Energy of Knots and Conformal Geometry

K(pE Series on Knots and Everyth ing - Vol. 33 ENERGY OF KNOTS AND CONFORMAL GEOMETRY JUN O'HARA World Scientific EN...

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K(pE Series on Knots and Everyth ing - Vol. 33

ENERGY OF KNOTS AND CONFORMAL GEOMETRY JUN O'HARA

World Scientific

ENERGY OF KNOTS AND CONFORMAL GEOMETRY

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Series on Knots and Everything — Vol. 33

ENERGY OF KNOTS AND CONFORMAL GEOMETRY

JUN O'HARA Department of Mathematics Tokyo Metropolitan University Japan

U S * World Scientific w l

New Jersey • London • Singapore •• Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data O'Hara, Jun. Energy of knots and conformal geometry / Jun O'Hara p. cm. — (K & E series on knots and everything ; v. 33) Includes bibliographical references and index. ISBN 981-238-316-6 (alk. paper) 1. Knot theory. 2. Conformal geometry. I. Title. II. Series. QA612.2.036 2003 514'.224-dc21

2003041104

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2003 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.

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.

Printed in Singapore.

& #L ® •![ u l£
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Preface

Energy of knots was introduced to produce an "optimal embedding" of a knot, a beautiful knot which would represent its knot type. The basic philosophy due to Fukuhara and Sakuma independently is as follows. Suppose there is a non-conductive knotted string which is charged uniformly in a non-conductive viscous fluid. Then it might evolve to decrease its electrostatic energy without intersecting itself because of Coulomb's repulsive force until it comes to a critical point of the energy. Hence we might be able to define an "optimal embedding" of a knot by an "energy minimizer", which is an embedding that attains the minimum energy within its isotopy class. Thus our motivational problem can be stated as: Can we define an "energy" on the space of knots so that its gradient flow can evolve a knot to an "optimal embedding" while preserving its isotopy type? For this purpose, our energy should blow up if a knot degenerates to a "singular knot" with double points since a knot might change its knot type through a crossing change. The first example of such an energy, E^2\ for smooth knots was defined as the renormalization of a modified electrostatic energy of charged knots under the assumption that the absolute value of the repulsive force between a pair of unit point charges is proportional to the inverse cube of the distance. In Part 1 we introduce several types of the knot energy and study the problem of whether or not there is an "energy minimizer" in each knot type. We first give a family of repulsive energies, which are obtained as the vii

Preface

Vlll

renormalization of modified electrostatic energy of charged knots. Then the answer to the above problem may depend on the following three conditions; the power exponent with which the distance between a pair of points on a knot are integrated, the primeness of a knot, which is a topological condition, and the metric of the ambient space, which is a geometric condition. We then introduce geometrically defined knot energies of another kind, such as ropelength. Roughly speaking they measure how much a knot can be fattened, or how tight a rope can be knotted. The interest in knot energies arose naturally when we regarded the knots as real objects. They have concrete backgrounds and applications in various fields of natural science. We introduce some results of numerical experiments. In Part 2 we study conformal geometry. The results in Part 2 were obtained in joint work with Remi Langevin, summarized in [LOl], from which we cite frequently. The first example of the knot energy, was eventually proved to be invariant under Mobius transformations of R3 U {oo} by Freedman, He, and Wang. Allowing the action of a large group as the Mobius group has both advantages and disadvantages. If a knot is an energy minimizer then its image by any Mobius transformation is again an energy minimizer of the same knot type (or possibly, its mirror image). On the other hand, conformal geometry could provide a new viewpoint to knot theory. We use the set of spheres in S3, which forms a 4-dimensional hyperbolic hypersurface in Minkowski space R 4,1 . Using a sphere that is tangent to a knot K at two points we introduce a complex valued (meromorphic) 2-form on K x K which will be called the infinitesimal cross ratio. This 2-form will give another interpretation of E^ and also it can express other conformally invariant energies. Finally an energy defined from an integral geometric viewpoint will be introduced. The energies presented in this book are defined geometrically and measure the complexity of embeddings. The study of our these subjects is called physical knot theory.

Acknowledgments The author is sincerely grateful to Gregory Buck, Jason Cantarella, Kenji Fukaya, Shinji Fukuhara, Yutaka Kanda, Nariya Kawazumi, Masanori Kobayashi, Rob Kusner, Tomotada Ohtsuki, Kaoru Ono, Makoto Sakuma,

Preface

IX

Jonathan Simon, Hiro-o Tokunaga, Andrzej Stasiak, Takashi Tsuboi, and especially, Remi Langevin, for helpful suggestions and conversations. The author deeply thanks his colleague Martin Guest for correcting the English translation. He also thanks Rob Kusner and John Sullivan for the permission of the use of their figures of the optimal knots from [KS1'] and from [Kusl], a web page of the Center for Geometry, Analysis, Numerics and Graphics (GANG), in Section 5.3, Chapter 6, and the cover page of this book. The author appreciates the kind hospitality of Laboratoire de Topologie, Universite de Bourgogne during his stay in Dijon, France to study with Remi Langevin. Some of the sections in Part 2, in particular Section 13.6, are based on the manuscript for [LOl] by Langevin.

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Contents

Part 1 knot

In search of the "optimal embedding" of a 1

Chapter 1 Introduction 1.1 Motivational problem 1.2 Notations and remarks

3 3 8

Chapter 2 a-energy functional E^ 2.1 Renormalizations of electrostatic energy of charged knots . . . 2.2 Renormalizations of r~ a -modified electrostatic energy, E^ . . 2.3 Asymptotic behavior of r~a energy of polygonal knots 2.4 The self-repulsiveness of £ ( a )

9 9 15 22 26

Chapter 3 On E^ 31 3.1 Continuity 31 3.2 Behavior of E^ under "pull-tight" 34 3.3 Mobius invariance 34 3.4 The cosine formula for E^ 35 3.5 Existence of E^2' minimizers 42 3.6 Average crossing number and finiteness of knot types 44 3.7 Gradient, regularity of E^ minimizers, and criterion of criticality 45 3.8 Unstable i?(2)-critical torus knots 54 3.9 Energy associated to a diagram 57 xi

xii

Contents

3.9.1 General framework 3.9.2 "X-energy" 3.10 Normal projection energies 3.11 Generalization to higher dimensions

58 59 60 60

p

Chapter 4 L norm energy with higher index 4.1 Definition of (a,p)-energy functional for knots ea,p 4.2 Control of knots by Ea'p {ea'p) 4.3 Complete system of admissible solid tori and finiteness of knot types 4.4 Existence of Ea'p minimizers 4.5 The circles minimize Ea'p 4.6 Definition of a-energy polynomial for knots 4.7 Brylinski's beta function for knots 4.8 Other L -norm energies

63 63 65

Chapter 5 Numerical experiments 5.1 Numerical experiments on E^ 5.2 a > 2 cases. The limit a s n ^ o o when a > 3 5.3 Table of approximate minimum energies

83 84 84 86

Chapter 6

91

Stereo pictures of E^

minimizers

Chapter 7 Energy of knots in a Riemannian manifold 7.1 Definition of the unit density (a,p)-energy E™^ 7.2 Control of knots by E°$ 7.3 Existence of energy minimizers 7.4 Examples : Energy of knots in S3 and H 3 7.4.1 Energy of circles in S3 7.4.2 Energy of trefoils on Clifford tori in S3

75 77 78 79 79 81

101 101 102 105 107 107 109

(2)

7.4.3 Existence of Eg£ minimizers 7.4.4 Energy of knots in H 3 Other definitions The existence of energy minimizers

109 110 114 115

Chapter 8 Physical knot energies 8.1 Thickness and ropelength 8.2 Four thirds law 8.3 Osculating circles and osculating spheres

117 117 118 119

7.5 7.6

Contents

8.4 8.5 8.6 8.7

xiii

Global radius of curvature 121 Self distance type energies denned via the distance function . . 127 Relation between these geometric quantities and ea'p 131 Numerical computations and applications 132

P a r t 2 Energy of knots from a conformal geometric viewpoint 133 Chapter 9 Preparation from conformal geometry 9.1 The Lorentzian metric on Minkowski space 9.2 The Lorentzian exterior product 9.3 The space of spheres 9.4 The 4-tuple map and the cross ratio of 4 points 9.5 Pencils of spheres 9.6 Modulus of an annulus 9.7 Cross-separating annuli and the modulus of four points 9.8 The measure on the space of spheres A 9.9 Orientations of 2-spheres

....

Chapter 10 The space of non-trivial spheres of a knot 10.1 Non-trivial spheres of a knot 10.2 The 4-tuple map for a knot 10.3 Generalization of the 4-tuple map to the diagonal 10.3.1 Twice tangent spheres 10.3.2 Tangent spheres 10.3.3 Osculating spheres 10.4 Lower semi-continuity of the radii of non-trivial spheres . . . . Chapter 11 The infinitesimal cross ratio 11.1 The infinitesimal cross ratio of the complex plane 11.2 The real part as the canonical symplectic form of T*S2 11.3 The infinitesimal cross ratio for a knot 11.4 From the cosine formula to the original definition of E^ 11.5 K^2)-energy for links Chapter 12 The conformal sin energy Es-mg 12.1 The projection of the inverted open knot 12.2 The geometric meaning of Es-ln g

135 135 143 144 147 151 156 158 165 168 175 175 177 179 179 183 184 186

189 189 . . . . 191 201 . . . 205 207 213 213 215

xiv

Contents

12.3 Self-repulsiveness of Esjn Q 12.4 ES[ng and the average crossing number 12.5 Esin6 for links

216 221 221

Chapter 13 Measure of non-trivial spheres 223 13.1 Non-trivial spheres, tangent spheres and twice tangent spheres 223 13.2 The volume of the set of the non-trivial spheres 224 13.3 The measure of non-trivial spheres in terms of the infinitesimal cross ratio 226 13.4 Non-trivial annuli and the modulus of a knot 231 13.5 Self-repulsiveness of the measure of non-trivial spheres 235 13.6 The measure of non-trivial spheres for non-trivial knots . . . . 237 13.7 Measure of non-trivial spheres for links 241 Appendix A Generalization of the Gauss formula for the linking number 243 A.l The Gauss formula for the linking number 243 A.2 The writhe and the self-linking number 245 A.3 The total twist 247 A.4 Average crossing number 249 A.5 The conformal angle and the Gauss integral 250 A.6 Mobius invariance of the writhe 252 A.7 The circular Gauss map and the inverted open knots 253 Appendix B The 3-tuple map to the set of circles in S3 B.l The set of unoriented circles in S3 B.2 The 3-tuple map to the set of circles

257 257 257

Appendix C Conformal moduli of a solid torus C.l Modulus of a 2-dimensional annulus revisited C.2 Conformal moduli of a solid torus

259 259 260

Appendix D

Kirchhoff elastica

263

Appendix E

Open problems and dreams

265

Bibliography

271

Index

285

PART 1

In search of the "optimal embedding" of a knot

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Chapter 1

Introduction

1.1

Motivational problem

A knot / is an embedding from a circle S1 into the 3-dimensional Euclidean space M3 or the 3-sphere S3. We always assume that a knot / is at least of class C 1 and its derivative / ' never vanishes. Its image K — /(S1) is also called a knot. A knot in a 3-dimensional manifold can be considered similarly. We assume that the circle 5 1 is oriented and hence a knot K = / ( S 1 ) is also oriented via / . Let M denote E 3 or S3. Two knots / and / ' in M are called isotopic if there is an isotopy ht : M —> M (t £ [0,1]) of the ambient space such that ho is equal to the identity map and that the map (x,t) >->• (ht(x),t) from M x [0,1] to itself is a homeomorphism. Then two knots are isotopic if and only if there is an orientation preserving homeomorphism h of M that satisfies / ' = h o / . A knot type [K] a knot K is an isotopy class of K. We wish to study how to identify the knot type of a given knot diagram. Unfortunately there are no knot invariants that have been proved to detect all the knot types completely. Sakuma [Sak2] and Fukuhara [Fukl] proposed a new geometric approach. Suppose we have a suitable functional on the space of knots, which will be called an "energy". Suppose the negative gradient flow of this "energy" can evolve a given knot to a critical embedding while preserving its isotopy type. If we make the optimistic assumption that a knot can always reach a local minimum position of the "energy", and that the number of the critical points, or at least, the critical values of the "energy", is finite in each knot type, then we can use this 3

Introduction

4

"energy" to detect the knot type of a given knot diagram. In order to guarantee the invariance of the knot type while a knot evolves itself, crossing changes should not be allowed. Thus we require that our "energy" should blow up if a knot degenerates to a "singular knot" with double points. In other words, our "energy" is infinite for immersions that are not embeddings.

e A

{immersion} Fig. 1.1

knot

type

A knot energy functional and a singular knot.

We present a conceptual figure in Figure 1.1. Each cell which corre-

Motivational

problem

5

sponds to a knot type is surrounded by infinitely high energy walls at the boundary, which is a subset of the set of immersions that are not embeddings. The "configuration" of the space of immersed circles depends on the topology. We work mostly with the C2-topology. If we work with the C°-topology we get into trouble because a knot may change its knot type continuously with respect to the C°-topology without having any selfintersections, as will be explained later. If a knot K attains the minimum value of the "energy" within its isotopy class [K], then we call it an "energy minimizer". It should be beautiful in some sense. When a knot type [K] has any symmetry, it is desirable that an energy minimizer within [K] has the same symmetry, although it is not always possible because there is a knot type which has at least two kinds of symmetries that cannot be realized at the same time. Definition 1.1

Let e be a real valued functional on the space of knots.

(1) A functional e is called self-repulsive if the value e(fn) of a sequence of knots /„ blows up if e(/ n ) converges with respect to the C°topology to an immersion with a double points. (This property is called charge in [DEJvR2].) (2) A functional e is called self-repulsive with respect to the C2topology, or C2 self-repulsive for short, if the value e(/ n ) of a sequence of knots /„ blows up if e(/„) converges with respect to the C 2 -topology to an immersion with a double point. (3) We call e a knot energy functional (or knot energy functional with respect to the C 2 -topology) when it is self-repulsive (or respectively, self-repulsive with respect to the C 2 -topology) and bounded below. We always assume that e is invariant under reparametrization of knots and the action of the group of the motions of the ambient space M3 or S 3 , R3 ix 0(3) or 0(4). Let IK denote the total length of a knot K and di<(x,y) denote the shorter arc-length between a pair of points x and y on K. Put 3 .., r ,. [ TJ. , x,y yE K such that 1 J(r{6,d) = {K: a, knot ) , , } I dK(x, y) =SlK and \x - y\ < dlk J

,, 1N (1.1)

for 0 < d < S < y , and JtTx = {/: S1 = [ 0 , 1 ] / ~ ^ M 3 : a knot : | / ' ( i ) | = 1 ( v f) and /(O) = 0 }.

6

Introduction

Since J^i is equicontinuous and uniformly bounded, any sequence of knots in Jt{ has a convergent subsequence by the Ascoli-Arzela theorem. L e m m a 1.1.1 Suppose e is a functional on the space of knots in M3 which is scale invariant, i. e. invariant under the homothety of M3. Then a functional e is self-repulsive if and only if lim

inf

d->+0

KeJ(T(S,d)

e(K) = oo

for any S with 0 < 5 < \ . Proof.

The "if part is obvious. Suppose lim d->+0

inf

e(K) < B

KeJC(S,d)

for some S and B < oo. It suffices to consider e restricted to the space Jif\. Then we can take a sequence of knots fn G J(T(S, -—) D J^[ such that e(/ n ) < B. The limit of a convergent subsequence of /„ is an immersion with a double point. Therefore e is not self-repulsive. • Definition 1.2 C 2 -topology).

Let e be a knot energy functional (with respect to the

(1) The energy e([if]) of a knot type [K] is given by e([K])=

inf

e(K').

If a knot K\ G [K] attains this infimum we call K\ an energy minimizer of a knot type [K] with respect to a knot energy functional e, or an eminimizer for short. (2) A knot energy functional e is called minimizer producing if there is an e-minimizer in every knot type. A knot energy functional is not necessarily minimizer producing (see Conjecture 3.1). Suppose e is a scale invariant functional on the space of knots in M.3. Suppose a sequence {-£Q} G J?[ in a given knot type [K] satisfies lim^oo e(Ki) e([K]). The difficulty lies in the fact that even if a subsequence Kni of Kn converges pointwise to a knot Koo, the limit K^ does not necessarily belong to the same knot type [K]. Definition 1.3 Let {K{\ be a sequence of knots of the same knot type [K] which converge pointwise to a knot Koo. Assume that the length l{Ki)

Motivational

7

problem

of Ki satisfies ^o < l(K-i) — ^o for all i for some positive constants IQ and LQ. Suppose Ki contains a 1-tangle (i.e. a knotted arc) T; of the same knot type [K'\ in a 3-ball Bi whose diameter decreases to 0 as i goes to oo. Then K^ belongs to a different knot type from the original one with [K\ = [K0O$K1]. Then we say that {Ki} is a pull-tight sequence or Ki pulls tight.

Fig. 1.2

Pull-tight.

We shall study if a knot energy functional e has the following properties. Definition 1.4

Let e be a functional on the space of knots.

(1) e is called basic if only circles attain the minimum value of e(K) ([DEJvR2]). (2) e is called tight if e(Ki) blows up for any pull-tight sequence ([DEJvR2]). (3) e is said to distinguish the unknot if inf ft': trivial

e{K) <

inf

e(K').

K': non — trivial

(4) e is said to be strong ([DEJvR2]) if for any given real number b there are only finitely many knot types [K] such that e([i^]) < b. In this case, we say that the upper bound for e induces the finiteness of knot types. (5) e is called rigid if there are only finitely many e-minimizers in each knot type. Our motivational problem is to look for a knot energy functional which produces a beautiful "optimal embedding" for every knot type as an energy minimizer.

Introduction

8

1.2

Notations and remarks

In order to simplify formulae, we often assume that knots are parametrized by arc-length. When we consider knots in R 3 , we often assume further that knots are rescaled to have length 1 by a similarity. In particular, whenever we denote a knot by a letter h we assume both, i.e. h is parametrized by arc-length and has length 1. The circle Sl is considered to be [0, l ] / ~ = R/Z = R mod 1. When a knot is given by an embedding / : S1 = R/Z -5- R3 ( o r S 3 ) , its lift / : R -» R 3 (orS*3) with period 1 will also be denoted by / . Since all the functionals we study in this book are invariant under re-parametrization of knots, we abuse notation by writing e(K) for e(/). We always assume that knots are at least of class C 1 with nonvanishing differentials. We may assume further differentiability according to the functionals we are interested in. We denote the standard norm of R3 by | • |. K denotes a knot as an image, f,g,h,--denote embeddings, x, y, • • • denote points on K, and s,t,- • • denote parameters in S1. In Part II we study conformal geometry using Minkowski space R 71-1 ' 1 . We denote by (, ) the Lorentzian pseudo inner product, whereas the standard Euclidean inner product is denoted by (, ). We use the symbol ± only in the case of the Lorentzian pseudo inner product. We also use the convention that x = f(s), x + dx = f(s + ds), y = f(t), and y+ dy = f(t + dt).

Chapter 2

a-energy functional

E^

One of the most natural and naive candidates for a knot energy functional would be the electrostatic energy of charged knots. Another candidate would be the bending and twisting energy of elastic knots which will be considered later in 9.2. The first attempt to consider the electrostatic energy of charged knots was made in the finite dimensional category by Fukuhara [Fukl]. He considered the space of the polygonal knots whose vertices are charged, and studied their modified electrostatic energy under the assumption that the absolute value of Coulomb's repulsive force F(P, Q) between a pair of unit point charges P, Q of distance r is proportional to r~m (m = 3,4,5, • • •). The reason for this modification is given in what follows. Fukuhara wrote a computer program to evolve a knot to decrease this energy.

2.1

Renormalizations of electrostatic energy of charged knots

Let us begin with some heuristic remarks. Let S1 = [0, l ] / ~ = R/Z = K mod 1 and h : S1 —> M3 be a knot that is parametrized by arc-length, namely, \ti{t)\ = 1 ( ^ e S 1 ) . Suppose that a knot K = h{Sl) is electrically charged uniformly. (We need this assumption of uniformity for the following reason. Suppose electrons can move freely along a knot. Consider a knot which is almost the same as a circle except for a small tangle. Then its "electrostatic energy" can be close to that of a circle if the tangle part carries a small charge. Thus every knot will degenerate to a circle in order to decrease its "electrostatic energy".) 9

E(a>

a-energy functional

10

Fig. 2.1

A knot which is barely charged at a small tangle.

Then the "voltage" at a point x — h(s) is given by

f

dy

dt

f

JK\x-y\

Jsl\h(a)-h(t)\'

and twice the "electrostatic energy" of K = ^(S 1 ) is given by

rr

dxdy

rr

=

JJKXK\X-V\

J

Jsixsi

dsdt \h(s)-h(t)y

(2-1)

These blow up for any knot because \h(s) — h(t)\ < \s — t\ implies f dt Js^\h{8)-h{t)\-

p du _ i_^\-°°-

The blow up of the "electrostatic energy" is caused by the contribution of the neighborhood of the diagonal A = {(x,y) G K x K \ x = y} to the integral. Thus we can obtain a finite valued functional by getting rid of it in the following way. (Our method is essentially based on a subtraction of type oo — oo. We will see later that this is not the only method.) Assume that the knot is charged except for a short subarc h((s — e,s + e)) = {y G K\dx(x, y) < e} around the point h(s), where dx(x,y) denotes the shorter arc-length (see Figure 2.2). Let V} '(K;x) — V} (h;s) be the voltage at the point x = h(s): V?\K;x) = VP{h-ts)= I JdK(x,y)>e

-A T =/" +1 " F ~ V\

Js-e

dt \h{s)-h{t)[

We call it the e-self avoiding voltage at x = h(s). We integrate this voltage VE{1)(K; x) over K to obtain the e-off diagonal electrostatic

Renormalizations

of electrostatic

energy of charged knots

11

charged

Fig. 2.2

energy E™ {K) = E^\K)

e-Self avoiding voltage.

E?\h):

= E^>{h)=

f V£{1\K;x)dx= JK

f JS1

V^\h-s)ds.

Let N%(A; S1) be the complement of the —^-tubular neighborhood of the diagonal A of S1 x S1: N^(A- S1) = {(a, t) e [0,1] x [0,1] : £ < \s - t\ < 1 - e}.

(2.2)

Then E[1]'(h) is the contribution of N£(A; S1) to the integral of the "electrostatic energy". Namely,

E™(h) =

dsdt

dsdt

/JV|(A ; SI) \Ks) ~ h(t)\

J y £ <| s _ 4 |
h(t)[

We show that although lim Ve{1\h; s) = lim E^(h)

£->0

£->0

= +oo

for any knot h and for any point h(s), the order of blowing up as e —> 0 is independent of the knot and the point on it. Let K = max t € 5i |/i"(£)| be the maximum of the curvature of a knot hiS1). Fix s £ S1 and define 9(u) > 0 by cos6>(u) = (h'(s),h'(s + u)).

E^a>

a-energy functional

12

Then 8(u) < n\u\ for any u. Therefore for 0 U

r-u

+ u)\ > / cos 0(t)dt > / cos ntdt Jo Jo

u > \h(s) -h(s

= — sinKu.

