# Ideal Knots

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<p>K(pE Series on Knots and Everything - Vol. 19</p>
<p>11] ^ m [ C M nEditors</p>
<p>A. Stasiak, V. Katntch and L H* Kauffman</p>
<p>World Scientific</p>
<p>IDEAL KNOTS</p>
<p>SERIES ON KNOTS AND EVERYTHINGEditor-in-charge: Louis H. Kauffman Published:Vol. 1: Knots and Physics L. H. Kauffman Vol. 2: How Surfaces Intersect in Space J. S. Carter</p>
<p>Vol. 3: Quantum Topology edited by L. H. Kauffman & R. A. Baadhio Vol. 4: Gauge Fields, Knots and GravityJ. Baez & J. P. Muniain Vol. 5: Gems, Computers and Attractors for 3-Manifolds S. Lins Vol. 6: Knots and Applications edited by L. H. Kauffman</p>
<p>Vol. 7: Random Knotting and Linking edited by K. C. Millets & D. W. Sumners Vol. 8: Symmetric Bends: How to Join Two Lengths of Cord R. E. Miles Vol. 9: Combinatorial PhysicsT. Bastin & C. W. Kilmister</p>
<p>Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics W. M. HonigVol. 11: History and Science of Knots edited by J. C. Turner & P. van de Griend Vol. 12: Relativistic Reality: A Modem View</p>
<p>J. D. Edmonds, Jr.Vol. 13: Entropic Spacetime Theory J. Armel Vol. 14: Diamond - A Paradox Logic N. S. K. Hellerstein Vol. 15: Lectures at Knots '96 edited by S. Suzuki Vol. 16: Delta - A Paradox Logic N. S. K. Hellerstein Vol. 19: Ideal Knots A. Stasiak, V. Katritch and L. H. Kauffman</p>
<p>K CE Series on Knots and Everything - Vol. 19</p>
<p>IDEAL KNOTS</p>
<p>Editors</p>
<p>A. StasiakUniversite de Lausanne</p>
<p>V. KatritchRutgers University</p>
<p>L. H. KauffmanUniversity of Illinois, Chicago</p>
<p>S World Scientific`t Singapore NewJerse Y London -Hong Kon 9</p>
<p>Published by</p>
<p>World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road , Singapore 912805 USA office: Suite 1B, 1060 Main Street , River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden , London WC2H 9HE</p>
<p>British Library Cataloguing -in-Publication Data A catalogue record for this book is available from the British Library.</p>
<p>Title of cover: Plato, Kelvin and ideal trefoil Cover done by Ben Laurie.</p>
<p>IDEAL KNOTS Copyright 1998 by World Scientific Publishing Co. Pte. Ltd.All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.</p>
<p>For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.</p>
<p>ISBN 981-02-3530-5</p>
<p>Printed in Singapore by Uto-Print</p>
<p>V</p>
<p>Preface In a strict topological sense all representations of a given type of a knot are equal as they can be converted into each other by a finite number of moves maintaining the original topology of the knot (moving of segments without passing through each other, translation and rotation, change of the scale). However, we intuitively avoid this extreme polymorphism of knots. Thus when we make a quick drawing of a trefoil we usually trace it with just three crossings and not with many additional unnecessary crossings. When we are asked for a really nice drawing of a trefoil we try to give it a very regular symmetrical shape. This is why the founders of the mathematical classification of knots represented different knots in the tables in such a way that each type of the knot was shown with its minimal crossing number and in addition had a regular shape with elements of symmetry (when this was possible). Furthermore, when several persons are asked independently for an extremely nice and elaborated drawing of a trefoil they will produce different drawings where the presented trefoils will usually maintain three-fold symmetry, but while one person will draw a trefoil with elongated blades another will give it almost circular shape. This is of course different from drawings of a triangle or of a tetragon where almost everyone will draw an equilateral triangle and a square where the only difference between individual drawings is their scale. Can we thus think about an ideal representation of a knot such that everyone would agree that it is the best representation of a given knot, like we all can agree that the square is the ideal representation of a tetragon? This question was raised at the physical knot theory session organized by Jonathan Simon and Gregory Buck at 1996 ASM meeting in Iowa and it turned out that there was no consensus. Background and the particular scientific interest of concerned scientists showed that they had in mind different ideal representations of knots. Some were applying simple geometrical criteria, some were minimizing certain physical or abstract energies, some looked for the simplest parametric descriptions of given knots, some wanted to have high symmetry, some hoped that the ideal shape would correspond to a "mean" trajectory of all possible representations of a given knot, some demanded that ideal knots should be built of a minimal number of straight segments on a particular lattice etc.etc. Despite this lack of consensus one thing become apparent: among an infinite sea of shapes of every type of knot only a small subset or even just one unique representation can satisfy certain criteria and is thus ideal according to these criteria. In this volume different authors characterize their approach to their favourite ideal knots and links. The following list of chapters with short descriptions of their contents should help to navigate between the chapters and to encourage readers to visit each domain of the ideality.</p>
