CONTENTS -- Preface -- Chapter 1 On the Theory of Solid Knots Otto Krotenheerdt and Sigrid Veit (translated by Ted Ashton) -- 1. Problem Statement and Results -- 2. A Lemma about Polygonal Knots -- 3. Solid Knots Made of Congruent Components -- 4. Solid Knots of Uniform Thickness d -- References -- Chapter 2 A Tutorial on Knot Energies E. J. Janse van Rensburg -- 1. Introduction to Knot Energies -- 2. Examples of Knot Energies -- 3. Properties of Knot Energies: Polygonal Knots -- 4. Thickness Energies -- 5. Conclusions -- References -- Chapter 3 Universal Energy Spectrum of Tight Knots and Links in Physics Roman V. Buniy and Thomas W. Kephart -- 1. Introduction -- 2. Review of previous physical results on tight knots and links -- 3. Exact calculations -- 4. Plasma physics -- 4.1. Magnetic relaxation -- 4.2. Abelian helicity -- 4.3. Non-abelian helicity -- 4.4. "Freeze-in" condition -- 5. QCD and Glueballs -- 5.1. QCD -- 5.2. Knot energies -- 5.3. Model -- 6. Discussion and conclusions -- References -- Chapter 4 Knot Dynamics in a Driven Hanging Chain: Experimental Results Andrew Belmonte -- 1. Introduction -- 2. Our Experimental System -- 2.1. Spontaneous Knots -- 3. How Long is a Knot? -- 3.1. Previous Work -- 3.2. Open Knot Length in Our Chain -- 4. Untying Dynamics: Dependence on Knot Type -- 5. Conclusions -- Acknowledgments -- References -- Chapter 5 Biarcs, Global Radius of Curvature, and the Computation of Ideal Knot Shapes (4 color plates) M. Carlen, B. Laurie, J. H. Maddocks and J. Smutny -- 1. Introduction -- 2. Criteria for the assessment of closeness to ideality -- 3. Why compute with Biarcs? -- 4. Simulated annealing with biarcs -- 5 . Results for the 3.1-knot -- 5.1. The numbers -- 5.2. The shape -- 5.3. The contact sets -- 6. Results for the 4.1-knot -- 6.1. The numbers -- 6.2. The shape -- 6.3. The contact sets -- 7. Discussion.

References -- Chapter 6 Knotted Umbilical Cords (2 color plates) Alain Goriely -- 1. Introduction -- 2. Description -- 3. Knots -- 4. History -- 5. Knotting frequency -- 6. Contributing factors -- 7. A simple model -- 8. Clinical significance -- 9. Complex and multiple knots -- 10. Handedness and perversion -- 11. Conclusions -- Acknowledgments -- References -- Chapter 7 Modelling DNA as a Flexible Thick Polymer: DNA Elasticity and Packaging Thermodynamics Cristian Micheletti and Davide Marenduzzo -- 1. Introduction -- 2. The model -- 3. Persistence length & Stretching curves -- 4. Loading curves & DNA packaging -- 5. Perspectives and Conclusions -- Acknowledgments -- References -- Chapter 8 Monte-Carlo Simulations of Gel-Electrophoresis of DNA Knots C. Weber, M. Fleurant, P. De Los Rios and G. Dietler -- 1. Introduction -- 2. Methods -- 3. Results -- 4. Conclusion -- Acknowledgments -- References -- Chapter 9 Atomic Force Microscopy of Complex DNA Knots F. Valle, M. Favre, J . Roca and G. Dietler -- 1. Introduction -- 2. Methods -- 3. Characterization of the conformation of a DNA molecule bound to a surface -- 4. Images of knotted DNA -- Acknowledgments -- References -- Chapter 10 Protein Folds, Knots and Tangles William R. Taylor -- 1. Introduction -- 2. The 'Topology' of open chains -- 2.1. Shrinking chains -- 2.1.1. Chain smoothing -- 2.1.2. A self-avoiding chain -- 2.1.3. Removing redundant points -- 2.1.4. A simple knot nomenclature -- 2.2. Knots in proteins -- 2.2.1. A protein trefoil knot -- 2.2.2. A protein figure-of-eight knot -- 2.3. Protein Pseudo-knots -- 2.3.1. A pseudo-knot an a SET domain -- 2.3.2. Generalased protein knots -- 2.3.3. Folding covalent and pseudo-knots -- 3. Protein fold complexity -- 3.1. Topological indices -- 3.1.1. "Tornado" plots -- 3.2. Topological accessibility -- 4. Random proteins.