(2.3)

Hence for any e\ and e 2 with 0 < E\ < e 2 < — K

2

du r'^
which implies 2(loge 2 - logei) < V$\h; „ /,

a) - V™{h; s) K£2

< 2 (Jogtan —

,

K£± \

log tan —j

.

Therefore 0 < (VeW(h;8) + 21og £ l ) - (Vg\h;

s) + 21og£2)

< 2 { ( l o g t a n ^ - l o g £ 2 ) - (log tan ~

- logei) } .

(2.4)

Since ,.

(,

l £ J^+o

K£

\

K

Vlostan y ~ loseJ = los 2

Ve(1)(/i;8) + 21oge converges uniformly as e decreases to 0. Put VW(h; s) = Urn (Ve(1) (h; s) + 2 log 2e) . We remark that we added log2 to make it match the formula (2.14). The uniform convergence implies that V^(h;s) is a continuous function of s and e

lim ( [ V}1){h;s)ds + 2\og2£) = f ^ + ° \Jsi ) Js*

V{1){h;s)ds.

Let us denote it by E^\h), and call it the renormalization of electrostatic energy or . E ^ - e n e r g y for short. Then E{l\h)

= lim(£W(/i) + 21og2e). e—>0

Renormalizations

Example 2.1

of electrostatic energy of charged knots

13

Let h0(t) = ( ^ cos27rf, ^ sin27ri, 0) be a circle. Then Js+£

\h0(s) - h0(t)\

s+1 e

_ r Js+£

ndt sinir(t-s) ns = —2 log B tan — . 2 Hence E^(h0)

= lim e

-.

^ ° J JE<\s-t\
:

-=21og-.

SmTC\s ~t\

7T

We shall introduce a double integral formula for E^\ h0 as a test function. Then EW(h)

= lim^i1)^) -

- EW(h0)

Let us fix the circle

E^{h0))

e—J-0

^°J

Je<\s-t\
-h(t)\

sin7r|s-t'

Then (2.3) implies 1

TT

u

1

1

|^(s) --h(s + u)\

sin7ra

K

\h0(s) - h0(s + u)\

7T

<

sin7ru

sin/cw for 0 < u < ?-. Since —

—

K

/l

u^O \_ U

\

7T

sillTTuJ

lim

( 1

K

I

B-toVsinra

\

1

„ f)

sin7rw/

the integrand 1

\h(s)-h(t)\

1

\h0(s) - h0(t)\

is bounded on S1 x S1 \ A. Thus we have

E^\h)-E{-1\h0)= V

'

V

ff

(.,, . * L, .,

1

r)dsdt.

' J Jsixsi \\h(s)-h{t)\ smn\a-t\J We obtain a double integral formula for E^ by subtracting a counter term to cancel the blow up of the integral around the diagonal. But the

E^a'

a-energy functional

14

counter term is not unique. Any term which blows up in the same order as \h(s) — /i(t)| _ 1 at the diagonal can serve as a counter term. For example, we can take .{dx(h(s), h(t))}~1 as a counter term as is explained in the next section, where d,K(h(s),h(t)) is the shorter arc-length between h(s) and h(t). Thus

(,,, . l , . ., - , „ , \ ,, ^dsdt.

E^(h)=ff

JJs^sA\Hs)-h(t)\

(2.5)

dK(h(s),h(t))J

The integrand is non-negative as \h(s) — h(t)\ < d(s,t) and not identically 0. Therefore E^^h) > 0 for any knot h. We remark that d/f (/i(s), h(t)) is equal to the arc-length d(s, t) between s and t on S 1 : d(s, t) = min{|s — t\, 1 — \s — t\} since the knot h(S1) is parametrized by arc-length. Theorem 2.1.1 (1) E^ is not self-repulsive, i. e. it takes a finite value on a "singular knot" with a self-intersection. (2) E^1' cannot distinguish the unknot, i.e. inf

E(1\K)=

K: trivial

E(l\K').

inf . K': non—trivial

Proof. (1) Suppose h is a "singular knot" with a right-angled double point P.

f(t,o,o) h(t) = \ .

x x

Si

(-\
i

n

(2-6)

Then the contribution of the neighborhood of P to the integral is given by

f i r\ + \ dsdt _ [TT j-T dsdt J-\J±-\ \Ks) - h(t)\ ~ i - i i - i V^TW

f"Tr rf rd0dr ~ Jo Jo ~r~

<0

°'

(2) Consider a pull-tight sequence of knots Ki of the same non-trivial knot type [K] which converges pointwise to a trivial knot K^. Suppose Ki has a tangle Tj of knot type [K] which is contained in a 3-ball of diameter 2~l. Assume that the Tj's are similar. Then the contribution of Tn x Tn to the integral of Ef'(Kn) is half of that of T n _i x T„_i to Ef'{Kn_i) as

Renormalizations

of r "-modified

electrostatic

energy,

E^a'

15

h(0) = ha)

Fig. 2.3

A planar figure eight singular knot.

the integrand is doubled, although Area(T n x Tn) = Area(T n _i x T ra _i). Therefore l i m ^ ^ E^{Kn) = £«(#oo)U 2.2

Renormalizations of r "-modified electrostatic energy,

In order to obtain a repulsive functional let us increase the power of \h(s) — h(t)\ in the integral. Definition 2.1 We make a non-physical assumption that the absolute value of Coulomb's repulsive force between a pair of unit point charges of distance r is proportional to r~^a+1^ where a > 1. Let the r~"-modified £-self avoiding voltage at a point x = h(s) Ve (K; x) = V}a (h; s) be the "voltage" at a point x = h(s) the subarc h([s + s, s + 1 — e]) = {y S K : dx{x,y) > e} is charged. V£(-a)(K;x) = V£{a\h;s)

s+l-e

dy dK(x,y)>e F ~~ Vr

s+e

dt \h(s)-h(t)\" (2.7)

{ Integrate V.(<*) e ' (K; x) over K to obtain the r "-modified £-off diagonal electrostatic energy E{£a){K) = E{£a)(h):

Eia){K)

= E^(h)=

f

Js1

V}a)(K;x)dx=

f

Js1

V£(a\h;s)ds.

Then E(£a) (h) is the contribution of the complement N£(A; Sl) of the (-^=-)

a-energy functional

16

E^a'

tubular neighborhood of the diagonal A of S 1 x S1 to the integral of the "electrostatic energy". Namely, Ei£a)(K)

=

E{ea\h)

_ f f

dxdy

rr

dsdt

(2-8)

Let us consider the blowing up behavior of V}a' (h; s) as e decreases to 0. To see this, let us fix s G S1 and give an expansion formula of \h(s+u) — h(s)\~a in terms of u. Let h^ denote h^k\s). We expand \h(s + u) - h(s)\2 first. Since (h'(t),h'{t)) = 1 for alU 6 S\ dn~2 o (1) If n = 2m + 1 for some m > 1 then E 2 m - i ^ _ 1 ( / i ( f c ) , / l ( 2 m + l - f c ) ) = o, fc=l

hence m

(h',h^)

= -J22m-lCk-1(h(k\h^m+1-k'>).

(2.9)

fc=2

(2) If n = 2m for some m > 2 then m— 1

2 £ 2m- 2 C fc -l(ft (fc) , /l( 2m " fc )) + 2 m - 2 C m _ 1 ( / l W , / l ( m ) ) fc=l

=

0,

hence m-l

(h',^2™-1)) = - £

„ 2m fc - )) 2m - 2 c fc _ 1 (/i«,/i(

2m

~ 2 ° m - 1 |fe^)| 2 .

fc=2

(2.10)

Renormalizations

a

of r

-modified

electrostatic energy, #(<*)

17

On the other hand, by Taylor's theorem \h(s + u)~ h(s)\2 = {h(s + u)-

/ m (/l (fc) )/l ( 2m +i-fc)^ V ^ \10' ' " • ' J_ \ „,2m+l I V ^TTTTT : 7T7- I U

I 2 _L 2 V ^ = U~ + 2 V

'

r ro_1

4^1 777, — J,

h(s),h{s + u) - h(s))

(/l)

\h^\2\

A!(2m-ifc)!

(m!) 2

^ V.

2m

r C — J.

+0{uM+z).

(2.11)

Note that the coefficient (h', h") of t3 is 0. Using (2.9) and (2.10) to delete the terms of the form (h', h^n~l^)tn in (2.11), we have \h(s + u) - h{s)\2 = u2

V> 1 J 9 V^1 ^ 2 m - l i ^ m=2

I,

fc(2m-fc)-2m fe!(2m-fc)! v

fe=2

+ l

E m11 1ZI ^^,

m=2

J. J ^

m

1 -- ft-; k) -- 4in 2mfu((k) fk(2m i ^ f i - l+ - -1-

Lfc=2

fc!(2m fcl^m

'

(2m _ fe) , j

y

+i /

(fc) l

+ l-fcM l-fe)!

^

w!i/i(m)i2r2m (2m+l-fc)\ I „ 2 m + l

'

M

+0( M 2 <+ 2 ). Calculation of the first few terms gives ..2 \h(s + u) -h(s)\2 2 _=u 4

l^"|24

12 -u

3

(^"^(3))«

12

2

(/i",^ )) 40

|/i( )| \ 45 /

(/i",/i(6)) 1008

2(M3>,fe(5>) 315

6

((h",hW) V 180

(h^,h^) 36

u7

|feW|2' 448

Next, we give an expansion formula of \h(s + u) — h(s)\~a for a > 0. Define

18

a-energy functional

E^a>

at = a,i(h;s) (i > 4) by

h(s)\2 - u2 - ^2 anUn-

\h(s + u)-

n—4

Since

/,

A»

,

a a

J.

(

2

(i + o = i + < + for |£| < 1, \h(s + u) — h(s)\

\h{s + u) -

~ 1) , 2

g +- — ^

2

a

a ( a — l ) ( a —2)

A,

Z+

can be expanded in u as

h{s)\'a

= u-a {1 - (a 4 u 2 + a 5 « 3 + • • • + a ^ u ™ " 1 + 0 ( u m ) ) } ~ f

=u a 1 +

~{

+

Y ^4"2 + asu3 + "'+ am+ium~1 + 0 (" m )) a^a

J ^2^4

+

2a 4 a 5 U 5 + [a\ + 2a 4 a 6 )u 6 H

hO(t

8 +

a ( a + 2)(a + 4) ^

= U " ^ 1 + — a4U

+ TT a 5M

+

+

. _. +

Q{um+A))

TTa6 + ~ ^

+

~a4

... |

u

, a a ( a + 2) \ * +1 y a 7 H T 0405 I u . a a(a + 2), , , a ( a + 2)(a + 4)L 3 \ 66 + ( y«s + > s + «4a6) + 7J a\\ u g

+--- + 0(um)

Renormalizations

+

" +

a

of r

-modified electrostatic

+

24

a f(/i",/iW) 2 \ 40

+

energy, #(<*)

19

24

|fe(3)|2 (a + 2)|h"| 4 ) + 45 576 J

4-a

I " f(fe(2)'fe(5)) | (^3)^(4)) , (a + 2)\hr(h",h^)} u5~a 72 ^

5

8

J

+ --- + 0(um-a).

(2.12)

Since

r

p u

du i =< +oo oo

p < —1 p > —1

we can take the sum of the terms in (2.12) whose exponent is smaller than or equal to — 1 with as a counter term to cancel the blow up of the integral of \h(s) - h(s + u)\~a. Definition 2.2 ([08]) Let ip^\s,t) nent at most —1 in:

be the sum of the terms with expo-

\h(s) - h(t)\-a - d(s,t)-a -

24

a(h"(s),hW(*))Jf_^-a 3 d(s,t) 24 (4) (fc"(s),ft (*)) , \h^(s)\2 a + 40 ^ 5 ^ "~2~ a

'72

+

^Mld(s,tf-

, (a + 2)\h"(s)\ 576

\4 —a

W*>

|(^) ( y)( S )) +(fe(3)(sU(4)(s)) (a +

2)\h"(S)\Hh"(s),hW(s)).5_a

(2.13)

Namely, ^ (s,£) be the first max{2, [a]} terms of (2.13), where [a] is the greatest integer that does not exceed a. Define the renormalization of r _ a - m o d i f i e d voltage by

V™(h;s) = f1 ipia)(s,t)dt, Js

a-energy functional

20

E^a>

and the renormalization of r "-modified electrostatic energy , or a-energy for short, by f V{a\h;s)ds Js1

E^{h)= Therefore E^(h)

ip{ha\s,t)dsdt.

= ff J JSixS1

is given according to the value of a as follows.

(1) When 1 < a < 3 V ( a ) ( M ) = /" ( .,. . 1 , . ., 1 Jsi \\h(s) - h(t)\<* E{-a\h)=

f f

( ,,,

1 x

_!_)<&, d(s,t)»J

, / .,

(2.14)

-^—)dsdt.

(2.15)

Since the integrand is non-negative and not identically zero E(a\h) for any knot h when 1 < a < 3.

>0

(2) When 3 < a < 4

F («> m _

ff

(

l

WW2 y . , .

i

(3) When 4 < a < 5 F(«)(hs = 1J

ff ( i L _ "l ft// ( sa)l22 yy^xsAl^)-^)! 01 d(S)*)« 24d(s7t) ~ a(h"{s),h<-3\s)) 24d(s,i)°- 3

dsdt.

E^(h) for a > 5 is given similarly. We call a the index of E^a>. The integrand of E^(h) is divided into the principal term \h(s) — h(t)\~a and the counter term to cancel the blow up of the integral. We remark that the counter term is determined up to a constant. Observe that the counter term is independent of the knot if a < 3. When 1 < a < 3 one can integrate the counter term on S1 \(s — e,s + e) to obtain

V^(h;s) = lim (Ve(°)(h;s) E-t-o \

~-^) [a — l)ea

+- ^ L

J

a —I

(2.16)

Renormalizations

of r

a

-modified electrostatic energy, E(a)

21

which implies E^(h)

= Viral e-s-0 (_

2a

E^ (h) e

[a

a

l)e ~

(2.17)

n(«) Thus the order of blowing up of E^ (h) as e —> 0 is independent of the knot h when a < 3. In this case the renormalization can be interpreted as taking the difference of the "extrinsic energtf based on the distance measured in the ambient space (chord length) and the "intrinsic energy" based on the distance measured in the knot itself (arc-length).

d(s,t)

\h(s)-h(t)\

Fig. 2.4

Arc-length and chord length.

When a > 3 the renormalization needs additional counter terms which depend on the curvature and higher derivatives. The order of blowing up of Ee (h) as e —> 0 gets complicated. For example, when a — 3 E^(h)

lim

[E?\h)-^

+

^jsi\h"{s)fds}+A.

e->-0

We remark that, for simplicity's sake, we have assumed higher differentiability of h here than is necessary for the well-definedness of the functiona l . In fact, for example, being of class C 2 is sufficient to define E^-a\h) (1 < a < 3). Remark 2.2.1 In defining voltages and energies one can use the distance in the ambient space instead of using arc-length to avoid the diagonal. Assume the knot K is charged outside the 3-ball Be{x) with center x and radius e. The voltage Ve(K; x) = V£(h; s) at point x = h(s) is then given

E^a>

a-energy functional

22

by V£(K;x)

dy

= V£(h;s) = f

\x ~ V\

J\x|ac —y|>e

Define E£(K) = E£(h) = f V£(K;x)dx = JK

dxdy

[[ J

\X-V\a

J\x-y \>e

The order of blowing up of Ee (h) as £ —> 0 can be calculated similarly. When 1 < a < 3 the order is the same as that of E£'(h). This is because if we put e = \h(s + e) — h(s)\ then e — £+ -—^- £ 3 + 0 ( e 4 ) , which implies 3 a and logs = logs + 0(e2). In particular, £ - ( a - i ) = g - ( a - i ) + 0(e ~ ) when 1 < a < 3 E™(h) = lim lE("\h) - 7 v ' e^o \ e K ' (aand when a = 3

2.3

Asymptotic behavior of r

a

^ l)ea~l

r

l + — J a - 1

energy of polygonal knots

Numerical calculations of the values of E^ have been carried out. In connection with this, we give formulae for the asymptotic behavior of the energy of charged n-gons that converges to a smooth knot as n increases to infinity. Definition 2.3 Let h : [0, l ] / ~ —> M3 be a smooth knot. Take n points equally scattered along h(Sx), and assume a point charge — at each point. n Assume (non-physically) that the absolute value of Coulomb's repulsive force between a pair of unit point charges of distance r is proportional to _a r -(a+i) w n e r e a > 1. Then the doubled r - m o d i f i e d electrostatic 1 n—1 energy of the polygonal knot with n vertices, h(0), h(-), • • • h( ), is n

given by

K°'m = Ti v

Si*m-*(o n j

n

Asymptotic

behavior of r

Fig. 2.5

a

energy of polygonal

knots

23

A polygonal knot.

En (h) blows up as n increases to oo. Let us study its asymptotic behavior. Let C be Euler's constant C = lim

1

1

1

logn oo

,

1

and ((a) be Riemann's zeta function ((a) = Y J —• n=l

Theorem 2.3.1 The asymptotic behavior of Ena(h) as n increases to oo is given according to whether a is an integer or not by the following. Here the ~ 's mean that the differences between the left hand sides and the right hand sides decrease to 0 as n increases to oo. (1) When a = 1 £( 1 >(/i)~£( 1 >(/ l ) + 2 ( l o g - + C). (2) When 1 < a < 3

E^(h)~E^(h)

+

2U(a)na-1

")« — 1

a- 1

a-energy functional

24

E^a>

In particular, we have E™(h)~E™{h)

+

^n-4.

(3) When a = 3 E?\h)

~ E™{h) + 2(C(3)n2 - 2) + 1 ( ^

\h"(s)\2ds^

(log |

+ C).

(4) When 3 < a < 4

E^\h)^E{-a\h)

+ 2(c{a)na-1

2 " a-1,

+ g (/ si l^'WI2^) (c(« - 2K- 3 -

a-3

a —3

(5) When a = 4 £( 4 )(/i) ~ £;(4)(ft) + 2(C(4)n3 - | ) + 1 ( ^ +

\h"(x)fdx^

(C(2)n - 2)

1 (fsi(h"(x)>h{3)(x))dx) 0°gy+
2

wftere £(4) = — , , 30 and ((2) =^r, and so on. Euler's constant C appears in these formulae only when a is an integer other than 2. Proof. Since Vi is integrable on S1 x Sl \ A

Eia){h)=

/L* ^ ' ^ = ^ i x>*°) / ±n ' i_n

Then Ena (h) corresponds to the contribution of the principal term to ~2 X/i^7 V'ft (~> ~ ) - Therefore the proof is reduced to the following lemma which gives the asymptotic behavior of the contribution of the counter term to ^ E ^ j V ^ C ^ . ! ) • Lemma 2.3.2

(1) Mien a = 1 i y ^ ^ ~ 2 ( l o g ^ + C).

Asymptotic

behavior of r

a

energy of polygonal

knots

25

(2) When a > 1

Proof. We give the proof in the case when n is an odd integer n = 2m + l. (1) The left-hand-side is equal to m

ET

-.

m

-.

2

l / = 2

= E7

|

m

i

\

(E--logmj+logm

which is asymptotic to the right hand side. (2) The left-hand-side is equal to

j = l Vn '

\j

= \ Vn '

j=m-\-l

v

n /

which is asymptotic to

D Proof of Theorem 2.3.1. We assume a = 3. A similar argument works when a / 3 . ff

/

1

1

|/i"(x)| 2 \

a-energy functional

26

E^a>

which implies E^\h)=

lim ( E ^ \ h ) - 2(C(3)n 2 - 2)

-I(/sil^)|^).2(logf + c)}. D

2.4

T h e self-repulsiveness of

E^

Hereafter we consider E^ only when a < 3 for t h e sake of simplicity. Instead of E^ (a > 3) we will study ea,p in Chapter 4. Let us give t h e condition for E^a' (1 < a < 3) t o be self-repulsive. ([05]) (1) £ ( Q ) is not self-repulsive

T h e o r e m 2.4.1 a

ifa<2.

a

(2) E^ > is self-repulsive if 2 < a < 3 i/ien _£A ). To 6e precise, for any b £ R £/iere zs a positive constant C — Ca(b) such that if E^(h) Cd(s,t) for any s,t& S1. Proof. (1) Suppose h is a "singular knot" with a right-angled double point P given by (2.6) in the proof of Theorem 2.1.1. Then the contribution of the neighborhood of P t o t h e integral (2.15) can b e estimated by /•i

/-i + i

L J t - t

dsdt

_

f^

fTr

fc' C^ d6dr_

dsdt

\h(s)-h(t)\a~J-iJ-±(s2+t2)^

~

Jo Jo

^

'

which is finite if a < 2 and infinite if a > 2. This implies t h a t if a < 2 then E^ takes a finite value even for a "singular knot" with a self-intersection, and hence E^ is not self-repulsive. (2) We estimate

inf

E^a)(K)

(see (1.1)).

Let h b e a knot with \h"\ = 1 a n d b = E^a\h) (b > 0). Let s , i b e points in S 1 = [ 0 , 1 ] / - with 0<s
Assume t h a t d < —. IfO
\h(s + u) - h(t-v)\

— then 8

—5.

The self-repulsiveness

of E^a>

27

h(s + u)
Fig. 2.6

The estimate for the chord length and the arc-length.

Since the integrand of E^a> is non-negative we have -l

If

r\

1

1

d^drj

Jo Jo

>

>

K

Jo Jo

l Qa ^(dJ -+« u - t+ - ^v)

l ^(fS)a

dudv

ay^ ,-{.-(^n**.

, 5 Since a + u - r - t i < a H

5 d+u+v 2 < — we nave 5 < —. Therefore we have 4 2 4-S ~ 3

b>
f l l *

(d + u + v)~adudv

r— r —

I I {d + u + v)~2 dudv Jo Jo

><1

(d+i)2 l & ° d(d+i >

Min TT

Mog 16d'

(2.18)

a-energy functional

28

E(a>

which implies 1

d>—exp[ exp

^

/

b 7TTZ

l-r^(ir)«

)S.