<p>vi</p>
<p>Ideal knots and their relation to the physics of real knots by A. Stasiak, J. Dubochet, V. Katritch and P. Pieranski. This chapter presents the concept of Platonic, ideal geometric representations of knots and demonstrates their ability to govern the time averaged behaviour of randomly distorted knotted polymers forming corresponding types of knots. In search of ideal knots by P. Pieranski. This chapter describes a numerical algorithm used to approach ideal geometric representations of knots and shows fascinating transitions of knots on their way toward ideal configurations. Annealing ideal knots and links: methods and pitfalls by B. Laurie. A simulated annealing algorithm adapted to perform a search for ideal representations of knots and links is presented together with impressive graphics of the obtained configurations and a generous offer to share the code. Knots with minimal energies by Y. Diao, C. Ernst and E. J. Janse van Rensburg. The authors reach the conclusion that the best energy which could drive knots into their canonical representation is their "thickness", experiments with real knotted ropes are compared with numerical simulations of the thick knots which are the same as ideal geometric representations of knots. The writhe of knots and links by E. J. Janse van Rensburg, D. W. Sumners, S. G. Whittington. Certain geometrical properties of random knots in the cubic lattice show surprising correlations with corresponding properties calculated for ideal geometric representations of knots. Minimal lattice knots by E. J. Janse van Rensburg. An ideal knot on the cubic lattice can be defined as that which requires the smallest number of edges to form a given knot. A theoretical and numerical approach to find minimal lattice knots is presented. Basic motifs presented in more complex knots show striking similarities to the motifs present in ideal geometric representations of these knots. Minimal edge piecewise linear knots by J. A. Calvo and K. C. Millett. Knots presented in this chapter reach their specific ideality by requiring the smallest number of straight segments to form a given type of knot. Entropy of a knot : simple arguments about difficult problem by A. Yu. Grosberg. This chapter discusses the available conformational space for different types of knots and concludes that it is related to the length/diameter ratio of ideal geometric representation of a given knot. Approximating the thickness of a knot by E. J. Rawdon. The ratio between the length and the thickness of the shortest rope forming a given type of knot is a topological invariant, however defining the thickness for polygonal knots used in computer simulations is a complex problem which is resolved in this chapter. Energy functions for knots: beginning to predict physical behaviour by J. Simon. This chapter and its preface introduces the concept of novel type of knot invariants based on specific physical properties of idealized shapes of knots and discusses whether certain idealized shapes are more ideal than others. Physically -based stochastic simplification of mathematical knots by R. P. Grzeszczuk, M. Huang and L. H. Kauffman. This chapter describes optimization of a simulated annealing algorithm and compares it with the gradient descent method applied for the process of untangling very complex configurations of knots and unknots.</p>
<p>VII</p>
<p>Knots and fluid dynamics by H. K. Moffatt. This chapter considers relaxation of knotted or linked solenoidal vector fields embedded in a fluid medium and demonstrates that minimum energy equilibrium states are closely related to ideal geometric representations of knots. Knots in bistable reacting systems by A. Malevanets and R. Kapral. Flows of reagent in chemical reactions may form topologically stabilized patterns with two stable phases where one of the phases forms a knot immersed in the other phase. Relaxation of such patterns leads to assuming a nearly ideal shape by the knotted phase. New developments in topological fluid mechanics : from Kelvin's vortex knots to magnetic knots by R. L. Ricca. Starting from a discussion of Kelvin's original conjectures on vortex atoms, the author reviews some of his new results on stability of vortex knots and dynamics of. inflexional magnetic knots and considers possible implications for energy estimates. Hamiltonian approach to knotted solitons by A. J. Niemi. This chapter demonstrates that Kelvin's proposal of stable knotted vortexes achieving nice regular shapes is now confirmed by modern simulations involving complex nonlinear equations of motion. Energy of