Ideal Knots

426 Pages·1998·19.37 MB·English

by  Andrzej Stasiak

K(pE Series on Knots and Everything - Vol. 19
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Editors
A. Stasiak, V. Katntch and L H* Kauffman
World Scientific
IDEAL KNOTS
SERIES ON KNOTS AND EVERYTHING
Editor-in-charge: Louis H. Kauffman
Published:
Vol. 1: Knots and Physics
L. H. Kauffman
Vol. 2: How Surfaces Intersect in Space
J. S. Carter
Vol. 3: Quantum Topology
edited by L. H. Kauffman & R. A. Baadhio
Vol. 4: Gauge Fields, Knots and Gravity
J. 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
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 Physics
T. Bastin & C. W. Kilmister
Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics
W. M. Honig
Vol. 11: History and Science of Knots
edited by J. C. Turner & P. van de Griend
Vol. 12: Relativistic Reality: A Modem View
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
K CE Series on Knots and Everything - Vol. 19
IDEAL KNOTS
Editors
A. Stasiak
Universite de Lausanne
V. Katritch
Rutgers University
L. H. Kauffman
University of Illinois, Chicago
S World Scientific
`t Singapore•NewJerse Y• London -Hong Kon9
Published by
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Title of cover: Plato, Kelvin and ideal trefoil
Cover done by Ben Laurie.
IDEAL KNOTS
Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd.
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V
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.
vi
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.
VII
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.
Andrzej Stasiak
Vsevolod Katritch
Louis H. Kauffman
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ix
CONTENTS
Preface v
Chapter 1 Ideal Knots and Their Relation to the Physics of Real Knots 1
(6 colour plates)
A. Stasiak, J. Dubochet, V. Katritch and P Pieranski
Chapter 2 In Search of Ideal Knots 20
P Pieranski
Chapter 3 Annealing Ideal Knots and Links: Methods and Pitfalls 42
(2 colour plates)
B. Laurie
Chapter 4 Knots with Minimal Energies 52
Y. Diao, C. Ernst and E. J. Janse van Rensburg
Chapter 5 The Writhe of Knots and Links 70
E. J. Janse van Rensburg, D. W. Summers and S. G. Whittington
Chapter 6 Minimal Lattice Knots 88
E. J. Janse van Rensburg
Chapter 7 Minimal Edge Piecewise Linear Knots 107
J. A. Calvo and K. C. Millett
Chapter 8 Entropy of a Knot: Simple Arguments About Difficult Problem 129
A. Yu. Grosberg
Chapter 9 Approximating the Thickness of a Knot (4 colour plates) 143
E. J. Rawdon
Chapter 10 Energy Functions for Knots: Beginning to Predict Physical
Behaviour(2 colour plates) 151
J. Simon