We have obtained this inequality under the assumption that d < —. Therefore \h(s) - h(t)\ > min j l , ± exp (~ f_ ^ 1

S

/

a

) J <*(*, t)

£< a )(/i)

«p(-r^)rf(M) exp

l"H|)«

(2-19)

for all s,t

•

Theorem 2.4.1 implies that ft,-1 satisfies a Lipschitz condition. Since h is parametrized by arc-length this means that ft, is a bi-Lipschitz map. Definition 2.4

Gromov's distortion ([Gr2]) is given by Distor(ft) = sup , , , ? * ' ? , ... V ' s/t \h(s) - h(t)\

(2.20) '

y

The distortion is obviously a knot energy functional. The inequality (2.19) implies that E^ (2 < a < 3) bounds the distortion from above: Distort) < 1 6 e x p — 4 ^ . Theorem 2.4.2 a>2. We call E^

([05]) E^

(1 < a < 3) is self-repulsive if and only if

(2 < a < 3) the a-energy. We will study the simplest case,

^w=fLAw^-•%*?)**•

(2 21)

'

in the next chapter. Example 2.2

Let h0(t) = ( ^ cos 2nt, ^ sin27rf, 0) be a circle. Then its

The self-repulsiveness

of E^a'

29

energy is given by 1

—

\

E™(h0)= [ ds2 f2 ( Jo = 2

Jo

,*

\(±smnu)2

,2-~)du u2)

7T C O t TTU H

u Jo = 4. R e m a r k 2.4.3 (1) E^ is called the M o b i u s e n e r g y in [FHW; K S 1 ' ; Lin] et al. since it is invariant under Mobius transformations as is shown in Section 11.3. (2) The choice of the counter t e r m to cancel the blowing u p of the integral at the diagonal has the ambiguity of a constant. T h e idea of using the power of arc-length as a counter t e r m was due to Nakauchi [Nak]. It has the advantage of making the integrand non-negative when a < 3. On the other hand, when a = 2 it seems more n a t u r a l to consider £•(2) __ 4 instead of E^ from a conformal geometric viewpoint as we will see in Chapter 11. It corresponds to taking the counter t e r m to be the integrand of a circle h0, where E^2' attains its minimum value, 4. Let us denote E^ — 4 by Eo ' , where 0 indicates t h a t it takes the value 0 instead of 4 at a circle. T h e n E^f1 is equal to twice the "energy " defined in [02].

Ei2)(h) = lim J£i2)(/i) - - } = =

IL^

(2-22)

\Hs)-Ht)\2 - {irin^*-*)}>}** (2'23) E^(h)~E^(h0).

(3) The arguments in this chapter only use the properties of the Euclidean metric space. Hence the statements also hold for knots in R™. T h e dimension of the ambient space concerns the topology of the space of knots since any knot in M™ with n > 4 is trivial. If we want to specify the ambient space we write E^f or E^J. (4) An energy which is very close to E^> is used in the study of approximate solutions of the n-body problem in [Bu3].

This page is intentionally left blank

Chapter 3

On E&

In this chapter it is convenient to extend the domain of the knot energy functional. Let X be S1, R, or an interval in K. Let / : X —> K3 be an embedding of class C 2 which is not necessarily parametrized by arc-length. Let / be the normalization of / after re-parametrization and rescaling so that / is parametrized by arc-length and that / has length the unity when X = S1. ^Define £ ( 2 ) ( / ) by the same formula as (2.21) and define £ ( 2 ) ( / ) by E^(f):

(3.1) where d«-(/(s),/(t)) is the shortest arc-length between f(s) and f(t) along the knot K = f(X) ([FHW]). The term | / ' ( s ) | | / ' ( i ) | is the Jacobian of the change of variables from the arc-length to arbitrary parameter. Putting K = f(X), x = f(s), and y — f(t), this energy can be expressed by

Em{K)

(32)

= ILA\^W~^hr)^

'

These formulations are invariant under rescaling and re-parametrization.

3.1

Continuity

Proposition 3.1.1 ([Ol]) The knot energy functional E^ with respect to the C2 -topology. 31

is continuous

32

On

£( 2 )

Proof. We show that for any knot h which is parametrized by arc-length with total length 1, and for any positive number e, there are positive numbers 5Q, 62 such that if a knot g parametrized by arc-length with total length 1 satisfies \h(s) - g(s)\ < 50 and

\h"(s) - g"(s)\ < 52

(3.3)

for all s € S1 then E^{h) - E^{g) < e. Let K = m a x s 6 s i |/i"( s )| be the maximum curvature of h. Then u > \h(s) — h(s + u)\ >— sin/cw

[0
(see (2.3)). Hence for any ei and £2 with 0 < e\ < £2 <

du

2r^
J

U

JE!

I(-^—

sin KM)

which implies - V;(22)(/i;s) < 2(KcotK£i - KCotK£2).

2 ( — - — J < VW(h;s) Therefore

o
{(-

VgHh;s)-f-

£1

£2

COt K£i

(3.4)

K COt K£2 £l

£2

Put <52 = K, then |
1

< — and 4

Therefore for any s € S 1 we have

V^(g;s)-fv}p(g;s)-^-)<e.

KCOt(Kt)

<

33

Continuity

Put d0 =

min

\h(s) - h(t)\ > 0. Take 50 (o < 60 < ~ ) such that

It—s|>ci

V

i d02

2 /

i i < (d0 + 260)2 ~ (d0 - 250)2

Then VJP(h;s)-Vci?)(g;s)

i < e. do2

<e for any s. Therefore \V^(h;s)-

3e for anys, and hence \E^(h)

- E^(g)\

V^(g;s)\<

< 3e.

•

Proposition 3.1.2 ([01; FHW]) E^ is not continuous with respect to the Cl -topology. If a knot degenerates to an immersion which is piecewise of class C1, but not of class C1, then its energy blows up. Proof.

Let he : (—1,1) —> R3 (0 < 9 < ir) be a curve segment defined by ( - 1
'(*,0,0)

hg(t)

(0
(t cos 9, t sin 9,0)

1).

Consider the cross term contribution

FW(D-2 r^1 [ln>

to E^(hg)

Z

J_^

r*

i

(

i

—

dsdt

2

J^yihe^-he^W

(t-s)

of the subarc

rn = hg((- — ,—^—\ U [—^,— )) (n = 0,1,2, •••). "• \ \ 2n 2n+1 J L 271+1 2n J ) Since Fn is homothetic to i~b, E^2\rn) is equal to E^(ro) for any n, which is positive if 9 > 0. Since hg((—l, 1)) D U^Lo -^i,

^2 Fig. 3.1

r0

The subarc rn and the cross term contribution.

E^(hg)>J2E{2)(rn) = oo re=0

On £<2>

34

for any 6 > 0. Therefore E1-2^ is not continuous with respect to the C 1 topology as £( 2 >(h 0 ) = 0 . • 3.2

Behavior of E^

under "pull-tight"

Proposition 3.2.1 ([01; FHW]) (cf. Theorem 4.2.8) Consider a line segment. Put a small 1-tangle T in the center of the line segment to make an open knot. Shrink the tangle T to a point while keeping its shape similar. Under this pull-tight process the value of E^2' converges from below. Since E^ is 0 for the straight line segment, E^ respect to the C°-topology.

3.3

is not continuous with

Mobius invariance

Freedman, He and Wang showed that E^ formations of R3 U {oo}.

is invariant under Mobius trans-

Definition 3.1 (1) Let E be a 2-dimensional sphere in R 3 with center P and radius r. The inversion Is of R3 U {oo} in E is given by Is{P) = oo, IE(OO) = P, and IS(X) = P + k(X - P), where k > 0 satisfies \IS(X) P\\X-P\ = r2. Namely, p +

Iz(X) = { oo P

\x-p\*{x~P)

if

XeM?

if

X = P,

if

X = oo.

When S is a 2-plane, which can be considered as a 2-sphere with center oo and radius oo, the inversion in E is understood to be a reflection in E. (2) A transformation of R 3 U {oo} which can be expressed as a composition of inversions in 2-spheres is called a Mobius transformation (see Definition 9.1 (7)). Remark 3.3.1 (1) A Mobius transformation T of R3 U {oo} maps a 2sphere into a 2-sphere and a circle into a circle, where 2-planes and lines are considered as 2-spheres and circles respectively, although the centers of spheres or circles are not preserved by T in general.

The cosine formula for E^2'

35

» WX)

Fig. 3.2

An inversion in a sphere.

(2) A Mobius transformation T preserves angles between smooth curves. In that sense it is also called a conformal transformation. Theorem 3.3.2 ([FHW]) Let f : S1 -> M3 be a knot and let T be a Mobius transformation o/R 3 U {oo}. (1) / / T o / ( 5 1 ) CM 3 then E™(T o f) =

E™(f).

1

(2) IfTof^S ) is an open knot which passes through oo then /)nM3) = E(2)(/)-4.

fi'2'((To

We will give the proof of (1) in the next section. Remark 3.3.3 As we remarked in Remark 2.4.3 we can define the energy E$(f) of an embedding / : X -> W1 by the same formula (3.1). A similar (2)

proof shows that £^„ is Mobius invariant. Among all the open knots only the straight lines give the minimum value of E^2\ which is 0. Therefore Corollary 3.3.4 knots, which is 4.

3.4

Only circles give the minimum, value of E^2' of closed

The cosine formula for

E^

From a conformal geometric viewpoint it is more natural to consider El (K) = E^ (K) — 4 as we mentioned in Remark 2.4.3. Let us make a quick review.

36

On

Fig. 3.3

E^

An inverted open knot.

Let K be a closed knot of length 1 or an open knot. Then (2.7), (2.8), (2.16), and (2.17) can be reformulated as dy

V™(K;x)=[ JyeK,d

:(x,y)>e

V0{2)(K; x) = lim (vP{K;

I 12 ' \x ~ U\

x) - - Y

Then E(2) (K) is given by

Ei2\K)

= f

V?\K-x)dX

= lim ( (I

^ %

- *-) . (3.5)

We remark that E{2) (K) = E^ (K)-4

if K is a closed knot, and E?) (K) =

The cosine formula for E^2'

EW(K)

37

if K is an open knot.

Let us give a proof of the Mobius invariance of EQ . Proof,

(a) We first observe that the 2-form

dxdy _ \f'(s)\\f(t)\ 2 l*-vl ~l/(*)-/(t)l2 as the principal term of the integral (3.1) is invariant under Mobius transformations. (Another proof can be found in Chapter 11.) Put (pel3).

|T'(p)| = | d e t d T ( p ) | *

We remark that when T is a Mobius transformation of W1 U {oo} the following argument works if we put |T'(p)| = | det dT(p)\^. In particular, if T is an inversion in a sphere of radius r with center 0 then |T'(p)| = —-.

H

Then

\T(p)-T(q)\ for p,q £R3

=

\T'(p)\\T'(q)\\p-q\

and |(To/)'(s)| = |T'(/(S))|2|/'(S)|

for / : S1 or M -> R 3 . Therefore \(Tofy(s)\\(TofY(t)\=

\f(s)\\f(t)\

|To/(s)-To/(i)P

|/(s)-/(i)|2"

(b) Let us introduce the cosine formula for E^2\ Let K = h(Sx) be a knot which is parametrized by arc-length. Put x — h(s), y = h(t), and dxdy

W^yJ=

dsdt

\h(s)-h(tW

in what follows. Fix e (0 < e a n d p±(e) = Ix(p±(£))- We call Kx the inverted open knot of K at x. We

On E<2>

38

remark that the asymptote lx of Kx has —h'(s) as its direction vector. By (a) \ix'{y)\

l

\x-y\

and \dy\ = \d(Ix(y))\ = \Ix'(y)\2\dy\

=

My|

x-y\2'

Therefore V2){K-x)=

T-^-T2=

Jp+(e) \X-y\2

\dy\

Jp+(e)

is equal to the arc-length of ii"x (p + (e),p_(e)) = Ix(h(s + e, 1 + s — e)) which is a subarc of the inverted open knot Kx between p+ (e) and p_ (e). Let k = \h"(s)\ be the curvature of K at x and n = h"(s) be the unit principal normal vector of K = /i(5'1) at x = h(s). Put n = 0 when |/i"(s)| = 0. Then p±(e) can be expressed by e2k p±{e) = x ± e/i'(s) + — n + 0(e 3 ), which implies 1 /fc p±(V) = a; ± — h'{s) + —n + 0(e). Let n be the orthogonal projection from M3 to the asymptote lx and g±(e) = ir{p±(s))- Then \q+(e)-q-{e)\

= j + 0(e).

Therefore

Jp+(e)

\x-y\2

2 fP-(£) - —= / l # l " l9+(£)-5-(e

Length ( i ^ (p + (e),p_(e))) - Length (TT (tf x (p + (e),p_(e)))) + 0(e).

The cosine formula for E^2'

39

Since V0 (K; x) is given by

Jf

V™(K;x) = ]im(f

y£K,5(x,y)>s

j ^x — , \

~ V\

x) can be interpreted as the asymptotic difference of the lengths of the inverted open knot Kx and its asymptote lx. (This is called the wasted length or the exess-length of the inverted open knot.) Definition 3.2 Let C(x,x,y) be the circle tangent to the knot K at x that passes through y with the induced orientation from that of K at the point x, and C(y,y,x) be the circle tangent to K at y that passes through x with the induced orientation (see Definition 8.3). Let 9K = 8K{X,U) (0 < OK < 7r) be the angle between C(x,x,y) and C(y,y,x) at x or at y. We call it the conformal angle (between x and y). Let C(oo,oo,y) — Ix(C(x,x,y)). Then this is a straight line through y which is parallel to the asymptote lx of Kx. As Ix is a conformal mapping, the angle 6K (oo, y) between C(oo, oo, y) and Kx at y is equal to the angle 6K{X,V) between C(x,x,y) and K at y. Therefore

V0^(K;x) = Mm[

f

^

\Jp+

= lim {Length (KX (p+(e),p_(e))) - Length (TT (KX (p+(e),p_(e)))) } /•P-( £ )

= lim /

(l — cos#£(1 -

—

n r n

e

;

^°JyeK Hence

(oo,y))dy

cose K (x,y))dy i

lo

\x-y\* (2)(TS\-

E?>(K)

ffff

l-cos6 (x,y) - ™eK \:>y)Kdxdy.

l C

(3.6)

This is called the cosine formula by Doyle and Schramm (see [AS] and [KS1']). According to Kusner and J. Sullivan, this formula is useful in numerical experiments. (c) Now the Mobius invariance is straightforward. Any Mobius transformation T keeps the principle part ^ of the integrand unchanged. It \x-y\-*

On EW C(oo,oo,y) --Ix{C(x,x,y))

9+(e)

OK(X,V) 9-(e)

Fig. 3.4

The inverted open knot, the conformal angle, and the exess-length.

also keeps the conformal angle OK unchanged. Therefore the cosine formula (2)

implies that it keeps Eo unchanged, which completes the proof of Theorem 3.3.2 as E?\K) = E^(K)-4 UK is a, closed knot and E^]{K) = E™{K) if K is an open knot. • Let us call Eo (K) the conformal (1 — cos 0) energy of a knot K. Definition 3.3 ([LOl]) Let vx be the positive unit tangent vector to the knot K at x. Then vx is the positive unit tangent vector to C(x, x, y) at x. Let vx(y) denote the positive unit tangent vector to C(x, x, y) at y. We call vx(y) the unit tangent vector at x conformally transported to y, and call the correspondence vx H-> vx(y) the unit conformal transportation from x to y. Remark 3.4.1 vx(y) and vy. (2) Let w

(1) The conformal angle 9x(x,y)

\*-y\

is the angle between

. Since vx(y) is obtained by the reflection of —vx in

The cosine formula for E^>

41

the vector w, vx{y) = 2(vx,w)w

-vx

= 2 I vx,-

X — XI \

X — XI

\x-y\J

r ~vx. \x-y\

(3.7)

(3) The conformal angle OK(X,H) is equal to the argument of the infinitesimal cross ratio QCR which will be given in Definition 11.4. (4) As we will see in Lemma 12.1.2, the conformal angle 9 K satisfies 6K = 0{\x — y\2)- The proof implies that if the following is a well-defined functional

SL

KXK

for some function

F(6K),

F(0K)-, '

dxdy '\x-y\2

then it is Mobius invariant.

The conformal angle and the unit conformal transportation can be generalized as follows. (1) Let x,y ER3 {x ± y) and u € TXR3. Put

Definition 3.4 /N

1 v-y\2

f„ ( I \

x — w\a; — w \x-y\; \x-y\

and call it the vector v conformally transported to y; call the correspondence TXR3 3u^

u(y) e TyR3

the conformal transportation from x to y. (2) Let i . j e l 3 (x ^ y), u e TXR3, and v G TVR3. The conformal angle 9 = 6(u,v) between u and v is given by the angle between u(y) and v. It is preserved under any Mobius transformation. Remark 3.4.2 (1) Let C(y,x;u) denote the oriented circle that is tangent to u at x and that passes through y whose orientation comes from that of u. Then u(y) is a positive tangent vector to C(y,x\u) at y. (2) Being a unit vector is not a conformally invariant condition. Our definition will make the inner product conformally invariant. Namely, we have I T*v,T*u(T(y)) J = (v,u(y)) for any Mobius transformation T of M3 U {oo} and for any u G TXR3 and v € TyR3.

On £ ( 2 )

42

3.5

Existence of E^

minimizers

A composite knot is a knot which is a connected sum of two non-trivial knots. A prime knot is a knot which is not a composite knot. Theorem 3.5.1 of a prime knot.

([FHW]) There is an

minimizer for any knot type

Outline of proof. Let [K] be a knot type of a prime knot. Take a sequence of knots {/i, /2, • • • } in [K] such that lim E^(fn)=E^([K}) ra->oo

= _inf

E™(f).

fe[K]

By the Ascoli-Arzela theorem there is a convergent subsequence after suitable re-parametrization, rescaling, and congruent transformation of M3. Take the limit /oo of a convergent subsequence. There is a possibility that /oo is the trivial knot because a tangle in /„ might pull tight to a point as n goes to infinity. Take a new sequence {f\,fi,---} in [K] by applying Mobius transformations if necessary to make each fn in a "relaxed position" so that we do not have pull-tight any more. Take the limit /oo of a convergent subsequence. Then /oo belongs to the same knot type [K] and -E^2'(/oo) = E^([K]) by Fatou's lemma, which means that /oo is an E^ minimizer. • Here are some remarks. Remark 3.5.2 later).

(1) E^

minimizers are smooth by Theorem 3.7.7 (see

(2) The number of E^ minimizers for each non-trivial prime knot type is not finite. Let f[x] be an E^ minimizer and T be any Mobius transformation. Then T o /[#•], or its mirror image if necessary, belongs to the same knot type and takes the same value of E^2' as f[K], a n d therefore is again an E^2' minimizer of the same knot type. (3) It is not known whether there exist any critical points in the trivial knot type except for the circle. Up to now numerical experiments suggest that there are none ([Lig-Se]). If this is correct, it would give an alternative proof of Hatcher's results that the space of trivial knots in S3 deformation retracts onto the space of great circles in S3 ([Ha]).

Existence

of E^

43

minimizers

On the other hand (see later) Kusner and J. Sullivan conjecture that there are unstable critical points in the (p, q) torus knot type if p or q is greater than 2. (4) It does not seem to be easy to find asymmetric E^ minimizers by numerical experiments. We cannot apply the proof of Theorem 3.5.1 to composite knot types. Let [Ki], [K2] be knot types of non-trivial knots and let [A'ljJ^] be the knot type of their connected sum. Take a sequence of knots {ffi,52)" - } m [K4K2] with l i m ^ o o E^(gn) = E{-2\[K4K2]). In this case we cannot prevent pull-tights of both components corresponding to [Ki] and [K2] at the same time. When we evolve one component by a Mobius transformation, it might make the other component smaller. In fact Kusner and J. Sullivan observed that both components pull tight to points in numerical experiments ([KS1']). Consider the corresponding open knot by an inversion in a sphere with center on the knot. Then the above phenomenon means that both components corresponding to [K\] and [K2] move in the opposite direction to have more distance from each other (Figure 3.5). If there is enough distance the interaction of both components in the integral would converge to 0.

^r-^3)-

•'&) \l)~

4mm[ o-o 1

Fig. 3.5

\

The energy minimization of a composite knot.

On E&

44

L e m m a 3.5.3

([KS1'])

E¥\[K4K2}) < EPdKtV +

E™^]).

C o n j e c t u r e 3.1 ([KS1']) (1) There are no E^ minimizers in composite knot types because all the tangles will pull tight. (2) T h e minimization of E^ leads a composite knot to a "conformal connected sum" of the E^ minimizers of the constituent knot types, and hence E^\[K^K2\)

3.6

= ^2)([id]) +

E?\[K2]).

Average crossing n u m b e r and finiteness of knot t y p e s

Freedman and He defined the average crossing number (see Definition A.5) [FH]. T h e n Freedman, He and Wang showed t h a t E^ bounds the average crossing number (c) from above. Namely: T h e o r e m 3.6.1 ([FHW]) (1) For any open knot f : K ->• R 3 we have (c)(f) < 2 T T £ ' ^ 2 H / ) - Therefore if we denote the minimum crossing number of a knot type [K] by c([K]) Theorem 3.3.2 implies c([K])<±(EW([K])-4). (2)For any knot f : S1 -)> R 3 we have ( c ) ( / ) < — £ ( 2 ) ( / ) + - . As the minimum crossing number of a non-trivial knot is not smaller t h a n 3, Corollary 3.6.2 ([FHW]) E^ can distinguish precise, ifE^(f) < 6TT + 4 « 22.8 then f(Sl) is

the unknot. unknotted.

To be more

It does not seem t h a t this quantity 6TT + 4 is sharp. Numerical experiments implies t h a t the minimum value of E^ of non-trivial knots is about 74, which is attained by a trefoil (Table 5.1). Since there are only finitely many knot types whose minimum crossing numbers are less t h a n a given number n, we have Corollary 3.6.3 ([FHW]) Only finitely many knot types can occur under any given E^2' threshold. The number of the knot types with a representative f with £ ( 2 ) ( / )
Gradient, regularity of E^2' minimizers,

3.7

and criterion of criticality

45

Gradient, regularity of E^2' minimizers, and criterion of criticality

The first variation or the gradient of E^ cannot be expressed by local data of knots because E^ is defined as an integral of global quantities. In this section we introduce the calculation of the gradient of E^ after [FHW]. Let / : S1 — [0,1] /~—• R3 be a knot of class C1,1 which is not necessarily parametrized by arc-length. Lemma 3.7.1

([FHW]) For any e > 0,

ff • I f ( \ flA,2\f(s)\\f(t)\dsdt=^-i J JE<\t-s\
+ 0(s).

(3.8)

e

Put e+(s)=dK(f(s),f(s

+ e)),

and e_(s) = dK(f(s

- e), f(s)).

Then s+ = e+(s) is given by

e+= r + V(oi# J s

= e f \f'(s + etM Jo

= -e[(l-Z){\f'\\a

+ e€)}]1o+e2

J\l-t){\f'\'{s

+ e0} d£

It follows that

£ = ^-|7W ( 1 -°{ | '' r ( ' + s 0 } * + ow and similar expression holds for — . Let L be the length of the knot K — / ( 5 1 ) . Then the left hand side of

On £ ( 2 )

46

(3.8) is equal to

J Si \Je<\t-s\
dK(f{s), 1

fit))2

J

4

/ s ,y?+^w]i/'<»>i*. /' s, i

—ds-4 ^

-f

f\l-0y^{\f'\'(s

+ e0-\f'\'(s-e0}dtds

+ O(e). (3.9)

Then the third term of the right hand side is at most 0(e). When / is sufficiently smooth it is because | / ' | (s + e£) — | / ' | (s — e£) is 0(e 3 )- For the general case, the reader is referred to Fr-He-Wa for the proof. • Define

%>(/)=//

l«|L**-I+4.

7yle<\t-s\< e
,3.10)

e

Then E(E)U) = II

(\f(\ * f(+W ~ A
2 M/(«)-/(*)l f(t)\

J Jd(s,t)>e\\f(s)-

2

dK(f(Sd),f(t)) {f(s),f(t)) J 2

+ 0(e)

K

and hence £(2'(/) = lim£(E)(/). We remark that E^ is different from E? ' given in (2.8). Let 7? : S1 = [0, l ] / ~ - » M3 be a C^-function. Let / , (qr e K) be a family of knots defined by fq(t) = f(t) + qri(t).

