ropelength, a n analytic approach......Page 342
5. Discussion......Page 348
Acknowledgments......Page 350
References......Page 351
1. Introduction......Page 354
2. Edge-Edge Checks......Page 356
3. The Octree Data Structure......Page 359
4. The Core of the Algorithm......Page 362
5. Implementation Issues......Page 363
6. Performance......Page 366
7. Conclusions and Future Directions......Page 369
References......Page 371
1. Introduction......Page 374
2.2. Entropic forces......Page 377
3.1. Definition of random linking probabilities......Page 378
3.3. Analytic expressions of linking probabilities......Page 380
3.3.1. Linking probability of the trivial link......Page 381
3.3.3. Linking probability of the Hopf link......Page 383
4. Topological entropic forces......Page 385
4.1. Entropic force for the trivial link......Page 386
4.2. Entropic force f o r the case of nontrivial links......Page 387
4.3. Entropic force for the Hopf link......Page 388
5.1. The mean square radius of gyration for a random link consisting of two random knots......Page 389
5.2. Evaluation of the average size of random links......Page 390
References......Page 392
1.1. The goal of this work......Page 394
2. Brief overview of our recent work......Page 396
2.1. Simulation methods......Page 397
2.2. Knot population fractions......Page 398
2.3.1. Scaling of the trivial knot size......Page 399
2.3.2. Corrections to scaling......Page 400
2.3.3. Averaged sizes of non-trivial knots......Page 402
3. Probability distributions of the loop sizes......Page 403
4. Concluding remarks......Page 406
Appendix A. Loop generation......Page 408
Appendix B. Probability distribution of all loops......Page 410
References......Page 414
1.1. Goal and plan of this work......Page 416
1.2. Why lattice model is natural for our purposes......Page 418
2. Brief overview of our recent results......Page 419
2.2. Topology......Page 420
3. Testing knot localization hypothesis by renormalization......Page 423
4. Conclusion......Page 427
References......Page 428
1.1. Motivation and contents of the paper......Page 430
1.2. Some theoretical results on the topological swelling of SAPs......Page 432
2.1. Method for constructing ensembles of cylindrical SAPs......Page 434
2.3. Characteristic length of random knotting N , for cylindrical SAPs and rod-bead model......Page 435
3. Finite-size behaviors of RK2 for some knots......Page 436
3.1. The radial distribution functions of random knots......Page 438
3.2. Decrease of the topological effect under the increase of the excluded volume......Page 440
3.3. Interpretations through the characteristic length......Page 442
4. Asymptotic behaviors of RK2......Page 443
4.1. The r-dependence of the amplitude ratio......Page 445
5. N-r diagram......Page 446
References......Page 448
1. Thermodynamics of Random Knot Diagrams......Page 452
1.1. Knots o n lattices: model and definitions......Page 455
1.1.1. Reidemeister moves and definition of the Kaujfman invariant......Page 456
1.1.2. A partition function of the Potts model as a bichromatic polynomial......Page 458
1.1.3. Kauffman invariant represented as a partition function for the Potts model......Page 460
1.2.1. The form of a lattice for the Potts model and the positions of ferro- and antaferromagnetic bonds......Page 465
1.2.2. The method of transfer matrix......Page 466
1.3.1. Correlations between the degree of Jones polynomial of the lattice knot and the minimum energy of the Potts model......Page 471
1.3.2. The probability distribution of the degree of the polynomial invariant and the minimum energy of the corresponding Potts model......Page 474
2. Physical Applications......Page 477
3. The model of densely packed knots on a lattice and the concept of “quasi-knots”......Page 479
3.1. Comparison of digerent definitions of the “knot complexity ”......Page 481
3.2.1. Unconditional distributions......Page 487
3.2.2. Conditional distributions (“Brownian Bridges”)......Page 488
3.2.3. Matrix representation of Kauffman knot invariants on the strips......Page 490
4. General conclusions......Page 494
Acknowledgments......Page 498
Appendix: Minimum energy distribution in the Potts model with random ferro- and antiferromagnetic bonds......Page 499
References......Page 501
1. Introduction......Page 504
2. Basic Concepts......Page 507
3. Algorithm One; Generating large, diagrammatically prime RP-graphs of knots......Page 510
4. Algorithm Two; Generating large, diagrammatically prime Hamiltonian RP-graphs of knots......Page 514
5. Data and Comparisons......Page 520
6. Conclusion......Page 523
References......Page 525
1. Introduction......Page 526
2. A Mathematical Model for Flat Knotted Ribbons......Page 527
3. The Trefoil Knot, the Pentagon, and the Golden Ratio......Page 530
4. Figure Eight Knot and a Hexagon......Page 533
5. Discussion......Page 536
References......Page 537
1. Introduction......Page 538
2. The knot 31 — the trefoil knot......Page 540
3. The knot 41 — the figure eight knot......Page 542
4. The knot 51 — the torus knot of type (5,2)......Page 545
5. The knot 52......Page 550
References......Page 554
1. Introduction......Page 556
2.1. Twist, writhe and linking number......Page 557
2.2. Basic relations......Page 558
3. Writhe of an arbitrary open segment......Page 559
3.1. Broken curve......Page 563
4. Writhe and the GauB integral......Page 564
4.1. Open curve and its closure......Page 565
4.2. Swirl......Page 568
4.3. Squint......Page 570
5. Example: helical shapes of arbitrary length......Page 572
5.1. Writhe of a single helix......Page 573
5.2. Writhe of a double helix......Page 574
Acknowledgements......Page 575
References......Page 576
1. Introduction......Page 578
2.1. Admissible mappings......Page 582
2.2. Global radius of curvature f o r surfaces......Page 583
3.1. Interior continuity of the normal......Page 585
3.2. Continuity of the normal at the boundary......Page 588
3.3. Structure of the image......Page 589
4. Convergence and compactness......Page 590
References......Page 594
1. Introduction......Page 596
2. Background......Page 597
3. Weighted clasps......Page 600
4. Clasps with parallels......Page 601
5. Conjectured minimizers for two cases......Page 605
6. A chained clasp......Page 606
7. The granny clasp......Page 608
References......Page 611
1. Introduction......Page 612
2. The Recursive Technique......Page 616
3. Extensions and Conjectures......Page 621
References......Page 626
1. Introduction......Page 628
2. A model for the enumeration of alternating links......Page 629
3. Some conjectures on asymptotic counting......Page 631
4. Numerical checks......Page 632
5. Virtual knot theory......Page 635
References......Page 636