knots by J. O'Hara. In this chapter equilibrium configurations of self repelling knots are considered where the repulsion is governed by arbitrary energies related to Coulomb's repelling force. Depending on the type of energies the knots could have or not have, a unique configuration minimizing a given energy. Mobius-invariant knot energies by R. B. Kusner and J. M. Sullivan. Among different arbitrary energies associated with trajectories of knots there is a class of energies which remain constant upon Mobius transformation (reflection in a sphere). Trajectories of knots which reach continuous minima of those energies are presented in this chapter. An introduction to harmonic knots by A. K. Trautwein. This chapter presents such configurations of knots that the x, y and z coordinates of their trajectories can be expressed by continuous harmonic functions with a possibly small number of harmonic components. These configurations have nice symmetrical shapes. Fourier knots by L. H. Kauffman. This chapter presents knots which have ideally simple and elegant parametrization of their trajectories via finite Fourier series, and thus have simple and elegant shapes. Symmetric knots and billiard knots by J. Przytycki. This chapter presents selected aspects of knots symmetry visible in the regularities and symmetries of their algebraic invariants or in symmetric trajectories of the actual knots.</p>
<p>Andrzej Stasiak Vsevolod Katritch Louis H. Kauffman</p>
<p>This page is intentionally left blank</p>
<p>ix</p>
<p>CONTENTS</p>
<p>Preface Chapter 1 Ideal Knots and Their Relation to the Physics of Real Knots (6 colour plates) A. Stasiak, J. Dubochet, V. Katritch and P Pieranski In Search of Ideal Knots P Pieranski Annealing Ideal Knots and Links: Methods and Pitfalls (2 colour plates) B. Laurie Knots with Minimal Energies Y. Diao, C. Ernst and E. J. Janse van Rensburg The Writhe of Knots and Links E. J. Janse van Rensburg, D. W. Summers and S. G. Whittington Minimal Lattice Knots E. J. Janse van Rensburg Minimal Edge Piecewise Linear Knots J. A. Calvo and K. C. Millett Entropy of a Knot: Simple Arguments About Difficult Problem A. Yu. Grosberg Approximating the Thickness of a Knot (4 colour plates) E. J. Rawdon</p>
<p>v 1</p>
<p>Chapter 2</p>
<p>20</p>
<p>Chapter 3</p>
<p>42</p>
<p>Chapter 4</p>
<p>52</p>
<p>Chapter 5</p>
<p>70</p>
<p>Chapter 6</p>
<p>88</p>
<p>Chapter 7</p>
<p>107</p>
<p>Chapter 8</p>
<p>129</p>
<p>Chapter 9</p>
<p>143</p>
<p>Chapter 10 Energy Functions for Knots: Beginning to Predict Physical Behaviour (2 colour plates) J. Simon</p>
<p>151</p>
<p>X</p>
<p>Chapter 11 Physically-Based Stochastic Simplification of Mathematical Knots R. P. Grzeszczuk, M. Huang and L. H. Kauffman Colour Plates Chapter 12 Knots and Fluid Dynamics</p>
<p>183</p>
<p>207 223</p>
<p>H. K. MoffattChapter 13 Knots in Bistable Reacting Systems (1 colour plate) A. Malevanets and R. Kapral Chapter 14 New Developments in Topological Fluid Mechanics: From Kelvin's Vortex Knots to Magnetic Knots R. L. Ricca Chapter 15 Hamiltonian Approach to Knotted Solitons A. J. Niemi Chapter 16 Energy of Knots J. O'Hara Chapter 17 Mobius-Invariant Knot Energies R. B. Kusner and J. M. Sullivan Chapter 18 An Introduction to Harmonic Knots A. K. Trautwein Chapter 19 Fourier Knots L. H. Kauffman Chapter 20 Symmetric Knots and Billiard Knots J. Przytycki 234</p>
<p>255</p>
<p>274</p>
<p>288</p>
<p>315</p>
<p>353</p>
<p>364</p>
<p>374</p>
<p>1</p>
<p>CHAPTER IIDEAL KNOTS AND THEIR RELATION TO THE PHYSICS OF REAL KNOTS ANDRZEJ STASIAK, JACQUES DUBOCHET Laboratoire d'Analyse Ultrastructurale Universite de Lausanne1015 Lausanne-Dorigny, Switzerland. e-mail: Andrzej.Stasiak@lau.unil.ch Jacques. Dubochet@lau. unil. ch VSEVOLOD KATRITCH Department of Chemistry Rutgers the State University of New Jersey New Brunswick, New Jersey 08903, USA. e-mail: seva@dna2.rutgers.edu PIOTR PIERANSKI Department of Physics</p>
<p>Poznan University of Technology Piotrowo 3, 60 965 Poznan, Poland andInstitute of Molecular Physics, Smoluchowskiego 17, 60 159 Poznan e-mail: pieransk@phys.put.poznan.pl</p>
<p>We present here the concept of ideal geometric representations of knots which are defined as minimal length trajectories of uniform diameter tubes forming a given type of knot. We show that ideal geometric representations of knots show interesting relations between each other and allow us to predict certain average properties of randomly distorted knotted polymers. Some relations between the behaviour of real physical knots and idealised representations of knots which minimise or maximise certain properties were previously observed and discussed in Ref. 1-5.</p>
<p>2 1 Ideal knots We learnt at school that the circle is the ideal planar geometric figure, since, amongst planar objects, it has the highest area to circumference ratio. Similarly the sphere is the ideal solid, since it has the highest volume to surface area r...</p>

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