Gradient, regularity of E^2' minimizers,

and criterion of criticality

When q is close to 0, fq is a C ^ - k n o t . If a map q H> E^2\fq) tiable at q = 0, we write

47

is differen-

VE^(f)(V)=^\fq) q=0

Lemma 3.7.2 ([FHW]) Let f : S1 = [0,1] / ~ - > M3 be a C^-knot, and n: S1 = [0,1] / ~ ->• R 3 fee a C1'1-function. Then V-B(2)(/)(»?) exists and is given 63/

l«"ll«*>l 4 * .

(3.11)

I/W-/WI 2

where p.v. indicates the principal value in the sense of Cauchy: p.v.

Proof.

£

->+°J

e

We use -E<e>- We have

iw-HIL dqj

- dsdt Jd(s,t)>e\fq{s)-fq{t)\2

9ff

{(fq(s),v'(s))

JJd{Jd(s,t)>e s,t)>s\

(fg(s)-fq(t),V(s)-ri(t)))

\fq(s)\2

\fq{s)-fq{t)\2

\fq(s)\\fq(t)\

\fq(s)~fq(tW

dsdt.

J

(3.12)

Since fq and 77 are C 1 ' 1 functions,

/,(*) - /«,(<) = (a " W )

+ (« - *) 2 / (1 ~ 0 # ( * + £(* - *)) de, ./o

77(a) - 77(i) = (s - i)r/(i) + (S - i) 2 A l - Ov"(t + £(s - t)) d!i . Jo

On EW

48

Therefore

(fq(S) - fg(t),v(s) - V(t)) = (s- t)2{(/;w,„'(f)) +(s-t)

[\i-t){(f'q'(t

+ as-t)),r,'(t)) + (f'q(t),r]"(t + t;(s-t)))}dt;

Jo

+0(( S -i) 2 )}.

A similar expression holds for \fq(s)-fq(t)\2 It follows that

= (fq(s) - fq(t) Jq(s) -

fq{t)).

(Ms)-fq(t)Ms)-v(t)) \fg(s) - fg(t)\2 = (f'q(t),rf(t)) •

(S t]

~

l(+W2 \f'M\

f\l-0{(f'q/(t

+ as-t)),r,'(t))

+ (f>(t),r,"(t +

-2 (fq(t),r,'(t)) (j'q'(t + fa - *)), jjf^^j

as-t)))

} dC

+0((s-t)2).

\f(s)\\f'(t)\

|/(S)-/(i)|2

_

(s-t)*

, 0 ( ^ 1 , \\s-t

(3.13)

Gradient, regularity of E^2> minimizers,

and criterion of criticality

49

we have

'{f'q(tU(t))

(fq(s)-fq(t),r,(*)-ri(t))\

2

\W)\

!/,(*)-/,WI

\f'(s)\\f(t)\ 2

I/W-/WI 2

(*

+2 (/^(t), vet)) (#(<+as - *)), 1 ^ 4 ) } # +0(1).

(3.14)

Therefore, if we put u = s — t and v = t + £(s — t) = i + £u then we have

d(

l

J J Jv€S1,ue[-^-,i-h\u\>e,ie[o,i]

u

+ 2 (/> - ^),TJ'(« - e«)) ( / » , i f f j r ^ ) } ^ d u d u +0(1).

(3.15)

On £< 2 )

50

Since rf and /' ?' satisfy the Lipschitz condition,

\fq(v --£«)l2 \fq(v --e«)i

l / » l 2 ' °{U)'

2

ii»i2

+U[U)

>

\fq(v

[f>-

-£u),rf{v -tu))

= (fq(v),ri'(v))r,'(v) + 0(u)

Then by (3.15) !*«><'«>

«esi,ue[—i-,-i-]>|>e,£e[o,i]

(i-o

(

/r(u)) ??»

w

I

V9

' l/^)l 2

^"^ IW?) + 2 (/'(l°' ^ (W ' JOTrO } *dudw + ° (1) yyy„esi,«e[-i,i],|u|> e ,€e[o,i]

u dv

\fq\v)\

+0(1). (3.16) It shows that that -—E^(fq) satisfies the Lipschitz condition in e, and hence it converges uniformly as e decreases to +0. Then (3.12) implies the Lemma. • Let -P/'(S)-L : R3 —>• R3 be the orthogonal projection onto the plane which is perpendicular to f'(s): PfW(«) = «-$$$•?(*)• Lemma 3.7.3

(3-17)

([FHW]) Let f be a knot of class C3'a for some 0 < a < 1.

Gradient, regularity of E^

Define Gf.S1^

minimizers,

and criterion of criticality

51

M3 by

Gf(s) Pns)±(f(t)-f(s)) \f{t)~f{s)?

1 d ( f'(s) \f'{s)\ds\\r{s)\J\

Then for any C2'a function n such that (r](s), f'(s)) have V ^ 2 ) ( / ) ( r ? ) = lim

EW(f

+

\f'(t)\ rdt. \f{t)-f{s)\*

= 0 for any s € S1 we

qr,)-EW(f)

g->0

- f {Gf{s)^{S))\f'{s)\ds. Js1 Proof.

(3.18)

By (3.12)

dE (e){fq) dq

=2 /

(/

L , , .J',^

Jt€Si l > V d ( s , t ) > A I / ( s ) l l / ( s ) - / ( * ) l

0

+4

fm{2,v'(s))ds\\f'{t)\dt

/

J

(/(S)-/W,^))|r(s)||/,W|^ !/(*)-/Wl4

/d(s,t)>e

Using integration by part for the first integral on the right hand side, E

dq

(e)Uq) q=0

:

/ {/

/'(*)

tes^Jd{s,t)>s\\f'(s)\\f(s)-f(t)\

Jtes1

Jdi.

d ( f'(s) \ tesi{Jd(s,t)>e\ds \\f'(s)\J

+4

2,r,(s))dsUf'(t)\dt

tesi\jd(s,t)>e

1 ,V(s))ds\\f'(t)\dt \f(s)-f(tW

(f'(s)J(s)-f(t))(f(s)Ms)) l/'(s)||/(s)-/WI4

1

^WW ^'^^

ds}\f(t)\dt

On £ ( 2 )

52

The first term of the right hand side vanishes because (rj(s), f (s)) = 0. The second term is equal to

The sum of the third and the forth terms is equal to

'/ {/

JseS1 (Jd(s,t

1

f,tu,

im

\m - mw \ l/,(i)l

1/00-/(*)l2

r ,„„

-

(f(t)-f(s)J'(s))

m)

\FW—Ms)

dt\\f\s)\ds.

Then the Lemma follows.

• r

Remark 3.7.4 The gradient of E^ at a knot of class C (r > 4) is of class C""™2. The differentiability drops by 2. Lemma 3.7.5 ([Hel]) Let f be as above. Then for any scalar function A of class C2, we have V£(/)(A/')=0.

(3.19)

Proof. For any sufficiently small q > 0, there is a function r\q : S1 —> K3 as in Figure 3.6 such that: (1) f + qXf and / + q2r]q are the same knot with different parametrizations; (2) ( 7 ? g ( S ) , / ' ( S ) ) = 0 f o r a n y S e 5 1 ; (3) |r/ 9 (s)| is uniformly bounded for sufficiently small q. Then Lemma 3.7.3 implies E^(f

+ q\f')

= E^(f

+ q2rlq) = E^(f)+q2

[ V'E^(f

+ Tq2ilq){nq)dr.

Jo

Analytic consideration shows that Vi?'2-1 (f+T~q2r]q)(riq) is uniformly bounded as q decreases to 0, which completes the proof. • As T / ( S ) K 3 - r / ( l ) ( / ' ( j ' ) ) e r / ( l ) ( / ' ( S ' ) ) 1 we have: Corollary 3.7.6 ([FHW],[Hel]) The formula (3.18) holds even if(n(s),f'(s)) does not necessarily vanish everywhere. Namely, Gf is the L2-gradient gvadf(E^) ofEW at f.

Gradient, regularity of E^2> minimizers,

and criterion of criticality

/(*) + ?A/'(s) = f(s') +

53

q\(s')

Fig. 3.6 r,q

By solving the Euler-Lagrange equation V£ ( 2 ) (/)(7?) = 0

(VJJGC00(S1,]R3))

locally, He obtained Theorem 3.7.7

([Hel]) (1) E^

(2) The circle is the only E^ curvature.

minimizers are smooth. critical planar loop with cube integrable

The gradient of E^ considered as a functional on the space of knots which are parametrized by arc-length can be obtained as the limit of the gradients of the discretized energies on the space of equilateral polygonal knots ([06]). Let Confn(M3) = {X = {Xu • • • , Xn) e (K3)™ | X% ± X3 (i ± j)} C (M 3 )" be a configuration space and let L„ = j x = (Xlt • • • ,Xn) e Confn(M3)

\Xt - Xj\ = ^ |

be the set of equilateral polygonal knots. Then Lra is a 2n-dimensional regular submanifold of Conf n (R 3 ). Define En2) : Conf„(K3) ->• R by Conf n (K 3 ) 3 X = (X 1 ; • • • , Xn) -> £<2>(X) = - I £ »^j

iy.-Y.l2

e

R

On EW

54

Let 7J X :T X (M 3 )"->T X (L U )

(Xel,)

be the orthogonal projection. Then the gradient of En ' |L n : L„ —> M. at X is gradx(^2)|L„)=iTx(gradx(K

(2)

If we take the limit as n increases to oo we obtain a formula for the gradient of E^\^{i), where Jf^ is the set of knots of class C4 parametrized by arc-length.

3.8

Unstable E^ 2 )-critical torus knots

In this section we introduce Kim and Kusner's results on the energy of torus knots on Clifford tori in S3 ([KK]). The energy can be generalized to an energy functional E^l for a knot in R —> R n by the same formula (3.1). As is mentioned in Remark 3.3.3, Eyl is invariant under Mobius transformations of Rn U {oo}. Let a be a stereographic projection from S3 C R4 onto K3 U{oo}. Then a can be extended to a Mobius transformation of R4 U {oo}. Therefore if a knot / in S3 is mapped to a closed knot a o / in K3 then E^l(f) = E^J(crof) = E^(crof). Thus, considering ivR4 for knots in S3 is equivalent to considering E^> for knots in M3. We denote the restriction of Ewl for knots in S3 by E^l , or simply by E in this section. We note that Eo ' can also be generalized by (3.6). EQ is suitable for a clear understanding of conformal geometric properties (see Part 2). (2)

If we use Eo we do not have to worry about whether a knot is closed or open. On the other hand, E^ is suitable for the study of its metric aspects including the computation of its gradient. We use the parametrization S 1 = [0, 2TT}/~ in this section. Let Tr be a Clifford torus of radius r (0 < r < 1) in S3 C C 2 S R 4 : Tr = Uz,w)eS3

CC2

\z\ =r,\w\ = y/l-r2

\ ,

Unstable E^2' -critical torus knots

55

and let r r = r r;Pj9 be a (p, q) torus knot on Tr: Tr.Pjt} :

[0,2TT]

30^ (repei, \ A - r2 eq0i\ G S 3 C C 2 ^ R 4 .

(3.20)

S1 acts isometrically on 5 3 by ipO

3 a^'Be^l^^l^f^leS ]. w I \ elqSw

Then E(R4'\

is invariant under this 5 1 -action, and T^S1)

is an orbit of

the point (r, 0, \/\ — r 2 ,0) since at(Tr(s)) = Ty(s + £). Therefore Palais' principle of symmetric criticality ([Pa]) implies: Lemma 3.8.1 If d_ n(2)/ E$(Tr) dr

=0

(3.21)

(2) I

i/ien r ro is critical for E^l

3.

The principle of symmetric criticality states that if a group G acts on a manifold M and if a function / : M —> R is G-invariant, then a symmetric point p (i.e. gp = p for all j € G) is a critical point of / if it is a critical point of f\s : £ —> R, where £ is the fixed point set of G, £ = {q G M : gq = q ^g G G}. This principle unfortunately is not valid in complete generality, and so we need a proof of the Lemma. (2) I

Proof. Let us denote £^4 by E in this proof. We first remark that the gradient gradT E of E at rr is also invariant under this 5 1 -action, i.e. (a t )*(grad Tt ,£(s)) = grad T r £(s + t). Let fe : S1 —> S3 be a smooth family of knots with /o = Tr, and let S : S1 —>• R4 be its tangent vector:

5(s) = yef£(s)

eTf S 3

On E&

56

Since \fd(s)\ = |T-;(0)| for all s e S1 we have d_ -E(fe) de'

= f e=0

(&adfE(a),8{s))\ti(8)\ds

Jsi

0

:Jsi((as)^adfoE(0),5(S))\fd(0)\ds (gradfE(0),(a-a),6(s))K(P)\ds

Let S be an ^ - i n v a r i a n t variation given as the average of 5 by the 5 J - a c t ion,

S{t) =

i Is* L (a-)*J(* + S^S E T /o(*)^

Then ^ ^ ( / e ) | e = 0 = (grad / o £(0),<J(0)) • K ( 0 ) | . T h e gradient decomposes according to the direct sum decomposition T

f0(0)S3

where (XV imTr)1-

= Tfo(0)Tr

©

is spanned by Tr-r r (0).

(Tfo(0)Tr)±, Since is1 is constant on the

set of the S^-orbits of points on TV, grad^ _E(0)L1 fore -K-E{fe) t0

JwE(Tr)

= 0 if grady £ ( 0 ) 1 , „

e=0

0.

„,

/„ (0) J >•

= 0.

There-

_ ._,_ = 0 , which is equivalent

•

Since a (p,q) torus knot is prime, Theorem 3.5.1 guarantees t h a t there is an minimizer. Using numerical experiments Kim and Kusner deduced the following conjecture. C o n j e c t u r e 3.2 ([KK]) T h e critical torus knot on a Clifford torus a o r ,p,q where r 0 is obtained by (3.21) is a stable local minimum for E^2' if and only if p = 2 or q = 2. T

On the other hand, when (p,q) = (2,3), i.e. in the trefoil case, numerical (2)

computation implies t h a t ER-I (rr-,2,3) takes the minimum value approxi-

Unstable E^2'-critical

torus knots

57

mately 74.41204 at ro ~ 0.880, which is close to the minimum energy of the trefoil obtained by numerical experiments. Kusner and Stengle showed that E^ of (p, q) torus knot on a Clifford torus can be expressed by a residue formula. Lemma 3.8.2

(see [KK])

^4)(rr;p,9)=4 + 4 7 r 2 ^ R e s - 5 — -

Z1-P(ZP

r2)q2}

- { r V + (1 - 1)2

+

(! _

r

2)zl-9(z9 _

X )2

'

where the summation is taken over the residues inside the unit disk. (2) I

Let us write E = EyRJ

Proof.

2 2

T'r(t) = ^/r p great circle

2

+ (1 - r )q

2

3

and rr = T r;Pig in this proof. Since

and E(TT) = 4 + E(rr) - E{C0), where C 0 is a

r 2 p 2 + (1 - r2)q2 E(rr)-2TT

J ^ (^ ^ I2 gl i p s __ |2 - l|2

+

1

,( l __ r^2 i) 0| e« i ,9 sS __ i]_|2 i2 ~

2 Q^

rfs

)

. 4 + 2. / f rV + ( l - r V ' _w V^lev* - 1|2 + (1 - r 2 ) | e ^ - 1|2

1 _ \eis - l l n '

Since \elt - 1|2 = - e " ' ' ^ * - l ) 2 for t e M, the substitution z = e i s , dz = izds gives

2 ^ ~

4

) ( r V + (1 - r 2 )g 2 )

„. y r 2 e - i p s ( e i p s __ 1)2 + ( 1 -

|z|=1

r2-}e-iqs(eiqs

. _ 1)2

i(r2p2 + (1 — r2)<72) V r 2 ^ 1 ^ ^ ~ l ) 2 + (1 - r 2 ) ^ 1 - ^ ^ - l ) 2

e-»s(e»s

i (z - If

, ^ _ 1)

ds.

Then Cauchy's residue formula implies the Lemma as the order-two singularity at z = 1 of the first term in the integrand is cancelled by the second term.

•

On £ ( 2 )

58

3.9

Energy associated to a diagram

The cosine formula (3.6) shows t h a t El (K) can be expressed as a double integral on the space of pairs of points on a knot K as

Fo V

E?\K)=[[ J JKXK

^ '^dxdy,

\x-y\2

where F0 is a function of a pair of points x and y and the tangent vectors vx and vy at these two points. T h e right hand side of the above equation can be generalized in two ways: One is to use another function F t h a t satisfies F(x,y;vx,vy)/\x — y\2 — 0(\x — y\~1+e) (e > 0) near the diagonal, and the other is to consider an integral with a greater multiplicity by increasing the number of points on the knot or by taking points from outside the knot. For the second method, it is natural to use so-called chord diagrams, as in the study of the Kontsevich integral of the Vassiliev invariant.

3.9.1

General

framework

Let pi, • • • ,pn be points on an oriented (closed) curve 7. We write pi -< • • • -< pn if the (cyclic) order of p\, • • • , pn coincides with the orientation of 7Let D be a planar diagram which consists of an oriented circle S1, vertices p i , - • • ,pn on S"1 w i t h p ! -< • • • -< pn a n d p „ + i , • • • ,pn+m i n M 2 \ 5 1 , and edges e\, • • • , e ; with dei = {ps(i),Pt(i)} ( s (*) 7^ t(i))- We assume t h a t there is at most one edge with the same pair of end points. P u t Confnim(K,S3)

= { O n , • • • ,xn+m)

GKnx

UD = { ( z i , - - - ,Xn+m) S Gonintm(K,

(S3)m

\ Xj jt xk (j ± k)} ,

S3) I xi -< ••• -
.

Let us consider a functional associated to the diagram D which is given by ED,
= / #(
,
for some functions ^ and (pi(xs^,xt^), where ifi(xs^,xt^) also on t h e tangent vectors vXg(i) and vXt(i of K at xs^ exist.

may depend and xt^ if they

Energy associated to a diagram

59

If we take a "^-diagram" with a pair of points on S1 as vertices and a unique edge joining them and put $(w(x, y; vx,vv)) — , K: '-, then \x-yY we get EQ . The other examples of a function {ip{x,y;vx,Vy)) associated to the same '^-diagram" will be given in Section 3.10 and Part 2. 3.9.2

"X-energy"

Lin denned "X-energy" for knots ([Lin]) which is defined by 2 pairs of points which cross each other. Let X denote a diagram which consists of S1, four points pi,P2,P3,P4 on S1 as vertices and two edges ei3 and e24 with deis = {pi,p$} and 9e 2 4 =

{p2,Pi}-

Fig. 3.7

The diagram X.

Let 0i3 = 9K{X\,X^) (#24 = 0K(X2IX
([Lin]) Put

TP ns\ f Ex,cOS{K) = / i

JK ;x1^x3^X2^x4

T? iis\ = /f Ex,sin(K) K4;

Xi^.X3^(.X2^,X4

(l-cos0i3)(l-cos024) , , , , —; j2| 1^—dx 1 dx 2 dx 3 dx 4 , \Xl ~ Xs\

\X2~X\\

sin6>24 ~ , dxidx , , dx ,dx , i sin021377T. 2 3 4 \xi -x3\ \x2 -xA\2

and call them the conformal (1 — cos) X-energy and the conformal sin X-energy respectively. These energies are Mobius invariant by definition.

On E(2>

60

3.10

Normal projection energies

Let vx be the unit tangent vector to K at a point x, vx be the orthogonal 2-plane to vx through x, and let PVx± : M3 = TXK © vx —» vx be the orthogonal projection. Instead of considering a (modified) Coulomb repulsive force F(x, y) we can consider its normal projection PVi±(F{x,y)) to vx. Let r\x = r)x(x,y) (or rjy = r/y(x,y)) (0 < T]X,T)V < IT) be the angle between vx and the chord (y — x) joining x and y (or respectively, between vy and (x — y)). We remark that r)x(x,y) = 0(\x — y\) near the diagonal. Definition 3.6 (1) The normal projection energy of Buck and Orloff ([B02]) is given by £nP(#) = / / J

sin rjx(y) -. '^dxdy. JKxK

(2) The symmetric normal projection energy of Buck ([BS3]) is given by =

rr J Js>xsi

sm,x(v)^vy(x) \x-y\2

Clearly Enp(K) > Esnp(K) for any knot K. average crossing number implies Proposition 3.10.1 crossing number (c):

The formula (A.5) of the

([BS3]) The projection energies bound the average

^np(^)>^s„p(^)>47r(c)(^).

3.11

Generalization to higher dimensions

Several generalizations of E^ to higher dimensions which keep the Mobius invariance property have been carried out in [AS] and [KS1']. Let us introduce Kusner and Sullivan's method. Suppose IIi and II2 are two oriented fc-planes in M™ spanned by ordered orthonormal bases «i, • • • ,Uk and vi, • • • ,Vk, which give their orientations. The combined angle 9 between II1 and II2 is given by cos# = det((ttj, Vj)).

Generalization

to higher

dimensions

61

We remark that the right hand side does not depend on the choice of ordered orthonormal bases u\, • • • ,Uk and v\, • • • ,Vk- When 77i and II2 are two 2planes in R3 the combined angle 9 is the angle between JJi and II2 in the ordinary sense. Let M be fc-dimensional oriented submanifold of M n . Let x,y G M be x—y a pair of points in M and let w = -. Let Sx(y) be an oriented kF-J/l sphere tangent to M at a; which passes through y, where the orientation of TxSx(y) is consistent with that of TXM. The combined angle 9 = 9M(X,U) between Sx (y) and Sy (x) is defined as the combined angle between the two tangent spaces at the points of intersection. Let u\, • • • ,v,k and i>i, • • • ,Vk be ordered orthonormal bases of the tangent spaces TXM and TyM. Let Hi G TyM be the image of U{ by the unit conformal transportation from x to y, i.e. v,i — 2{ui,w)w — m. Then the tangent space TySx{y) is spanned by an ordered orthonormal basis u\,- • • , ilk, which gives (—l) fe+1 times the orientation oiTySx(y). Thus the combined angle 9 is given by cos 9 = ( - l ) f e + 1 d e t ( ( u i , f i ) ) . The right hand side does not depend on the choice of ordered orthonormal bases u\, • • • ,Uk and v\, • • • ,VkDefinition 3.7 given by

([KS1']) The conformal (1 - cos6>)fe energy of M is

£(i-cos0)»=(M~)= / /

-j

J JMXM

r^f-dvol M (a;)dvol M (y).

F

_

y\

c.

,, 1 /i J dvol M (a:)dvolM(y) . . . A/r.., . Since the angle 9 and j —nrr—-— are Mobius invariant, so is \x — y \ Z k

£ ; (l-cos6l) fe (-^")-

Remark 3.11.1 When M is a knot then 9i<;{x,y) = 0{\x - y\2) by Lemma 12.1.2, but, in general, the combined angle 9M(x,y) is of order \x — y\ near the diagonal. For example, suppose M is a helicoid in R3 given by f(u,v) — (ucosv,us'mv,v), and x = /(0,0) and y = f(0,v). Then Sx(y) is the x^-plane and Sy{x) is the plane {j/ = tanwx}, and hence 0 = v = |x — y|. Therefore, unlike in the case of knots, when M2 is a surface inM 3 , sin M2XM?