4.1. Constructing Semi-random Folds -- 4.1.1. Self-avoiding Random Walks -- 4.1.2. Secondary structure lattice folds -- 4.1.3. Off-lattice folds -- 4.2. Analysis of semi-random folds -- 4.2.1. Semi-random Walks -- 4.2.2. Off-lattice folds -- 4.2.3. Secondary structure lattice folds -- 5. Conclusions -- Acknowledgements -- References -- Chapter 11 Tying Down Open Knots: A Statistical Method for Identifying Open Knots with Applications to Proteins (7 color plates) Kenneth C. Millett and Benjamin M. Sheldon -- 1. Introduction -- 2 . Random Walks and Knotting -- 3. Visualization of the Knotting Spectrum -- 4. Applications to the Identification of Knotting in Proteins -- 5. Conclusions and Speculations -- References -- Chapter 12 Scaling of the Average Crossing Number in Equilateral Random Walks, Knots and Proteins Akos Dobay, Jacques Dubochet, Andrzej Stasiak and Yuanan Diao -- 1. Introduction -- 2. Simulation methods -- 3. Results -- 3.1. (ACN) scaling in linear and closed random walks -- 3.2. ( A C N ) scaling in the individual knot types -- 3.3. The equilibrium length of a knot -- 3.4. Scaling of (ACN) in natural protein structures -- 4. Conclusions and Outlook -- Acknowledgments -- References -- Chapter 13 Folding Complexity in a Random-Walk Copolymer Model Gustavo A. Arteca -- 1. Introduction -- 2. Simulation method and shape characterization -- 3. Scaling behaviour as a function of the bond-length ratio -- 4. Conclusions -- Acknowledgments -- References -- Chapter 14 Universal Characteristics of Polygonal Knot Probabilities Kenneth C. Millett and Eric J. Rawdon -- 1. Introduction -- 2. Models of Knotting -- 3. Generation and Analysis of Knot Probability Data -- 4. Knot Probabilities and Associated Functional Models -- 4.1. The Exponential Decay Model (ED) -- 4.2. The Deguchi-Tsurusaki Model (DT) -- 4.3. The Dobay et al. Model (DSDS).

4.4. The Quadratic Variation (QV) -- 4.5. The Full Variation Model (FV) -- 4.6. Applications to Unknot Data -- 5. Analysis of Functional Models of Non-trivial Knot Probability -- 5.1. The Trefoil Knot -- 5.2. The Figure-Eight Knot -- 6. Conclusions and Speculations -- Acknowledgements -- References -- Chapter 15 The Average Crossing Number of Gaussian Random Walks and Polygons Yuanan Diao and Clam Ernst -- 1. Introduction -- 2. The Gaussian Random Walks and Polygons -- 3. The Main Results and their Proofs -- Acknowledgments -- References -- Chapter 16 Ropelength of Tight Polygonal Knots Justyna Baransku, Piotr Pieranski an and Eric J.Rawdon -- 1. Introduction -- 2. Computing upper bounds for smooth knots inscribed in tight polygons -- 2.1. Ropelength of knots tied on the perfect rope -- 2.2. Ropelength of polygonal knots -- 3. Interpretation of simulations performed with the SONO algorithm -- 3.1. Basic procedures of SONO -- 3.2. Physical sense of the SONO algorithm and practical details of simulations -- 3.3. The problem offinding the right ropelength, an experimental approach -- 4. Ropelength of SONO knots -- 4.1. The problem of finding the right ropelength, a n analytic approach -- 5. Discussion -- Acknowledgments -- References -- Chapter 17 A Fast Octree-Based Algorithm for Computing Ropelength Ted Ashton and Jason Cantarella -- 1. Introduction -- 2. Edge-Edge Checks -- 3. The Octree Data Structure -- 4. The Core of the Algorithm -- 5. Implementation Issues -- 6. Performance -- 7. Conclusions and Future Directions -- Acknowledgements -- References -- Chapter 18 Topological Entropic Force between a Pair of Random Knots Forming a Fixed Link Tetsuo Deguchi -- 1. Introduction -- 2. Statistical mechanical background of entropic forces -- 2.1. Probability and the free energy of a macroscopic state -- 2.2. Entropic forces -- 3. Random links.

3.1. Definition of random linking probabilities -- 3.2. On simulation results of random linking probability -- 3.3. Analytic expressions of linking probabilities -- 3.3.1. Linking probability of the trivial link -- 3.3.2. Probability of being a nontrivial link -- 3.3.3. Linking probability of the Hopf link -- 3.3.4. A consistency check -- 4. Topological entropic forces -- 4.1. Entropic force for the trivial link -- 4.2. Entropic force f o r the case of nontrivial links -- 4.3. Entropic force for the Hopf link -- 5. Average size of random links -- 5.1. The mean square radius of gyration for a random link consisting of two random knots -- 5.2. Evaluation of the average size of random links -- Acknowledgements -- References -- Chapter 19 Under-Knotted and Over-Knotted Polymers: 1. Unrestricted Loops Nathan T. Moore, Rhonald C. Lua and Alexander Yu. Grosberg -- 1. Introduction -- 1.1. The goal of this work -- 1.2. Some terminology: non-phantom polymers and self-avoiding polymers are two different things -- 2. Brief overview of our recent work -- 2.1. Simulation methods -- 2.2. Knot population fractions -- 2.3. Average size of diferent knots -- 2.3.1. Scaling of the trivial knot size -- 2.3.2. Corrections to scaling -- 2.3.3. Averaged sizes of non-trivial knots -- 3. Probability distributions of the loop sizes -- 4. Concluding remarks -- Acknowledgments -- Appendix A. Loop generation -- Appendix B. Probability distribution of all loops -- References -- Chapter 20 Under-Knotted and Over-Knotted Polymers: 2. Compact Self-Avoiding Loops Rhonald C. Lua, Nathan T. Moore and Alexander Yu. Grosberg -- 1. Introduction -- 1.1. Goal and plan of this work -- 1.2. Why lattice model is natural for our purposes -- 2. Brief overview of our recent results -- 2.1. Generation of compact loops -- 2.2. Topology -- 3. Testing knot localization hypothesis by renormalization.

4. Conclusion.