\x-y\

d volM2 (x)d volM2 (y)

62

is well-defined only if e > 0.

Chapter 4

2/norm energy with higher index

As we saw in the last section there are E^ minimizers in prime knot types (Theorem 3.5.1) but it is conjectured that there are no £(2) minimizers in composite knot types (Conjecture 3.1). There are potentially two ways to produce energy minimizers for each knot type. One is to make the index of the energy greater than 2, and the other is to change the ambient space. In this section we consider the first approach. Since it does not seem easy to study E^a' when a > 3 we consider the following family of knot energy functionals instead. In this section we assume that a knot h is parametrized by arc-length.

4.1

Definition of (a,p)-energy functional for knots

Definition 4.1

ea,p

Let a > 0. Define ip^ : S1 x S1 \ A -» R by

ihw-hwr 1 fihoo-hmr 1

Jd(s,t)-i d(s,t)

1 / 1 a a \h(s)-h(t)\ * \h(s) - h(t)\ 63

1 d(s,ty

(a > 0),

'

64

L norm energy with higher index

Let ea'p(h) denote the L -norm of ip^' f° r 0 < p < oo ([03]), and put Ea'P{h) = (aea'p)p for 0 < p < oo: Ea'p(h)=ff

( ,, . .

1

, / ..

s-^-1

dsdt.

(4.1)

We call the product ap the index of ea,p and Ea'p. When (a,p) = (0, oo) we define the index ap to be 2. Since the integrand is non-negative and not identically zero, ea,p{h) and Ea,p{h) are positive for any (closed) knot h. We can define ea'p and Ea'p similarly for an open knot, i.e. an embedding from R or an interval into R 3 , when ea'p and Ea'p are non-negative. Remark 4.1.1

(1) Define the distance function of a knot h by Ah : S1 x S1 3 (s,t) ^ \h(s) - h(t)\ € K.

The position of a point in M3 is determined by the distances to three prefixed non-collinear points up to the mirror symmetry in the plane through the three points. Therefore the distance function, and hence the function (fh , determines the knot up to a motion of R 3 . (2) We have aea'x £ ( ) = £ 2 ' 1 = 2e2>1.

= E01-1 = E^

for 1 < a < 3, and in particular

2

(3) Holder's inequality implies that if 1 < pi < pi then e a ' p i (h) < e ' 2 (/i) for any a and h. ap

(4) When (a,p) — (0, oo) we have e0>oo(/l) = log(Distor(/i)), where Distor(ft.) is Gromov's distortion given by (2.20). (5) Since (2.12) implies l i m ^ 2 ) ( s , f ) = ^\h"(s)\2, we have 2 e2'°°(h) > — max|/i"(s)| . 24 ses1

(6) Let / be a knot with length If which is not necessarily parametrized by arc-length. Then ea'p(f) and Ea>p(f) can be defined as ea'p(f) and Ea'p(f) respectively, where / is the normalization of / that is parametrized

Control of knots by Ea'P

(ea'P)

65

by arc-length with length the unity:

Ea
Problem 4.1

(1) Does the map Ah : R>o —^ R>o given by

R 9 l 4 Area {(s, ( J G S ' X S 1 : \h(s) - h(t)\ > b} e R determine the knot ft. up to a motion of R3 ? (2) Put a = 2 for example. Does the map B^ = e2'' : M>o —>• R>o given by R > 0 9 p ^ e 2 ' p ( / i ) GR determine the knot ft- up to a motion of 1 4.2

Control of knots by Ea*

?

(e a 'P)

Theorem 4.2.1 (1) ([05]) The Junctionals ea'p andEa'p if and only if a and p satisfy

are well-defined

a < 2 or p <

(2 < a < 4), a —2 and they are self-repulsive if and only if ap > 2. In other words, Ea,p well-defined knot energy functional if and only if a and p satisfy 2 1 2 V > — (0 < a < 2) or > p > — (2 < a < 4), a a —2 a

is a

(4.2)

and ea'p is a well-defined knot energy functional if and only if a and p satisfy (4.2) or p = oo (0 < a < 2). (2) Let J£^2 denote the set of knots of class C2 with length 1 that are parametrized by arc-length. Then the map e*''(.) : [0,2] x (0,oo] is continuous, where Jt2

x X ^ M

is endowed with the C2 -topology.

(3) ([03]) Suppose a andp satisfy the above condition (4.2). If a knot h is a C1'^-embedding, where (3 > — , then ea'p(h) and Ea'p(h) are finite. 2p

66

L norm energy with higher index

Proof. (1) Well-definedness: The formula (2.12) implies that the integrand of (4.1) is 0(\s - i|( 2 -"' p ) near the diagonal A, hence ea'p and Ea<*> are well-defined if and only if (2 — a)p > — 1. Self-repulsiveness of Ea'p: The proof of Theorem 2.4.1 go parallel if we substitute a by ap to show that ea,p and E a,p do not blow up even if a knot has a self-intersection if ap < 2, and that they bound Gromov's distortion from above if ap > 2. To be more precise, \h(s) - h(t)\ > ^exp (-,f_a*£LP)

d(s,t)

(4.3)

for any s,t £ Sl. A direct calculation shows that e°'p is finite for a sigular knot given by (2.6) if p is finite, whereas ea'°° is obviously self-repulsive. (2) Generalize the map (fj^ to the diagonal A by

{

0

if 0 < a < 2 ,

W'(s)\2

i f a =2

24 Then the proof of Proposition 3.1.1 implies that the map M is continuous outside {2} x J£^2 x A. (3) If h is of class C 1,/3 then there is a positive constant A such that \h'(s) — h'(t)\ < A\s — t\P near the diagonal. Then for small u pu

\h(s + u) - h(s)\ >

r-u

(h'(s),h'(s+

£))<%> cos(lAA^)d£,. Jo Jo Therefore there is a positive constant A' such that \h(s) - h(t)\ >\s-t\A'\s - t\2l3+1 near the diagonal. Hence the integrand of ea'p is bounded by A"\s — £|(2/3-a)p n e a r the diagonal for some positive constant A". • Although the functional ea'p can connect E^ and the distortion continuously, we will forget about the distortion and study Ea'p in what follows because the distortion always needs a different argument and the inequality estimates will look simpler with Ea'p.

Control of knots by Ea
(ea'P)

67

So we call Ea,p the (a,p)-energy functional for knots if a and p satisfy the condition (4.2) in the above Theorem. The properties of Ea,p depend on whether the index ap = 2 or ap > 2: E ' (ap = 2) bounds a Lipschitz constant of h~l whereas Ea'p (ap > 2) controls the behavior of hi. If Ea,p(h) with ap > 2 is finite then h cannot have a sharp turn. ap

Let us fix a and p with ap > 2 in what follows. We refine Theorem 2.4.1 to have: Lemma 4.2.2 ([03]) Suppose ap > 2. There is a positive constant A\ — Ai(a,p) such that 1

ap

\h(s)-h(t)\>A1(E ' (h))

P+2

\h(s) -

h(t)\\^

d(s,t)

J

1

for any s,t G S and for any knot h which is parametrized by arc-length. Proof. Let hbea knot with \h"\ = 1 and B = £( Q '(/i) (B > 0). Let s,t be points in S1 = [0, l ] / ~ with 0 < s < t < s+ \ and let S = d(s,t) and d = \h(s) - h(t)\. Put A = 5 - d and £ = 1 - j = - . IiO
< d + u + v,

3 d(s + u,t — v) = S — (u + v) > d + — A. Therefore B>

I

I

( —,

(d + u + v)a

—

5^-r- I dudv

(d+f\)a,

rr—n*m'}'

>l

I

{d + u + v)~ap I 1 - 1 "V " 3 ' . " ) [ dudv.

Since d + u+ v < rf+|A_i d+fA " d+fA

*4-H)TfI

2£ < 4-£ -

x

_£ 2

(d + u + v)

ap

dudv.

68

L norm energy with higher index

Similarly, since

d + t+±

~ d+\\

8-6£~

8'

we have

^(<(+.+ „)-*,._LI{1-(3J±^)""J((,+.)— which implies

Jo Jo

Hence

J5>

H'-f)TH-fr'}{-(^fr}

j 2 — ap

(cup — l)(ap — 2)

As (1 - 0 ° < 1 ~ min{l, a}C for 0 < £ < 1 and a> 0, , a p _2 > ( n u n { l , g } ) P m i n { l , a p - 2} ~ 2e+P(ap-l)(ap-2)

R-irp+2 C

P+2

A^B-1

i-4V SJ ,

, v h r r r yl _ / ( m i n { l , a } ) " m i n { l , a p - 2 } \ ^ ! ^

wnere ^ i - ^ Corollary 4.2.3

26+P(Qp-i)(ap-2)

j

u

•

(1) Let B > 0. Define a function FB by FB{d)

d

gp-2

1- (yli-i.B^d) "

+2

Control of knots by Ea'"

(ea'P)

69

for d E O ^ S ^ ^ M . put

(d,5)c

0

r

11

ii —

L ' 'l J

2

d < S < FB(d)

dx Fig. 4.1

The domain Sin-

Then if Ea>P(h) < B then (\h{s)-h(t)\,d(s,t))E®B. (2) Define dx = dx{B) by FB(dx) Then if 0 < d < d\ then /

FB{d)
+ 2A1

= 2dlt i.e. dx =

ao-2

,

p+2

B^=^d~^

2~^A1B~^.

„ \

I < 2d.

Let Zvi • v2 denote the angle between two vectors v\ and v-i. The key lemma of this section is:

(4.4)

70

L norm energy with higher index

Lemma 4.2.4 ([03]) Suppose ap > 2. There exists a positive constant A = A(a,p) such that A{Ea'p(h)}^^d(s,t)1S^

Zh'(s) • h'(t) <

for any s,t £ S1. Therefore, if a knot h satisfies Ea'p(h)

< B for some B then h is

gp-2

i

2

2

a C ' (p+ ) -embedding, more precisely, h' satisfies the Holder condition of order —fwith a universal coefficient which depends only on ct,p, and 2(p + 2) B, but not on the knot h. Proof. The first half is a consequence of the following Lemma. The second half is implied by the first half since since \h'(s) — h'(t)\ < Zh'(s) • h'(t). U Lemma 4.2.5 Suppose ap > 2. There is a positive constant Ai = A2(a,p) such that for any B > 0 and 0 (0 < 9 < -|-) and for any knot h with Ea'P(h)
d(s,t)

< A2B

1

2(p + 2)

"p-2 0 «p-2

then Zti(s) • h'{t) < 9. Proof.

Put q = —^— — ? < 2. Suppose \h(s0) - h(t0)\ < ^ . Then, by (4.4),

for any s,t G[ s0,t0] d = \h{s) - h(t)\ < d(s0, t0) < di and hence d{s,t) < d + a3d1+q, ap-2

1

where a% = as(B) = 2AX p+2 B»p- 2 . (We will use the lower case a for constants which depends on B.) Therefore the curve segment h([s, i\) is contained in the solid cylinder with axis h(s)h(t) and radius re(rf) which is given by rB(d)

=

where a 4 = a4(B) = ^-A1

—*—^d 2

B2
+2

= a4d

+

•

Control of knots by Ea
Fig. 4.2

(ea'p)

A subarc contained in a solid cylinder.

Put d0 = \h(so) — h(to)\. Let us rescale the subarc /i((so — £,to + e)) by the homothety to obtain h = d0~xh so that \h(so) — h(to)\ — 1. Then if x, y € /i([so, io]) then the subarc of ft([so, to]) between x and y is contained in the solid cylinder with axis xy and radius a^d^ \x — y| 1+ "=\ Lemma 4.2.6 Suppose 0 < 9 < -|-. There is a positive constant As = A5(9) such that iff : [0,1] —> M3 is a C1 -immersion such that j-y(O) —T(^)I = 1 and that any subarc 0/7 between x and y is contained in the solid cylinder with axis xy and radius A$\x — y\1+^, then Zj'(0) • j'(l) < 9. Proof.

We show that — tan — 2 2

A*

{l + (tanf) 2 } J 2*

1

has the property of the Lemma. Assume 7(0) = (0,0,0) and 7(7) = (1,0,0). Put

$71

if{s : 7i(s) = xn},

Xn = (xn,0,0), yn = \h(sn) - Xn

for n £ N . Define zn inductively by z\ = A, Zn+1

|

+ J 4 5 (v££+1*")

'Hv^+4

If we apply the condition of the Lemma to the subarc 7([0, sn]) we can 1

ft

show inductively that yn < zn for all n 6 N. Since Ah < "T tan TT we have

L norm energy with higher index

72

Fig. 4.3

Solid cylinders.

< tan —. Inductively we have — < tan -^ for all n s N since

1+-:

=

V 2A5-Ln 5xn 2 < 1 +

— 2^1+1

< — + 2A52~^

{ 1 + ( tan 1+-:

< 2A5 + 2A5 ( V ^ + . . . + 2 - * W 1 + (tan | -

{l + (tan|)2}1+' <2A

2*-l

< tan •

Therefore the angle between 7'(0) and 7(0)7(1) is not greater than -w. D

Control of knots by Ea'P

(ea'P)

73

Proof of Lemma 4.2.5. Lemma 4.2.6 implies that if d = \h(s) - h(t)\ satisfies d < ^ ^ and a4{B)d^ < A5(8) then Zh'(s) • h'(t) < 9. Then di{B)

A*

a4{B)

6 i (tan • ^4iB

min <

«"-2

{l + (tan|)2} 2*

> min

-f

6"^2"T

-i+i

>yllJB"

(4.5)

2i+i

1

since 1 > t a n - | > - | and 2 " 1 _ i > 2 ~ 1 _ i ( y ) " 0^. Now we can choose A2 so that the right hand of (4.5) is equal to A2B~ We remark that A2 B ^ 2 ^ f < *_ for any 6 (0 < 6 < -§-). D Thus Ea'p (ap > 2) can control a knot locally by Lemma 4.2.4 and globally by (4.3). Namely, if Ea'p (ap > 2) is bounded above, then a knot cannot have a sharp turn in a small neighborhood and a pair of distant points by arc-length cannot be too close. Lemma 4.2.7

Let h be a knot. Put R

A2(Ea*(h))-^^)^,

=

where A2 is the constant given in Lemma 4.2.5, and

3. exp 16

Ea'P(h) 2

\a\P

{!-(!)"}

74

L norm energy with higher index

(see (4.3)). Then h(S1)nBp(h(s)) is connected for any s. Let it be denoted by [s—ti,s+t2]- Then \h(s+t) — h(s)\ (or \h(s—t) — h(s)\) is a monotonically increasing function oft € [0, £2] (or [0,£i] respectively). As a consequence, Bp(h(s)) D h(S1) is an unknotted arc. Proof. Assume s = 0. Divide the knot into two subarcs, Fi = h([—R, R\) and T2 = h((R, 1-R)). Then (4.3) implies that T2 n Bp(h(0)) = 0. On the other hand, if 0 < t0 < R then Zh'(0) • h'(t) < -|- for any 0 < t < t0, and hence Zh'(0) • (h{t0) - h(0)) < f. Therefore Zh'(t0) • (h(t0) - h{0)) < \ , which completes the proof. • Lemma 4.2.4 shows that ea,p (cap > 2) blows up if a knot has a pull-tight. The behavior of ea,p when a knot has a pull-tight is given as follows. Theorem 4.2.8 ([03]) Consider a line segment. Put a small 1-tangle T in the center of the line segment to make an open knot. Shrink the tangle T to a point while keeping its shape similar.

Fig. 4.4

Pull-tight of a tangle.

Under this pull-tight process the value of Ea
2 then the value of Ea,p

blows up.

(2) If ap = 2 then the value of Ea'p converges to a positive constant which depends on the shape of the tangle T. (3) If ap < 2 then the value of Ea'p converges to 0 which is equal to the value of Ea'p of the straight line segment. Note that in this case Ea'p is not self-repulsive. Outline of proof. Let us give a brief explanation. Suppose a tangle T shrinks to half its size. Then the integrand becomes 2ap times larger on T x T, whereas the area of T x T becomes 2~2 times smaller. Therefore the contribution of the tangle T to the integral (4.1) under the pull-tight process becomes 2 Q p _ 2 times larger. Thus the behavior of Ea,p depends on whether ap — 2 is larger, smaller or equal to 0. •

Complete system of admissible solid tori and finiteness of knot types

C o r o l l a r y 4.2.9

4.3

If ap > 2 then Ea'p

is not Mobius

75

invariant.

C o m p l e t e s y s t e m of a d m i s s i b l e solid tori a n d of k n o t t y p e s

finiteness

L e m m a 4.2.4 implies t h a t only finitely many "shapes" of knots can occur under any given Ea'p {ap > 2) threshold. Let us fix a,p with ap > 2 and B > 0 in the rest of this chapter. Definition 4.2 ([05]) (1) A homeomorphism F : D2 x S1 —> T G R 3 is called an a d m i s s i b l e ( £ - ) s o l i d t o r u s if the following three conditions are satisfied. (i) Let f(t)

= F(0,t)

be t h e core of T . T h e n | / ' | = 1 and Ea>p(f) 1

(ii) T contains the e-tubular neighborhood NE(f(S ))

< B.

of the core knot.

a,p

(iii) If a knot h which satisfies E (h) S1 is a homeomorphism, where pr 2 : D2 x S1 —> Sl is the projection. Namely, h intersects each meridian disc transversely at one point. (2) A set of finitely many admissible (e-)solid tori & = {T\, • • • , Tn} is said to be a c o m p l e t e ( e - ) s y s t e m of solid tori if any knot h t h a t satisfies Ea'p(h) 0.

([05]) There is a complete

(e-)system

of solid tori for

Roughly speaking, we construct a complete system of "thickened polygonal knots". It suffices to work with JCQ = {h : \h'\ = 1, /i(0) = 0, /i'(0) = ( 1 , 0 , 0 ) , and Ea'p(h)

< B}.

Definition 4.3 Let J V E N and r > 0. By an i V - p o l y g o n a l k n o t we mean a sequence of N distinct points K — ( 0 , X \ , • • • , X N - I ) in R 3 . (1) A knot h 6 J^o is said to be r - c a p t u r e d by a polygonal knot K=(0,Xir-,XN^) if hi-fc) e Br(Xj) for all j . (2) A polygonal knot K = (0,Xi,--, X J V - I ) is called a d m i s s i b l e (JV, r ) - p o l y g o n a l k n o t if there is a knot h € J^o which is r-captured byK.

L norm energy with higher index

76

(3) A set of finitely many admissible N, r-polygonal knots

S= {Ki = {0,Xl--- ^ J r . J l l 0 there is a complete of (TV, r)-polygonal knots.

family

Proof. We do not need t h e condition Ea'p(h) < B of J^o here. Since J#6 is equicontinuous and uniformly bounded, it is relatively compact by t h e Ascoli-Arzela theorem. T h e n t h e conclusion follows if we define t h e distance of J^o by dist(/i,g) = max s ( E si \h(s) — g(s)\. D T h e proof of Theorem 4.3.1 reduces t o t h e following Lemma. L e m m a 4 . 3 . 3 ([05]) Th ere is No 6 N, ro > 0, and so > 0 such that for any admissible {NQ^Q)-polygonal knot K, there is a homomorphism FK : D2 x S1 -> T C M3 such that: (i) IfheJ^o

by K then Neo{h(Sv))

is recaptured x

C T,

(ii) If g e J^o satisfies g(S ) C T then pr 2 o FK~

X

o g is a

bi-Lipschitz

map. Sketch of proof. Let K be an admissible (N, r)-polygonal knot. By thickening edges of K by R > 0 we get a piecewise cylindrical solid torus TK{R). L e m m a 4.2.4 implies t h a t for sufficiently large N and for sufficiently small r there is a positive constant R = R(N, r) which is independent of t h e polygonal knot K such t h a t any knot h £ J^o t h a t is r-captured by K is contained in

TK(R)-

Fig. 4.5

A piecewise cylindrical solid torus.

We then choose N, r, and e so t h a t TK(R + ff) is a solid torus without self-intersection for any admissible (N, r)-polygonal knot K. We use

Existence

of Ea'p

77

minimizers

Lemma 4.2.4 for local control of the tangent vector h' and (4.3) for global control of the closest approach of a pair of points which are distant from each other by arc-length. If we deform a homeomorphism FK of a piecewise cylindrical solid torus TK(R) by transforming the core from the edges of the polygonal knot K into a smooth knot in J^o which is r-captured by K, we obtain an admissible e-solid torus as desired. • As a corollary of Theorem 4.3.1 we have: Corollary 4.3.4 Only finitely many knot types can occur under any given Ea,p threshold if ap > 2. More precisely, there is a constant C > 0 such that the number of knot types with a representative h with Ea'p(h) < B is smaller than exp for any B E I . Remark 4.3.5 (1) It is not known whether the Corollary holds for with ap = 2 and 0 < a < 2.

Ea'p

(2) When (a,p) = (0, oo) Gromov ([Grl]) showed that there is a Bo £ M. such that there are infinitely many knot types that satisfy e 0,o ° = log(Distor) < B0.

4.4

Existence of E a,p

minimizers

Theorem 4.3.1 implies the existence of energy minimizers. Suppose ap > 2 Let [K] be any knot type. Take a sequence {hi, /12, • • • } in [K] such that lim Ea'p(hn)

= Ea'p([K})

n->oo

= inf

Ea'p(h).

h£[K]

By the Ascoli-Arzela theorem there is a convergent subsequence {hni}. Theorem 4.3.1 ensures that there is a solid torus Tj such that infinitely many /i n i 's are included in Tj in a good manner. Let h^ be their limit. Then / i ^ is also contained in Tj in a good manner and hence belongs to the same knot type [K]. By Fatou's lemma Ea'p{h00) = Ea>p([K]) hence hoo is an Ea'p minimizer for \K\. Thus we have Theorem 4.4.1 ap > 2.

([05]) There is an Ea'p

minimizer for any knot type if

L norm energy with higher index

78

We conjecture t h a t the number of Ea,p minimizers (ap > 2) for each knot type is finite. We remark t h a t the number of the solid tori associated to Ea'p minimizers (ap > 2) in the sense of Theorem 4.3.1 is finite. Buck pointed out t h a t an energy critical knot for a composite knot type may not be unique. For example, Ki^K^K^ and Ki^K^Kz in this order may produce different knots t h a t a t t a i n the local minimum of the energy. On the other hand, when the index of the energy is 2, it is not known whether there is an ea'p minimizer for any knot type if ap = 2 ( a / 2) or (a,p) = (0, oo).

4.5

Ea,p

T h e circles m i n i m i z e

Let h0{t) = ( ^ cos27rr,, -^ sin2-7rt,0) denote a circle as before. Then the chord length between h0(t + s) and h0(t) is given by \h0(t + s) — h0(t)\

— — sin7rs TV

1

for any t £ S . Using ideas from Liiko Gabor ([LG]) and Hurwitz's proof of the isoperimetric inequality, A. Abrams, J. Cantarella, J. G. Fu, M. Ghomi, and R. Howard showed the following. L e m m a 4.5.1 on (0, -|-) then

([ACFGH]) Suppose ip : M —)• R is decreasing and

Lrp{\h(t+ )~ h(t)\ )dt > v 2

i

S

convex

( ( ^ y

with equality if and only if h is a circle. I

/ / ip satisfies an additional

condition

that

I

I (Sill

ip I I

7TS \

1

j

I ds is finite

then

4>(\h(t + s) - h(t)\2)dtds > [

2\

v f f 8 " ^ds)

with equality if and only if h is a circle. T h e o r e m 4.5.2 ([ACFGH]) Suppose on (0, -j) for any s € 5 1 and f -00 f

J f

V (s, ( ^ ^ )

2

) ds

with equality if and only if h is a circle. Proof.

ip{£,) = ^ ( 5 , 0 satisfies the condition of the above Lemma.

n

If we put d(s,0)a

^ we obtain

Corollary 4.5.3 ([ACFGH]) Suppose p > 1 and 0 < a < 2 + ±. Then only the circle gives the minimum value of Ea'p. 4.6

Definition of a - e n e r g y polynomial for k n o t s

Definition 4.4 For 0 < a < 2 define the a-energy polynomial Pg(h; u) of a knot h by a generating function of Ea'n{h): E exV(uvla\s,t)),dsdt

P%\h;u)=

•jj

l

s l x

1 exp
a1 l + Eicitu^. ' (h)u

Ea 2 +, ' W..2

\nvn-^T7Z;)\**dt d(s,t)°

, ^ W

2!

.3

3!

is a well-defined knot energy functional for any u > 0.

4.7

Brylinski's b e t a function for k n o t s

Brylinski considered the principal term of E^ complex number a.

as a complex function of a

Definition 4.5 ([Bryl]) Suppose ft is a knot which is parametrized by arc-length. Let z be a complex number. The b e t a function Bh(z) of h

L norm energy with higher index

80

is defined by

\h(s) - h(t)\zdsdt.

Bh(z) = f f Since

S1 x S1 3 (s,t) ^ \h(s) - h(t)\z e C is continuous for !Re (z) > 0 the integral converges in that domain. Example 4.1

Let ha be a circle. sin — ) dt 2 = 2 2+3 TT f2

(sinx)zdx

Jo change of variable sin x = ^/y~ implies j

2

(smx)zdx

1 r1 *=i, = - j yz~S{l-y)

,-x, „,z 1 1 I dy = B(~ + - , - ) ,

{siax)"dx = r's beta function where B(p, q) is Euler's beta function dx

Jo Using the functional equation

r(p)r(q)

B(p,q)

r(P +
where P(p) is Euler's gamma function /•OO

\W) = I/ r(w)=

Jo

W

t ~h e-'e-'dt

we have

Therefore Bho (z) is a meromorphic function of z. As r ( ^ — ) takes its poles at

= 0 , - 1 , —2, • • • the poles of Bh0 (z) are z = — 1, —3, —5, • • •.

Other L -norm

energies

81

Theorem 4.7.1 ([Bryl]) For a smooth knot h the function C 9 z >->• Bh{z) 6 C extends analytically to a meromorphic function on C with only possible poles at z = — 1, — 3, — 5, • • •. The residues can be computed explicitly by integrating a polynomial in the curvature, the torsion and their derivatives. When z = - 2 , Bh{-2)

E{?\h).

= E™(h) - 4 =

P

4.8

Other L -norm energies

The function ip^*' is not the only candidate to produce L -norm knot energies. We can modify the Mobius invariant knot energy functionals which will be given in the second part of this book. Let 9h(s,t) be the conformal angle between h(s) and h(t) (see Definition 3.2). It is of the order of \s —1\2 near the diagonal as will be shown in Lemma 12.1.2. We showed in Section 3.4 that EW satisfies

\h(s)-h(t)\2

Jisixsi

We will also show in Chapters 12 and 13 that v ru\ f f Esme(h) = / /

-j—

sin6h(s,t) rjw^dsdt

and sin uh ~ Vh cos Vhdsdt Emnts{h) = / / —rrv-, ,,.,,„ 1 J , sJS^xS ixS1 \h(s)-h(tW are well defined knot energy functionals (with respect to the C 2 -topology). These functionals are invariant under the Mobius transformations since the angle Oh and the 2-form

^ are both Mobius invariant. \x - yY

We can take the L -norm of the integrands of these functionals to obtain

82

L norm energy with higher index

new knot energy functionals (with respect to the C 2 -topology): 'l-coe*h(M)\ s i x S i \ \h(s) -

'

h(t)\*

sineh(s,t) y d s M ,sixSi

'

K\h(s)-h(t)f

(smOh-9hcoseh\p lslxS1{

\h(s)-h(tW

\$

J

1

'

although these new functionals are not Mobius invariant any more. Another L -norm energy is given in Definition 8.7.

Chapter 5

Numerical experiments

Numerical experiments aimed at finding beautiful "optimal embeddings" as energy minimizers have been carried out by evolving polygonal knots so as to decrease their discretized energies. T h e energy of polygonal knots has been studied in [BOl; BS1; F u k l ; K S 1 ' ; Lig-Se; S i m l ; Sim2]. As for computer programs and the results of numerical experiments on E^2' and E^ with higher index, the reader is referred to [Fuk2; Gun; Lig-Se; K S 1 ' ; Sim2; S u l ] . Some of the E^ minimizers were visualized in a video by A h a r a [Ah] (part of whose results were reported in [04]). A knot table of the minimizers of E^ was given by Kusner and Sullivan [KS1']. Some of their result can be found in Section 5.3 and Chapter 6. T h e program "KnotPlot" by R. G. Scharein is available on the web at [Sc]. T h e program "KView" by K. Hunt is available at [HS]. T h e program "MING" by Y.-Q. Wu, which is based on the minimal distance energy by J. Simon, is available at [Wu]. T h e program "Surface Evolver" by K. Brakke is available on t h e web ([Bra2]); this was further extended by J. Sullivan t o deal with other energies like repulsive-charge knot energies. Informations on computer programs, pictures of beautiful knots, and moving pictures (mpeg files) of knots evolving themselves to decrease their energy are available at [HS] [Sc], [Su3], and [Wu]. Results on "optimal embeddings" in various senses are reported in [SKK]. Numerical computations of the relations among E^2\ the absolute value of the writhe (see Definition A.2), the average crossing number (see Definition A.5), and the minimum crossing number by Stasiak et al. can be found in [CS; KBMSDS; KOPDS]. Approximately affine relations have fre83

Numerical

84

experiments

quently been observed in numerical experiments. For example, for certain families of knots which differ only by the number of twists they contain, like a family of (2, k) torus knots (k = 3, 5, • • •), E^ and the minimum crossing number of E^2' minimizers are related in approximately affine manner with the same slope ([HKS]). The applications to the study of DNA can be found in [KBMSDS; LKSKDS; SKBMD]. 5.1

Numerical experiments on

Among knot energies, E^ has been studied most intensively. There are several ways to discretize E^2\ Ahara [Ah] considered the energy on the space of equilateral polygons. Kusner and J. Sullivan [KS1'] computed experimentally the approximate minimum values of E^ for knot types with the conjugate gradient method, testing three kinds of dicretization and concluded that the cosine formula (3.6) is better than the other two. Simon's UMD is based on the approximation of the minimal distance between a pair or edges, and Ligocki and Sethian used the annealing method to deal with very complicated knots. No examples of a pair of knot types of prime knots with the same minimum value of E^2> have been reported up to now (November 2002). The order of knot types according to the minimum values of E^ might depend on computer programs. In fact Simon's UMD implies 7i < 819 < 72 < 73 < 820 < 74.

The reader is referred to Remark 3.5.2, Conjectures 3.1 and 3.2. 5.2

a > 2 cases. The limit as n —> 00 when a > 3

In order to produce beautiful knots as energy minimizers, it is better to use the energy with index greater than 2 for the following reason. It is conjectured that there are no £(2)

minimizers in composite knot types whereas there are always energy minimizers when the index is greater than 2. The Mobius invariance property implies that there are infinitely many -minimizers for any prime knot type whereas the number of the 'shapes' of the energy minimizers is finite in the sense of Theorem 4.3.1 when the index is greater than 2. Since E^ does not blow up even if we have a pulltight, E^ cannot evolve a small tangle ([Lig-Se]) whereas the energy with

a > 2 cases.

The limit

as n —f oo when a > 3

85

index greater than 2 can. The larger the index of the energy, the larger the order of blowing up as a knot degenerates to have self-intersections, hence the smaller the possibility of a knot to change its knot type during the knot-evolving process. This is why a higher index is used or the total squared curvature functional is added in such computer programs as [Fuk2; Gun; Sc]. Since we studied only E^ with a < 3 in Chapter 2 let us study E^ with a > 3 here. Take a polygonal knot h in a knot type [K] with n vertices, Xi, • • • , xn, with total length L. Assume a point charge |XJ+I — Xi\ at each point Xi, where the suffix numbers are taken modulo n. Evolve h so as to decrease the value of the principal term of the discretized a-energy: \Xi+l

-Xj\\Xj \Xi

+ l

-Xj

X 7"

where we put La 2 to make it scale invariant, until it comes to a critical point. Then increase the number of the vertices.

increase the number

evolving !iliMIl!|*

/

to decrease

;'

the energy

i-._

evolving •Ililltliilj^.

of vertices

F i g . 5.1

?

to decrease the energy

K n o t evolving p r o c e s s .

By repeating this process the polygonal knot converges to a continuous (a) map /loo S1 Let us consider the characterization of h If we expand En '(h) in n then the dominating term among those terms which are dependent of the knot h is the total squared curvature functional for any a > 3 by Theorem 2.3.1. On the other hand Langer and Singer showed that there is no stable critical point with respect to the total squared curvature functional in any non-trivial knot type, as is stated in Theorem D.l later. Thus we are led to Conjecture 5.1 (1) n£

If a > 3 then n£' satisfies the following conditions

is independent of a.

Numerical

86

experiments

(2) hx attains the local minimum value of the total squared curvature functional among [K], where [K] is a closure of [K] in the set of immersions with respect to the C2-topology.

(3) ht] e[K]\[K]5.3

Table of approximate minimum energies

We end this chapter by introducing Kusner and Sullivan's result on numerical computation ([KS1']). Figures 5.2 - 5.5 show the approximate energy minimizers and their values of Eo = E^ — 4 of the first 36 knot types of prime knots ordered according to their energies computed by the cosine formula.

Table of approximate

minimum

50

unknot

-

energies

100

87

150

0

-

70.4 (2 3)-torus knot

3i

lligure eight " 5i

10-1.9

& (7

0

(2."j)-torus knot

126.8.

I:M.H

102.8

1(58.5

172.9

63

7i

Fig. 5.2

(2,7)-lorus knot

181.0

Approximate energy minimizers and their values of E^'

200

Numerical

0

72

50

experiments

100

150

200

mm 192.7

{'AA) torus knot

>19

74 '

"^

197.0

197.7

199.7 ^

2(W.7

w 520

g

207.1

77

8

2CW.9

-0^ Fig. 5.3

209.5

Approximate energy minimizers and their values of E,(2)

250

Table of approximate

0

50

minimum

energies

100

150

89

200

220.')

224..1

22(i.2 ^yd

c/-C

228.(1

A>

231.3

232.4

£x

233.2

233.7

>n

Fig. 5.4

Approximate energy minimizers and their values of EQ

250

90

Numerical

0

50

experiments

100

150

200

250

234.0

(2,9)-torus knol

9i

>12

23-1.(5

235.1 '"X

>13

237.9

Ho

23*.

>14

a

213.1

515

>16

U7

238.7

/Cx

2()l.-r)

eg

2().">. I

Fig. 5.5

Approximate energy minimizers and their values of E„

300

Chapter 6

Stereo pictures of E^

minimizers

Figures 6.1 - 6.16 show the stereo pictures of the approximate E^ minimizers obtained by Kusner and Sullivan. To see the stereoscopic effect, look at the left figure with the right eye and the right one with the left eye (by crossing the eyes). These figures are taken from [Kusl], where the full archive, including their results for links, is available. Note that E^2' minimizers are not unique because of the Mobius invariance.

91

92

Stereo pictures of E^2'

Fig. 6.1

minimizers

Trefoil-left. Look with the right eye.

Fig. 6.2

Figure eight-left.

Stereo pictures of E^2>

Fig. 6.3

minimizers

Trefoil-right. Look with the left eye.

Fig. 6.4

Figure eight-right.

93

94

Stereo pictures of E^2'

Fig. 6.5

minimizers

52-left. Look with the right eye.

Fig. 6.6

6i-left.

Stereo pictures of E^

Fig. 6.7

minimizers

52-right. Look with the right eye.

Fig. 6.8

6i-right.

95

96

Stereo pictures of E^2'

Fig. 6.9

minimizers

62-left. Look with the right eye.

Fig. 6.10

82-left.

Stereo pictures of E^2'

Fig. 6.11

minimizers

62-right. Look with the right eye.

Fig. 6.12

82-right.

97

98

Stereo pictures of E^2'

Fig. 6.13

minimizers

63-left. Look with the right eye.

Fig. 6.14

819-left.

Stereo pictures of E^2'

Fig. 6.15

minimizers

63-right. Look with the right eye.

Fig. 6.16

819-right.

99

This page is intentionally left blank

Chapter 7

Energy of knots in a Riemannian manifold

An alternative approach to produce energy minimizers for knot types is to change the ambient space. 7.1

Definition of the unit density (a,p)-energy

E^

Definition 7.1 ([07]) Let / : S1 —• M be a smooth embedding into a Riemannian manifold M = (M,g). Let dM(f(s),f(t)) be the innmum of the lengths of the paths which join f(s) and f(t) and let dn(f(s),f(t)) be the shortest arc-length between the two points. Define the unit density (a, p)-energy functional for knots in M, £ ^ , by Ea'p (7.1) and put e Q / ( / ) = ±{Ea*(f)}l P>—

2 a

( 0 < a < 2)

We call ap the index of E^f

for or

1 2 >p > — a —2 a

(2 < a < 4).

and e ^ - If p — 1 we write E^

=

E°^.

This condition for.a and p is the same as that in the Euclidean case, (4.2) in Theorem 4.2.1. When p = 1 E^' = Ea^ can be interpreted as the renormalization of r"a modified electrostatic energy of charged knots under the assumption that the electric density is constantly equal to 1 so that the total quantity of electric charge is equal to the length If of a knot f{Sl). 101

102

Energy of knots in a Riemannian

manifold

The well-definedness of E°^f is a consequence of Nash's theorem [Nas] which guarantees that any manifold M can be embedded isometrically into the Euclidean space M.N for sufficiently large N. Note that Eapf(f) > 0 for any knot / since the integrand is nonnegative. The Jacobian term | / ' ( s ) | | / ' ( t ) | makes E^f invariant under re-parametrization.

7.2

Control of knots by E °$*

We assume that knots are parametrized by arc-length. Theorem 7.2.1 E^f is self-repulsive. More precisely, for any i £ K there is a positive constant C = C(a, b, I) such that if a knot f of length I satisfies Ea*{f) CdK(f(s)J(t)) for all s,t e S1. Proof. The argument in the Euclidean case works as well for M since the estimates involved can be derived from the triangle inequality of the metric space and the fact that <1M{X, y) < dx(x,y). As we cannot normalize knots to have total length 1 because there is no standard homothety, we use the change of variables of the integral, s — ls',t = It', instead. Therefore the coefficient C depends on the total length I of the knot. • We establish a similar estimate for the distance of a pair of points on a knot in terms of the arc-length given in section 4.2 by embedding M isometrically into some Euclidean space. We fix a and p (ap > 2). Let us fix an isometric embedding i : M —>• M^ and identify M with its image L(M). Lemma 7.2.2 If M is compact then there are positive constants c\ and C2 such that if dM{P,Q) < c\ then dM(P,Q) < \P — Q\+c2\P — Q\s, where \P — Q\ is the distance in M.N. Proof. It is shown in [Miln2] that there is some EQ > 0 such that M can be covered by finitely many open sets C/j's such that, for each Ui, (i) any pair of points in [/» can be joined by a unique geodesic of length < £o,

(ii) for each p € Ui the exponential map exp maps the £o-ball in TpM diffeomorphically onto expp(£?£o) D Ui.

Control of knots by E

103

Therefore if CIM{P,Q) < £o then p and q can be joined by a unique geodesic 7. Assume t h a t d,M{p,q) = t and 7 is parametrized by arc-length with 7(0) = p, j(t) = q. By Taylor's expansion

b _,i_«_M,. for some 0 < C3 < 1. P u t C4 =;

, l7"(^)P

max

PeMn'{0)£S(TpM),At£[0,eo]

{

7 is a geodesic such t h a t | 7 ' | = 1,7(0) =p

24

Then \p — q\ > t — c^t3. As there is a positive constant £\ such t h a t if t < E\ then (t — C4i 3 ) 3 > \t3, if t < m i n { £ 0 , £ i } then c4t3) + 2c{t - c4t3)3

dM(p, q)=t<(t-

2c4\p - ql3

<\p-q\+

a A knot in a manifold M cannot necessarily be normalized to have the unit length. But Lemma 4.2.4 also holds for Ea^f as it can be proved without any condition on t h e t o t a l length of a knot. T h e n Corollary 4.2.3 can b e paraphrased to L e m m a 7.2.3

Put q = — as in Chapter 4. p+ 2

(1) Let b > 0. Put C5 = Cs(b) = A ^ B P T 5 , where A\ is given in 2

4.2.4, and di = c 5 " = A\B

forde

[0,d2).

"p~ . Define a function

Lemma

FB by

Put &B = {(d,5)

c [0,oo) 2 \d<

5

Then if a knot f with \f'\ = 1 satisfies EaAf(f) (dM{x,y),dK(x,y)) (2) Define d1 = di(B) 0 < d < d\ then FB(d)

then

G 2>'B.

by ^ ( d i ) = 2dx, i.e. di = (2c 5 )~^". Then if

< d(l

+ 2c5d9)

< 2d.

Energy of knots in a Riemannian

104

manifold

Proposition 7.2.4 There is a positive constant a knot f satisfies Elf(f) < B then dM{x,y)

<\x-y\

CQ

=

C6(J3)

such that if

(1 + c6\x - y\q)

for any pair of points x,y on /(S*1) with dM{x,y) < rnin{di,ci}, where C\ is the constant given in Lemma 7.2.2. Proof. Put 5 = d,K{x,y),d = d,M(x,y),dw = \x—y\, andcfa = min{di,ci}. Since CZR < 6M < d$ we have 6
+ 2c5d9)

< (du + c2dl) (1 + 2c5 (dR +
+ c2dl)(l

+ cfdl),

c2dl)q) where

c' = 2c 5 (l + c2d%)q

< dR (l + c'dl (l + Cjjd\-q + c2di)) , which completes the proof as q < 2.

•

As a corollary, Lemma 4.2.5 also holds for E^f,

and hence:

Lemma 7.2.5 Suppose ap > 2. Let us fix an isometric embedding t : M —>• M.N and identify M with L(M) C M.N and knots in M with knots in RN through t. There exists a positive constant A = A(a,p,M,i) such that if a knot f : Sf = [0, i ] / ~ -» M satisfies | / ' | = 1 then \f'(s) - f'(t)\

< Zf'(s) • f'(t) < A

{Ea*{f)}^md^t)^k

for any s,t 6 Sf, where d(s, t) is the arc-length in Sj and • is the standard distance ofM.N. Therefore, if a knot f parametrized by arc-length satisfies E^f(f) < B 1

a p —2

for some B then t o / is a C '2(p+2) -embedding, more precisely, (t o / ) satisfies the Holder condition of order — with a universal coefficient 2 ( p + 2)

which does not depend on the knot f. We remark that Lemma 4.2.7 also holds for E

•"

Existence of energy

7.3

minimizers

105

Existence of energy minimizers

Our approach to obtain an energy minimizer with respect to a knot energy functional e for a knot type [K\ is first to take a sequence {fn} of knots which belong to [K] with limn^oo e(/ n ) = e([.K]) and then to take a subsequence of {/ n } which converges to a limit /oo E [K]. The following difficulties might arise in our approach.

(a) The limit /oo may not exist. For example f^ may diverge when the ambient space is non-compact. (b) The limit /oo may not be smooth. The length of /oo may diverge like a Peano curve. (c) The limit /oo may converge to one point. (d) The limit / ^ may not belong to [K] because pull-tight has changed its knot type.

Let us give conditions to avoid such possibilities. As for case (a), the Ascoli-Arzela theorem implies Lemma 7.3.1 Let {fn} be a sequence of knots in M. Assume that there is a compact subspace W and a constant IQ > 0 such that fn(Sl) is contained 1 in W for all n and that the length of f^S ) is smaller than lo for all n. Then we can reparametrize fn to obtain a new sequence {fn} that has a converging subsequence to a continuous map /oo with respect to the C°topology. We remark that the limit /oo is not necessarily an embedding. As for case (b): Lemma 7.3.2 There is a positive constant l\ — h(W, a,p;B) for a compact subspace W of M, a constant b and a knot energy functional E^f such that if a knot finW satisfies E°j^'{f) *E(± •

>2//"4J-

-y1'^)

1 -

0

V,

— dt ta )

-m

dt

' 2l{:i-2L) 2 '(I -AL)

*('-*)'

LaP

which blows up as I increases to oo.

•

As for case (c), Lemma 7.2.5 implies Lemma 7.3.3 Suppose ap > 2. Then for a compact subspace W of M and a constant B there is a positive constant l2 = 1%(W, ct,p; B) such that if a knot in W satisfies E^f(f) l2As for case (d), Lemma 7.2.5 implies Lemma 7.3.4 Suppose ap > 2 and M is compact. If Ec^f(fn) formly bounded then pull-tight does not happen.

is uni-

Lemma 7.2.5 and the above lemmas allow us to use the same argument as in the Euclidean case (Theorem 4.4.1) to obtain Theorem 7.3.5 Let M be a compact manifold. minimizer for any knot type if the index ap > 2.

Then there is an Ea^

Sketch of proof. We identify M with t(M) C RN through an isometric embedding L, and / : S1 -> M with t o f : S1 ->• K^ as before. Let [K] be a knot type. Put B = E°${[K]) + 1 and JT = {/ : [0,/] -»• RN | I > 0, | / ' | = 1, E'if(f)

[K]} .

By Lemmas 7.3.2 and 7.3.3 there are constants l\ and I2 such that if / € J£T then its length If satisfies I2 < 1/ < h- Hence Jf is equicontinuous. The argument in section 4.3 is parallel. The difference is that we cannot

Examples : Energy of knots in S3 and H 3

107

normalize knots to have total length 1 in this case. We modify Definition 4.3 to the following form. Definition 7.2

Let p, r > 0.

(1) A knot / 6 J(f is said to be (p, r)-captured by a polygonal knot K = (Xi, • • • , Xk) in RN with k vertices if (A; - l)p
{K* = (Xi,---,Xl)\

\
is called a (p, r) complete family of polygonal knots if any / € Jtf can be (p, r)-captured by some Kl. Then Proposition 4.3.2 and Lemma 4.3.3 also hold in this modified situation, and hence we have a complete system of solid tori DN~l x Sl, which gives the proof. •

7.4

Examples : Energy of knots in S3 and H 3

When the index ap = 2 we conjecture that it depends on the metric of the ambient space whether there are energy minimizers or not. To see (2)

this, let us consider EM when the ambient space M is the 3-dimensional sphere S3 C M4 or the 3-dimensional hyperbolic space H 3 , which are manifolds with constant curvature ± 1 . In order to avoid confusion we write E]^ = E^ and E°^£ = Ea'p to specify the ambient space. Recall that (2)

in the Euclidean case there are infinitely many -Ej^ minimizers in prime (2)

knot types whereas it is conjectured that there are no E^

minimizers in

(2)

composite knot types. We conjecture that there are E's3 minimizers for any knot type whereas there are no E]^ minimizers for any knot type. 7.4.1 Energy of circles in S3 The restriction of the standard metric of M4 to S3 provides the standard metric of S3. Let x and y (x ^ y) be two points in 5 3 . The (shorter)

Energy of knots in a Riemannian

108

manifold

geodesic between x and y is contained in the great circle through x and y which is obtained as the intersection of S 3 and the 2-plane that passes through x, y, and the origin, which is uniquely determined unless x and y are antipodal. The spherical distance dg-i{x,y) between x and y is given by the angle between Ox and Oy, which satisfies d§3 (x, y) = arccos ((a;, y)) — 2 arcsin I — \x — y| We note that E"sf(K) < E^?(K) for any knot K in S 3 since dS3(x,y) > d^i{x,y) for any i / y , and dK(x,y) does not change whether we use the spherical metric or the Euclidean metric for the ambient space. Proposition 7.4.1 ([07]) Let Sr (0 < r < §) be a circle in S3 C R4 with geodesical radius r given by Sr : [0,27r] 3 9 H (sinrcos9,sinrsin9,0,cosr) € S*3. T/ien (a,p) energy E°gf(Sr) of r with

o/ SV is a monotonically decreasing function

£°?(Sf)=0,

^o

[oo

In particular, Eysi(Sr)

(ap>2),

is a monotonically decreasing function of r with

0 = Ef3(S%)

< \hn Ef3(Sr)

= E$(h0)

= 4,

(21

which implies that E sl is not Mobius invariant. Proof.

Put R = sin r. Then

=23-^TT

r R2-^ {(—^—x J0

[\ arcsm(7? sin t))

--Y ta J

Examples : Energy of knots in S

SinCe

M (arcBinCflsino) <

0 f

°r ^

and M

^ (° < ^ < l)

109

and

for

^

* > 0'

and 2 — ap <0, the integrand of the right hand side is a monotonically decreasing function of R for any t. Since lim 7.4.2

r = — , lim E " ? ( 5 r ) = lim R2-apEaJ(h0).

£

Energy of trefoils

n

on Clifford tori in S3

Let r r denote the trefoil on the Clifford torus Tr C S3 C C 2 of radius r (0 < r < 1) which is given as rr;2,3 in (3.20). Then the Euclidean distance between rr(s) and r r (i) is given by - y)f + ( V l - r 2 sin 5 i ^ _ J ^ .

| T r (a) - rr(t)\ = J(2rSm(x

On the other hand, since |r£(0)| = \/9 — 5r 2 the arc-length is given by dTr(Tr(s),Tr(t))

= ^ 9 - 5r 2 |s - t|

for 0 < Is — tl < 7r. Therefore

r - '' *g(V

W_*(;

4 arcsin Jr2 sin2 £ + (1 - r 2 ) sin2 §

Numerical experiment by Ligocki [Lig] implies that ESI{TT) is concave with respect to r and takes the minimum value approximately 54.3263 at r « 0.861388, while it blows up as r goes to 0 or 1. (2)

7.4.3

Existence

of E s£

minimizers

Let us consider whether case (c) or (d) happens with respect to E si. Recall that numerical experiments tell us that the approximate minimum value of E^p of a trefoil is about 74 (Table 5.1), which implies that the approximate minimum value of E^3 of an open trefoil is about 70 which is 74 — 4 by Theorem 3.3.2. (c) Suppose a trefoil in S3 converges to a point. Then the metric around the knot becomes almost Euclidean hence the value of Eysi is not smaller than 74.

Energy of knots in a Riemannian

110

manifold

(d) Suppose gpt is a trefoil which is almost equal to the great circle except for a small "pulled-tight" tangle T. Since the metric around T is almost Euclidean Eysi(gpt) > 70. In both cases the value of B g 3 is much greater than 54 which is attained by a trefoil on a Clifford torus. Thus we conjecture that the value of Egl increases as a tangle or a knot itself shrinks to a point. This can be explained roughly as follows. n(2) Let E ( 2 ) 1 denote the restriction of _E R4 to the space of knots in S C Suppose a tangle or a knot itself shrinks to a point by a Mobius transformation. Then E (R2 ) 1| S3 is invariant. As dgz > d®4 we always have 4

p(2) J 3 S

< E

(2)1

Is3"

Since the ratio ds^/d^A converges to 1 as d^* decreases P(2)

to 0, the contribution of the shrinking part to E sl converges to that to (21 I

(2)

EjgJ , which makes Eys£ greater. Therefore we conjecture that neither the case (c) nor (d) happens with (2)

respect to Esi.

Together with Lemmas 7.3.1 and 7.3.2 we are led to (2)

(2)

Conjecture 7.1 There are Eysi minimizers for any knot type. Since Esi is no longer Mobius invariant, the number of E^l minimizers for any knot type is finite. 7.4.4

Energy of knots in H 3

Define the Lorentzian pseudo inner product for a pair of vectors u = ( u i , u 2 , u 3 , u 4 ) and v = (vi,v2,v3,v4) G M4 by (u, v) = luJv

= U\V\ + U2V2 + M3W3 — U4V4,

where

J

vO ' - i , Let Q — 0(3,1) be the set of linear isomorphism of M4 which preserves this Lorentzian pseudo inner product (•, •): g = 0(3,1) = {A e M 4 (R) I lAJA = J}.

Examples : Energy of knots in S3 and H 3

111

This group is called the Lorentz group. Define the 3-dimensional h y p e r b o l i c s p a c e H 3 C M4 by H 3 = {u = (ui,112,113,114) £ K 4 I {11,11) = —1, U4 > 0} . T h e restriction of the Lorentzian pseudo inner product provides a positive definite inner product of H 3 as we will see in Chapter 9. This M3 C R 4 is called the hyperboloid model of the hyperbolic space. T h e Lorentz group 0 ( 3 , 1 ) acts on H 3 isometrically and transitively. Let x and y (x ^ y) be two points in H 3 . T h e n we have {x,y) < —1 (Lemma 9.1.1 (4)). T h e geodesic 7 between x and y is obtained as the intersection of H 3 and the 2-plane t h a t passes through x, y, and the origin.

Fig. 7.1

H 3 and a geodesic.

y

P u t y = y + {x, y)x then {y, x) = 0 and {y, y) > 0. P u t v =

.

v(y>y) then {v,v) — 1 and {v,x) = 0 . T h e geodesic 7 can be parametrized by 7(i) = (coshi)a; + (sinht)w, where t = 0 at the point x. Let to be the parameter at y. Since y = —{x,y)x + \J{x,y)2 — 1 v we have coshio = — {x,y). Then the h y p e r b o l i c d i s t a n c e dtp{x,y) is given by the length of the arc of the geodesic 7 between x and y: du?{x,y)

/VM*).7' (t)) dt =

Jo

t0.

Energy of knots in a Riemannian

112

manifold

Therefore the hyperbolic distance satisfies cosh dEP(x,y)

=

-(x,y)

which means dtp(x,y)

+ \/(x,y)2

= log[-(x,y)

Then d^p (z, y) < dR* (x, y) for any

- 1J.

x^y.

Problem 7.2 Do we have E^(K) > E%?(K) for any knot K in H 3 ? This is not straightforward as the arclength also decreases if we use the hyperbolic metric for the ambient space. Proposition 7.4.2 ([07]) Let {Sr} be a family of circles of geodesical radius r in the hyperbolic space given by Sr : [0, 2TT] E> 9 h-> (sinh r cos 0, sinh r sin 6,0, cosh r) <E H 3 . (1) E^J(Sr)

(7.2)

is a monotonically increasing function of r with \imE<§(Sr)^E^(h0) lim E\J(Sr)

= i, = oo.

(2) For each a,p with ap > 2, E^p(Sr)

takes a minimum value and

lim EaJ>(Sr) = lim EaJ(Sr) r—>0

= oo.

r—>oo

Proof. Put R = sinhr. Since (S'r(0), S'r(0)) = R2 the arc-length between Sr(s) and Sr(t) for 0 < \s — t\ < •n is given by dSr(Sr(s),Sr(t))=R\S-t\. On the other hand, since (Sr(a),Sr(t))

• 2 = - [2RZ•>2 „sin

\8-t\ 2

the hyperbolic distance between them is given by d^(Sr{s),Sr(t))

Examples : Energy of knots in S3 and H 3

113

Therefore — E ^HP ( S r ) is given by 1 sin2 5 + 1 + 2 i ? s i n | y / i ? 2 s i n 2 | + l ^ R log(2.R 2 sin 2 t + l + 2 R s i n t^/R?

sin 2 t + 1

Put

m,t)

= —,

r

2

(7-3)

v

log ( 2 # 2 sin2 i + 1 + 2 J Rsint\/i? 2 sin2 i + 1 J

Since -?TH ^(-R, i) > 0 for any R (0 < R < 1) and for any £ > 0, the integrand of ( ae^p (Sr))p is a monotonically increasing function of R if ap = 2. It is easy to shows lim \P(R,t) = oo and lim ^(R.i) — , which J

R^oo

y

'

;

R^O

V

' '

2sin*'

completes the proof of (1). If ap > 2 then lim £ ^ ( S r ) = lim i ? H,—HI

2

" ^ f (ft0) = oo,

R—>U

and lim E^p(S = oo as the integrand of E^p(S ' Hp'('S'r) E^p(r) r) R—>oo

creases to oo, which completes the proof of (2).

blows up as R in•

Thus we are led to Conjecture 7.3 There are no E^ minimizers in any knot type. Any knot will converge to a point if it evolves so as to decrease its value of E\p. Suppose fn is a convergent sequence in a prime knot type [K] with Hindoo Ey(fn) = Ey([K]). Then n o /„ converges to an Ey minimizer after rescaling where 7r : R4 —> R3 is the orthogonal projection. Conjecture 7.4 Among knots with length / in H 3 , circles with length I minimize E^ (ap > 2). If this conjecture is ture, then Proposition 7.4.2 (2) implies that if ap > 2 then for any d e l there is a positive constant 1$ = l${a,p;B) such that if E^p(f) < B then the length lf of the knot / ( S 1 ) satisfies lf < l3. Therefore there is a compact subset KB C H 3 such that if E^(f) < B 11 3 then /(5 ) is included in KB after an isometric transformation of H . Then

Energy of knots in a Riemannian

114

manifold

Theorem 7.3.5 implies that there is an E^p minimizer for any knot type if the index ap > 2.

7.5

Other definitions

Since there are no standard similarities in M, we cannot normalize a knot so that its length is 1. This allows us two other ways of generalizing (4.1) as is explained in [07]. One is the scale invariant (cc, p)-energy E^'ps which is scale invariant when M = M3 and the other is the unit quantity (a, p)-energy E^f which is defined under the assumption that the total quantity of electric charge is equal to 1 when p — 1. They are defined by

££*(/) = if 9 '

1

J Js xs

1

[ \

1^M

1 Q

aa

dx

] P | / , ( 8 ) l l / , ( f ) l J J Jt J

If

If

where CLM — d,M(f(s),f(t)) is the distance in M, CLK = djc(f(s),f(t)) is the arc-length, and If is the arc-length of the knot /(S1). They differ from the unit density energy E^f(f) only by a power of If with EZ(f)

=

EM'pq(f) =

hap-2EaMP(f) l 2

r EaMp(f).

We consider the unit density energy here because it seems most suited to the production of energy minimizers. In general, Lemma 7.2.5 fails for the scale invariant energy, and Lemma 7.3.2 fails for the unit quantity energy. Example 7.1 Let Sr be a circle in H 3 with geodesical radius r given in (7.2) and put R = sinhr as before. (1) The scale invariant energy is given by

^ s (5 r ) = 4(2,r- i | 2 j( SW ))"__L_j p dt) where \P(R,t) is given in (7.3). Since \P(R,i) is a monotonically increasing function of R, E^Ps(Sr) is a monotonically increasing function of r.

The existence of energy minimizers

115

(2) The unit quantity energy E^pP (Sr) (p > 1) is given by

•^BP .a ("-*»•)

8TT

'0

{ log (2R2 sin2 £ + 1 + 2Rsint^R2

sin2 t + 1) } 2

^ ^

A direct calculation shows that the integrand is a monotonically decreasing function of r ([07]), and so is E^ (Sr) with

limi? ]H p P o (5 r )(S' r ) = oo

lim E H p P q (5 r )(S , r ) = 0

and

Conjecture 7.5 (1) As for the scale invariant energy, there are no E^p minimizers in any knot type. Any knot will converge to a point if it evolves so as to decrease its value of E^p . Suppose /„ is a convergent sequence in a knot type [K] with lim E^Ps{fn)

= E^Ps([K\).

If an E^p

minimizer

p

for [K] exists then n o /„ converges to an E£3' minimizer after rescaling where 7T : R 4 ->• K 3 i s the orthogonal projection. (2) As for the unit quantity energy, there are no E^p minimizers in any knot type. Any knot will diverge to infinity if it evolves so as to decrease its value of E^,p . Suppose fn is a diverging sequence in a knot type [K\ with lim E^p

(fn)= E^p

([K]). If ap > 3 then there is a singular knot

2

on the limit sphere S ^ as the limit of / „ .

7.6

The existence of energy minimizers

Here is a table of the answers and conjectures to our motivational problems: Q : Can we define an "optimal embedding" as an energy minimizer? Q : If so, is the number of energy minimizers in each knot type finite?

116

Energy of knots in a Riemannian

manifold

Table : Are there energy minimizers, and if so, how many? Manifold Energy p(2) _ p2,l ^M — -^ M

Ea*{ap>2)

M = S3

M = R3

M = H3

(YES, Jt < oo ?)

Prime : YES, (j = oo (Composite : NO ?)

(NO ?)

YES (tt < oo ?)

YES after rescaling (« < oo ?)

(YES ?)

(?) indicates a conjecture. ft denotes the number of the minimizers for each knot type.

Chapter 8

Physical knot energies

In this chapter we study geometrically defined knot energy Junctionals which do not come from Coulomb potential energy.

8.1

Thickness and ropelength

In this section we consider a knot as a core of a "solid knot", by which we mean an embedded solid torus in M3 with uniform thickness. The ropelength measures how long a rope of unit diameter is needed to make a knot. To be more precise, the ropelength of a knot K is the minimum length of the core C of a "solid knot" of diameter 1 such that C is homothetic to K. The thickness measures how much a knot can be fattened, namely, it is the maximum radius of a "solid knot" whose core is equal to the given knot. Therefore the ropelength of a knot is the quotient of the total length of the knot by its thickness. The study of this area started with [KV], where, for example, the minimum length of a "solid knot" with unit diameter whose core represents a given knot type has been studied. The subject was rediscovered by a question by L. Siebenmann in 1985 ([LSDR]): Problem: Can you tie a knot in one-foot length of one-inch rope? We show that the thickness is equal to the minimum global radius of curvature, which is equal to either the minimum radius of curvature or one half of the doubly critical self distance. Definition 8.1

(1) The thickness ([LSDR],[BS3],[KS2]) of a knot K, 117

Physical knot

118

Thk(K),

energies

is given by Thk(K) = sup{r > 0 | vT{x) n vr{y) = 0 (x ^ y)},

where i/r(x) is the normal disk to K of radius r at x. (2) The ropelength ([KS1'],[LSDR]) of a knot K, R1(K), is given by R1(K)-

L ^ Thk(K)'

where L(K) denotes the total length of the knot. A minimizer of the ropelength is called an ideal knot. (3) The inflation radius iv{K) of a knot K is defined by ir(Ar) = sup{£0 | NE{K) is homeomorphic to D 2 x 5 1

(0 < v e < e0)},

where N£(K) is the e-neighborhood of K. In this case the normal disks are not necessarily embedded. In the rest of this chapter we assume that a knot K — h{Sl) is parametrized by arc-length and has total length 1. Clearly Thk(h) < ir(ft). The reader is referred to [ChuMof] and [Mof] for the 'thickness' obtained via the magnetic flux of a tubular neighborhood of a knot, and also to [KBMSDS] and [KOPDS].

8.2

Four thirds law

It was observed in [Bu2; KS1'; CKS1; BS3; CDGl] that some of the functional of the form

E(K)=f[

^y^pldxdy

J JKXK

F

—

y\

for a function F grow as the -j power of the ropelength, where vx and vy denote the tangent vectors of AT at a; and y. In fact: Theorem 8.2.1

([Bu2; BS3]) \\B1(K)$

>

Esnp(K).

Osculating circles and osculating

spheres

119

Combined with Proposition 3.10.1, the above Proposition implies Theorem A.5.2 (with the coefficient improved). The Y exponent is sharp. Let K be a knot with Thk(K) = 1. Let r = r(K) denote the smallest radius of a 2-sphere which contains K. Then the length of the knot, hence its ropelength, is of the order of r 3 , and the "voltage" at each point x on K,

"<*> = / „ 3

^

*

is of the order of r, since the integrand is of the order of r~2 on the average. Therefore E(K) is of the order of r 4 , which is the order of R1(K)~5~.

8.3

Osculating circles and osculating spheres

Definition 8.2 (1) An osculating circle C — C(x,x,x) of a knot K at x is the circle which is tangent to K dX x at least to the third order, and its radius is the radius of curvature, which we denote by p(x). (2) An osculating sphere E^'(x) = SK{X,X,X,X) of a knot K at x is the 2-sphere which is most tangent to K at x. It contains the osculating circle (not necessarily as its great circle) and is tangent to K at x at least to the fourth order. (3) A twice tangent sphere Sx(x,y) of a knot / at x and y is an oriented 2-sphere that is tangent to the knot K both at x and y. An osculating circle is always uniquely determined. A twice tangent sphere is uniquely determined unless there is a circle C which is tangent to K at both x and y. When such a circle C exists, any 2-sphere that contains C is a twice tangent sphere at x and y. In this section we take the smallest one as Si((x,y). Let k = k(x) be the curvature and r = T{X) be the torsion of K at x. Then p(x) = l/k(x). Proposition 8.3.1 (1) An osculating sphere is uniquely determined if the order of tangency of the osculating circle to the knot is just 3. (2) Suppose the order of tangency of the osculating circle to the knot is just 3. Then the twice tangent sphere Ex(x,y) is uniquely determined for y close to x, and Ex(x,y)

approaches TrK [x) as the limit as y approaches

Physical knot

120

energies

x. The radius of the osculating sphere at x is given

where p' denotes the differential of p with respect to the arc-length. When the order of tangency of the osculating circle to the knot is greater than 3, any 2-sphere through the osculating circle is tangent to the knot to the fourth order. But there might be a unique 2-sphere which is tangent to the knot with a higher order of tangency than 4. Proof. Assume that a knot K — f{Sl) is parametrized by arc-length and let x = /(0). (Then we remark that the curvature and the torsion are given by k = \f"\ and r = (/'", / ' x f")/\f x / " | 2 . ) If we take the Frenet frame at x then Bouquet's formula implies /

0 ( i 4 ) a n d /'(*)

^ 3

/(*)

6

—t3

V

/

6

\ i \

0(i 3 ).

kt + ^e

kr

V

k

/ 1

2 fcrf2 2

/

Then the osculating circle C at 0 has center f 0, — , 0 ] and radius Let it be parametrized by the arclength as C(s)

— sinks, — (1 — cosfcs),0 k k

(8.1)

to have /M(0) = C ^ ( 0 ) for i = 0,l, 2. Then C is tangent to if at 0 to the fourth order, i.e. / ( 3 ) (0) = C^(0), if and only if k' = 0 and fcr = 0. Suppose /( 3 )(0) ^ C( 3 )(0), i.e. fc' / 0 or h / 0. Let P(t) = (0,y(t),z(t)) be the center of the twice tangent sphere EK(Q, f(t)). Then |P(t)| 2 = | P ( < ) - / ( * ) | 2 which implies (P(t)-f(t))±f'(t), namely, ±t2 2 kt

+

J^3 6

-iz

U(t),p(t))-(f(t),p(t))

+ 0(t4) 0(t3),

Global radius of curvature

121

which implies

+o( )

+o(,

*))-(A) ' =(£) >\

k2r '

r

Thus t h e twice tangent sphere SK{0, f(t)) approaches t h e 2-sphere E0 with center PQ = 10, p, — J and radius r$ = \j p2 + ( ^—) , which contains t h e osculating circle C . It is easy t o show t h a t among all 2-spheres E with center P and radius r, \f(i) — P\2 — r2 near t = 0 takes its minimum a t EQ, when it is 0(t4). Therefore EQ is t h e osculating sphere at 0. W h e n / ( 3 ) ( 0 ) = C ( 3 ' ( 0 ) , i.e. k' = 0 and fcr = 0, any 2-sphere Ez with center Pz = (0, — , z J with radius r z = W ( - r - ) Pz\2 -r2z = 0(t4) for any z E R.

8.4

+ z2 satisfies \f(t) — •

G l o b a l r a d i u s of c u r v a t u r e

Gonzalez and Maddocks ([GM]) introduced t h e global radius of curvature by generalizing t h e circle of curvature. We will see later t h a t t h e thickness is equal t o t h e minimum global radius of curvature. Let us consider a knot K in R 3 . D e f i n i t i o n 8 . 3 Let C(x, y, z) denote t h e circle t h a t passes through x, y, and z in K, and let r(C(x,y,z)) be its radius. W h e n two (or three) of x, y, and z coincide C(x,y,z) means t h e tangent (or respectively, osculating) circle. W h e n t h e knot K is oriented, we assume t h a t C(x,y,z) is oriented in such a way t h a t x, y, and z are in t h e same cyclic order in C(x,y,z) as in K. W h e n x = y t h e orientation of C(x,x,y) will b e chosen t o coincide with t h a t of K at t h e tangent point x. D e f i n i t i o n 8.4 T h e g l o b a l r a d i u s o f c u r v a t u r e pK (x) of a knot K at x is defined by PK\X)=

inf y,z£K

r(C{x,y,z)).

Physical knot

122

energies

Since the map K3 3 (x,y,z) >->• r(C(x,y,z)) € R is continuous by Lemma B.l, the global radius of curvature is attained by some triple (x,y, z), and therefore PK(X)= Ju

)

mm

1 1 1 1 1 1

r(C(x,y,z)).

I

y,*eK 3

When it is attained by a triple (x, x, x), pJK) i(x) is just the radius of curvature in the ordinary sense. Thus pK (x) < p(x). Proposition 8.4.1

([GM]) This p^K (x) is attained by a triple

(x,y,y),

Proof. Suppose pK' (x) is attained by a triple (x, y, z) with y ^ z. There are two cases. Case I: Suppose x = y ^ z. Let E be the 2-sphere which has C(x,x,z) as its great circle. If K is tangent to S at z then C(x, z, z) C S. Therefore r(C(x, z, z)) < r(C(x,x,z)). Thus pK (x) is also attained by (x,z,z). If K is transversal with S at z then we can take a point z' £ K near z inside S. Then r(C(x,x,z')) < r(C(x,x,z)) which is a contradiction. Case II: Suppose x 7^ y ^ z ^ x. Let Z" be the 2-sphere which has C(x, y, z) as its great circle. Suppose K is tangent to E at y. Then as C(y,y,x) is contained in S', we have r(C(y,y,x)) < r(C(x,y,z)). Therefore pK (x) is attained by (x,y,y)-

If K is transversal to S at both y and z, we can take y' and z' on K near y and z respectively inside S, and a 2-sphere S" contained in S' with

S"r\K

D {x,y',z'}.

Then

r(C(x,y',z'))
•

Definition 8.5 Let cr(x,y,z,w) denote the smallest sphere that passes through x, y, z and w in K, where the points in a D K are counted with multiplicity according to the order of tangency, namely, when some of x, y, z7 and w coincide a(x,y, z,w) is understood to mean a tangent sphere. Let r(cr(x, y, z, w)) be the radius of a(x, y, z, w). Put P(K (x) =

inf

, r(a{x, y, z, w)).

y,z,w£K

Global radius of curvature

Fig. 8.1 r(C(x,y',z'))

123

< r(E") < r(E') = p ^ ( x )

The cr(x, y, z, w) is uniquely determined generically, i.e. when the concircular condition fails (see Section 10.3). Since r(a(x7y, z, w)) > r(C(x,y,z)) we have pK (x) > pK'(x). Proposition 8.4.2

([GMS]) (1) p{^\x)

= p(^\x)

(2) pK (x) is attained by a 4-tuple (x,y,y,w) miny,z,weK r(cr(x, y, z, w)).

for any x.

G K4.

Therefore pK (x) =

Proof. We show that pK (x) > pK (x). Suppose p"K' (x) is attained by a triple (x, y, y). Let S be the 2-sphere which has C(x, y, y) as its great circle. We show that K intersects E in at least 4 points counted with multiplicity. This implies that (3)

n p^(x)

= r(C(x,y,y))

=

(4), r(S)>p^(x).

There are two cases. Case I: Suppose x ^ y. If the knot K is tangent to S at x then E = o-(x,x,y,y). If the knot K is transversal to £ at x and if the order of tangency of K and S at y is 2, then K must intersect S (not necessarily transversally) at a third point z which is different from both x and y, and therefore S = a(x,z,y,y). If the order of tangency of K and £ at y is more than or equal to 3, then £ =
Physical knot energies

124

If it is 3 then K must intersect E (not necessarily transversally) at another point z (z ^ a;). Then E = <j(x,x,x,z).

x = y case. S =
1 / 5 case S — a(x, z, y, y)

Fig. 8.2 The sphere £ which has C(x,y,y) section point with K.

as its great circle must have a fourth inter-

If the order of tangency is more than or equal to 4 then E — cr(x, x, x, x).

• Remark 8.4.3 The map KA 3 (x, y, z, w) — i >• r(cr(x, y, z, w)) £ K. is lower semi-continuous, but not necessarily continuous, as will be seen in Proposition 10.4.1. Definition 8.6

Define

A (if) = in{x€K p{x\x)

= iniXiytZeK

D(K) = inixeK

= infXtyiZtWeKr(a(x,y,

p^\x)

r(C(x,y,z)), z,w)).

The above Proposition 8.4.2 implies that A(if) = O(K). implies that the infimum is attained: A (if) = minpyK' (x) = x£K

min

Lemma B.l

r(C(x,y,z)),

x,y,z£K

and therefore \3(K) = minpK(x) xQK

=

min

r(a(x,y,z,w)).

x,y,z^wdzK

Let us call it the minimum global radius of curvature of K. Remark 8.4.4 This O(K) is the infimum of the radii of the non-trivial spheres for K, which will be defined in Definition 10.1.

Global radius of curvature

125

Theorem 8.4.5 ([GM]) Let x be a point which attains the minimum global radius of curvature. Then A(if) = pK (x) = r(C(x,y,y)) for some ("ON

y-

(1) If y = x, i.e. A(if) is the minimum radius of curvature p(x), then the osculating sphere at x contains the osculating sphere at x as the great circle, and its radius is equal to A(K). Therefore A(if) = D(if) = r(a(x,x,x,x)). (2) If y ^ x and A (if) = r(C(x,y,y)) < p(z) for any z £ K, then the radius of the (smallest) twice tangent sphere £K(X,"II) — dD3 is equal to A(if). Therefore A (if) = d ( i f ) = r(cr(x,x,y,y)). Moreover, x and y are antipodal on EK{X, y) and IntD 3 Pi if = 0.

(1) A(iC) = r{a(x,x,x,x)), which is the minimum radius of curvature

Fig. 8.3

(2) A(if) = r(a(x,x,y,y)), which is the closest approach. x and y are antipodal A(K)

=

D(K).

Corollary 8.4.6 The infimum of the radii of the non-trivial spheres for K, I—l(if) = mmXtV}Z>W£K r(cr(x,y, z, w)), is attained by a twice tangent sphere or an osculating sphere. Proof. (1) Since the radius of curvature p takes its minimum at x, p'(x) = 0. Therefore Proposition 8.3.1 implies the radius of the osculating sphere SK (x) at x is p(x). (2) Put C\ = C(x,y,y). Since pK (y) is attained by Ci, Proposition 8.4.1 implies that C\ is tangent to if at x. Let E = dD3 be a 2-sphere with center Os which has C\ as its great circle. Then £ is twice tangent to if at a; and y. (As r(S) = D(if), £ is the smallest twice tangent sphere at x and y.)

Physical knot energies

126

Since p(x) > r(E) by the assumption, the knot K cannot have the tangency of order 3 with S at x, and hence K must lie in the outside of S near x except for the tangent point x. Suppose x and y are not antipodal in S. By moving the center a little in the direction (x + y)/2 — Os we can take a strictly smaller sphere than S which intersects K at four points, which contradicts to r(S) — O(K). Suppose Int£>3 n K ^ 0. Then we can take a strictly smaller sphere which is tangent to K at x which has two more intersection points with K, which is a contradiction. • R e m a r k 8.4.7 Although min^y r(a(x, x, y, y)) — m i n ^ r(C{x, y, y)), if we fix x min^ r(a(x, x, y, y)) = min y r{C{x, y, y)) does not hold in general. Since l/r(C(x,y,z)), l/r(C(x,x,y)), and l/p\J{x) p tinuous, we can take their L -norms.

are bounded and con-

Definition 8.7 ([GM]) Suppose a knot K is parametrized by arc-length and has total length 1. Let U3'P(K) and UX'P{K) be LP-norms (1 < p < oo) ofl/r(C(2;,?/,^))andl/^)(x):

U K)

^ -{IILK,KHcix^d^

U**{K)=([[ Ul*{K)

-—^—tedy

=

when 1 < p < oo, and U3'00(K) = U1'00(K)

A(KY

Similarly, as l/r(a(x,y, z,w)) is continuous at least almost everywhere, one can consider its L norm:

U4'P(K) =([[[[ \J J J

——— JKXKXKXK

-p dxdydzdw] '

r{(7{x,y,z,w)Y

J

when p < oo and U*,°°(K) = ess. sup — - — r(
<

O(K)'

Self distance type energies defined via the distance

function

127

We remark that Ul,2{K) bounds the total squared curvature from above. Proposition 8.4.8

U^P{K) is not self-repulsive if p < j (j — 2,3,4).

This was shown in [Su2] when j = 2. Proof. Let us give a proof when j = 3. Let c\ and C2 be two straight line segments that intersect in a right angle at P and let K\ = c\ U c
eKxxK^xKx

< ^

<2k —

2fc_1

Then 3 /r> M D _ / / /

P {U3>*'(K 1)}P

=

dxdydz

tfixifixi?!

< £

r{C(x,y,z))p

2fcPVol(Tfc).

k — — OO

Let T be a homothety of R3 by -|- with center P: T(X) = P + \{X P). Then there is fco e Z such that if fc > fco then (x,y, z) E 7fc if and only if (Tko-k{x),Tk°-k(y),Tk°-k(z)) G Tfeo, and hence Vol(Tfc) = 23(fco-fc)vol(rfco). Then fco-l {[/ 3 'P ( ^ l ) } " <

53

2 f c P V o 1

k— — oo

(^1

x f l X ^ j f ^

2 3fc 0

+ (p-3)fc V o l ( 7 i 0 ) ,

k=ko

which converges when p < 3.

•

Proposition 8.4.9 ([Su2]) U2'P(K) is self-repulsive with respect to the C2-topology if p > 2. Problem 8.1

8.5

Are W'P(K) (j = 1, 2,3,4) self-repulsive if p > j ?

Self distance type energies defined via the distance function

Another kind of geometric knot energy functional which does not come from Coulomb potential energy measures how close a knot approaches itself. The thickness, which is equal to the minimum global radius of curvature,

Physical knot

128

energies

is dominated locally by the radius of curvature and globally by the closest approach of a knot. Definition 8.8

Let

Ah : S1 x S1 \ A 3 (s, t)

H->

\h{s) - h(t)\ e R

be the distance function of a knot outside the diagonal A. (1) Define the minimum distance md(/i) of a knot h to be the smallest (possibly degenerate) local minimum value of Ah if one exists, otherwise put md(h) = maxx^yAh(s,t). (2) Define the doubly critical self distance dcsd(fa) of a knot h ([Sim3],[LSDR]) to be the smallest critical value of Ah, namely dcsd(» = min {\h(s) - h(t)\ : h'(s) _L (h{s) - h{t)),h'{t) Theorem 8.5.1

_L (h(s) - h(t))} .

The minimum global radius of curvature is given by A(if) = min < minp(x), — dcsd(if) > , [x£K 2 J

where p{x) denotes the radius of curvature at x. Proof.

Then Theorem 8.4.5 implies

A(K) = min < min p(x), min {r(Efc(x,y)) [x<=K

\ x and y are antipodal}

> min < mmp(x), — dcsd(if) >. [xeK 2 J Suppose ir dcsd(K) < A(K). Then -^ dcsd(K) < minx<EK p(x). Assume dcsd(.RT) is attained by a pair x,y G K. Let S be a 2-sphere which takes the chord xy as its diameter. Since its radius satisfies r(S) < p(x),p(y) the knot K lie outside the sphere E near x and y except for the two tangent points. Then r(C(x,x,y)) — r(E) < A(K), which is a contradiction. • Theorem 8.5.2 to the thickness.

([GM]) The minimum global radius of curvature is equal

Proof. It is clear that Thk(K) < mmx€K p{x), -w dcsd(K). Therefore Thk(K)
Self distance type energies defined via the distance

function

generality that \x — z\ > \y — z\. Then r(C(x,x,y)) which is a contradiction. Therefore Thk(K) > A(K). Corollary 8.5.3

129

< \x — z\ < A(if), D

([LSDR]) The thickness is given by Thk(K) = min \ minp(a;), -~-dcsd(/0 1. [XEK 2 J

Proposition 8.5.4 ([BS3]) The ropelength bounds the distortion from above and hence it is self-repulsive. Proof. Suppose the thickness of a knot K — h(S1) with total length 1 is r (r < r^). Suppose there is a pair of points x,y on the knot K with |sc — 2/1 ^ rd'K(x,y) < TT- Since the curvature of h is smaller than or equal to — both of the two subarcs of K between x and y have to leave the -^--neighborhood of {x,y}. Then r-tubular neighborhood of K has a selfintersection near x and y, which contradicts the definition of the thickness. Therefore, Distor < Rl.

• Definition 8.9

Let Ah,s be a distance function from a fixed point h(s):

Ah,s = Ah(s, ):S1\{s}3t^

\h(s) - h(t)\ e M.

Let Xh(s) > 0 be the smallest critical value of Ah,s- Then Kuiper's self distance sd(h) of a knot h ([Kui]) is defined by sd(/i)= inf1 Xh(s) ses = inf { \h(s) - h{t)\ : h'{s) _L (h(s) - h(t))} . Obviously sd(ii') < dcsd(if), and in general sd(ii") < dcsd(ii'). If sd(K) is attained by \h(s) - h(t)\ with h'(s) _L (h(s) - h(t)) then r(C(h(s),h(t),h(t))) = j;sd(K), and hence ±sd(K) > A(K). Then Theorem 8.5.2 and Corollary 8.5.3 imply: Theorem 8.5.5

([LSDR]) The thickness satisfies Thk(K) = min j min p(x), - ^ s d ( ^ ) 1 . I x£K

2

I

130

Physical knot

energies

Theorem 8.5.6 ([01; LSDR]) Only finitely many knot types can occur under any given upperbound for the self-distance. The same statement holds for the ropelength. Proof. We prove for sd~ using the same argument as in [Ol]. Suppose sd(h) = r. Let Bo.9r(^) be a 3-ball with center x = h(s) and radius 0.9r. Then the intersection of the knot with the ball, Bo.gr(x) fl h^1), is a connected subarc, which we shall denote by h([s — t\, s + ^D- Then, as \h(s+t)—h(s)\ (or |/i(s — t) — h(s)\) is a monotonically increasing function of t £ [0, £2] (or [0, ti] respectively), if p < 0.9r then the subarc Bp{x) n h(Sx) can be deformed into a straight line segment by an isotopy of Bp(x) which keeps the boundary sphere fixed. Put N = [1/1.8r] + 1, where [] is Gauss' symbol. Then the knot ^(-S1) can be covered by ./V such balls with radius 0.9r. Hence the knot h(Sl) can be ambient isotoped to a polygonal knot with TV vertices by a sequence of isotopies inside the 3-balls.

Fig. 8.4 Any knot with sd < ro is ambient isotopic to a polygonal knot with [1/1.8ro] vertices.

The theorem follows from the fact that there are only finitely many knot types of polygonal knots with N vertices. • The self distance and the doubly critical self distance are not continuous with respect to the C2-topology. For example sd(h) might be attained by an inessential critical pair of points which will disappear by a slight

ea'v

131

A4 ( 1 ) (M = {g •• S1 ->• M3 : \h(s) - g(s)\ < e, \h'(s) - g'(s)\

<£(^seS1)}

Relation between these geometric quantities and

perturbation of h. Let

be the e-neighborhood of h with respect to the C1-topology. Define the essential self distance ess.sd(/i) and essential doubly critical self distance ess.dcsd(fo) of a knot h by ess.sd(/i) = inf

sup

sd(
ess.dcsd(Zi) = inf

sup

dcsd(g).

Clearly ess.sd(h) > sd(h) and ess.dcsd(/i) > dcsd(/i). Let Ph(s) be the smallest (possibly degenerate) local minimum value of AhtS if exists, otherwise put (3h(s) = max^ggi Ah}S(t). Then the monotone radius or the beads radius of a knot h, rm(ft), is defined by rm(h) = miseSi

f3h(s)

- inf «,,n L ^ n -mstiSup|r0>U

B

Hs)(r)

n

HS1) is connected (0 < Bh{s)(r0)2h(Si)

v

r < r0) 1

/'

where Bp(r) C ffi3 denotes the 3-ball of center P and radius r > 0. If h is a non-trivial knot then Ah,s n a s a (possibly degenerate) local minimum for any s £ S1 because otherwise ^(S 1 ) bounds a disk. In this r 1 case the last condition uBM 4(ss)()(o) 2 h(S )" in the right hand side is not necessary. By definition we have sd(/i) < dcsd(h) < md(/}), sd(/i) < rm(/i) < md(/i), rm(/i) < 2ir(/i).

8.6

Relation between these geometric quantities and

ea,p

The relation between these quantities and ea,p is as follows. Lemma 4.2.7 implies that if ap > 2 then

sd(h) > A2 (—J °P"2 e a ' p ( / i r ^

(8.2)

132

Physical knot

energies

for any knot h, where A2 is the constant given in Lemma 4.2.5. Since sd(h) < min{ess.sd(/i),dcsd(/i),ess.dcsd(/i),md(/i),rm(/i),2ir(/i)} this means that ea'p (ap > 2) bounds these quantities from below except for the thickness. In general ea'p (ap > 2) does not bound the thickness from below because ea'p (ap > 2) does not bound the curvature \h"\ from above unless (a,p) = (2, 00). These geometric quantities as thickness and self distance decrease to 0 as a tangle pulls-tight, whereas ea'p (ap = 2) does not by Theorem 4.2.8. Hence ea'p (ap = 2) does not bound these geometric quantities from below. (8.2) means that ea,p (ap > 2) bounds sd~ from above. As we saw in Chapter 4 the distortion is bounded from above by a function of ea'p (by formula (4.3) when ap = 2 and by Corollary 4.2.3 when ap > 2). Conjecture 8.2 We conjecture that sd~x bounds the distortion from above, and hence that sd~ is self-repulsive. 8.7

Numerical computations and applications

Applications and numerical computations can be found in [SM] and [MMTB] (optimal packing of single and double helical tubes), [VCLPKDS] (the study of DNA replication), [PPS] (open knots), [DSDS] (optimal lengths of random knots), and in the book [SKK]. The reader is referred to [SKK] for the figures of ideal knots, i.e. minimizers of the ropelength.

PART 2

Energy of knots from a conformal geometric viewpoint

This page is intentionally left blank

Chapter 9

Preparation from conformal geometry

We showed in P a r t 1 t h a t E^ is Mobius invariant. In P a r t 2, we show t h a t the integrand of E^ can be interpreted from a conformal geometric viewpoint. This viewpoint provides a new interpretation of Mobius invariant knot energy f u n c t i o n a l . Using a sphere t h a t is tangent to a knot K at two points, we introduce the infinitesimal cross ratio OCR-, which is a meromorphic 2-form on K xK. We show t h a t t h e integrand of E^ can b e expressed in t e r m s of ficRWe work in Minkowski space M 4 ' 1 with Lorentzian pseudo inner product. We use the set of spheres which intersect a knot at four points, which is a subset of t h e set of spheres in S3 which forms a 4-dimensional hyperbolic hypersurface in M 4 , 1 . We introduce a new knot energy functional using integral geometry. Let us start with some basic facts in conformal geometry which will be used later.

9.1

The Lorentzian metric on Minkowski space

D e f i n i t i o n 9.1 (1) Define the L o r e n t z i a n q u a d r a t i c f o r m (or the Lorentzian norm) L and the associated L o r e n t z i a n p s e u d o i n n e r p r o d u c t (•, •) on R n + 2 by (u, v) = *uJv = mvi 2

H

L(u) = (u, u) = u i H

1- un+ivn+1 2

1- un+i 135

-

-

un+2vn+2 2

un+2 ,

Preparation from conformal

136

geometry

where

J = Jn+1,1 =

lo 'J Two vectors u,v in R r a + 2 are called .L-orthogonal if (u,v) = 0, also written u ± v. For a linear subspace W of R n + 2 we denote its L-orthogonal complement by W^: WL = {v € R n + 2 | (v,w) = 0 ww e W}. T h e Euclidean space R n + 2 equipped with the Lorentzian pseudo inner product (•, •) is called M i n k o w s k i s p a c e and denoted by R™+ 1 ' 1 . (2) Let Q = 0{n + 1,1) be the set of linear isomorphisms of R n + 2 which preserve this Lorentzian pseudo inner product (•, •): G - 0{n + 1, l) = {Ae

M „ + 2 ( M ) | lAJA

= j} .

This group Q = 0{n + 1,1) is called the (full h o m o g e n e o u s ) L o r e n t z g r o u p or the c o n f o r m a l g r o u p . An element of the Lorentz group is called a conformal transformation of M " + 1 , 1 . Let Q+ = SO(n + 1,1) denote the set of orientation preserving conformal transformations of IR n + 1 , 1 , and SOo(n + 1,1) denote the component of SO(n + 1,1) which contains the identity. (3) A vector v in R n + 1 1 is called s p a c e - l i k e if (v,v) > 0, light-like if (v,v) — 0, and t i m e - l i k e if (v,v) < 0. A line is called space-like (or time-like or light-like) if it contains a space-like (or respectively, time-like or non-zero light-like) vector. A linear subspace W of R™"1"1'1 is called space-like if every non-zero vector in W is space-like. T h e isotropic cone V = {»£l"

H 1

| (v,v)

=0}

is called the light c o n e . (4) Let S^ = {V\ {0})/M x be the set of the points at infinity of the upper half light cone V+ — V D {x £ R""1"1-1 | x n + 2 > 0}. Since it can be considered as the set of lines through the origin in the light cone, it may be identified with the intersection 5™ of the upper half light cone V+ and the hyperplane {x G R n + 1 - 1 | xn+2 = 1}: Si = {(xlr--

,xn+1,l)

| ( x 1 ) 2 + --- + ( a ; n + i ) 2 - l = 0 } ,

The Lorentzian

metric on Minkowski

space

137

which is congruent to the n-sphere Sn. The Lorentzian metric restricted to Sf coincides with the standard Euclidean metric of Sn. We identify Sn with 5™ in what follows. (5) Let A be the hyperbolic hypersurface of one sheet given by

A = {v e

jn+1,1

(«,«> = !}

It is called de Sitter space and is also denoted by S71'1. As an element of the Lorentz group Q preserves the Lorentzian pseudo inner product, it acts isometrically on A. This action is transitive. (6) Let Fn+1 = /{v « e P- ron+1-1 (v,v) = — l } be the hyperbolic hypersurface of two sheets and F™+ = Fn+1 n {u e jn+1,1 un+2 > 0} be its positive sheet. The restriction of the Lorentzian quadratic form L to each tangent space of F " + 1 is positive definite since TVF™+1 = (Span(v)) J - is space-like as we will see in Lemma 9.1.1. Then F " + 1 is called a hyperboloid model for the n - 1-dimensional hyperbolic space fn+1

de Sitter space A Light cone

Fig. 9.1

Light cone, de Sitter space, and the hyperbolic space.

A conformal transformation of M" +1,1 is called positive if it maps _F™+1 to itself. The set of positive conformal transformations is called the pos-

138

Preparation from conformal

geometry

itive Lorentz group and is denoted by PQ = PO(n + 1,1). T h e positive Lorentz group PQ = PO(n + 1,1) acts transitively on H™+1 as the group of isometries. Any geodesic of Mn+1 is obtained as the intersection of H n + 1 with a 2-dimensional vector subspace of K""1"1'1 which contains a time-like vector. If x,y G H P + 1 (x ^ y) t h e n (x,y) < —1 by Lemma 9.1.1. Then, as was explained in Subsection 7.4.4, the geodesic between x and y is obtained as the intersection of H r a + 1 and the 2-plane Span (x, y), and the hyperbolic distance dn(x,y) between x and y is given by c o s h d ^ ( x , y ) = —(x,y). Each sphere £ in Sn is the "boundary at infinity" of a totally geodesic subspace h of HP , + 1 . (7) T h e Lorentz group Q acts transitively on S^ = Sn as the action on the set of lines in the light cone V. This action is called a M o b i u s t r a n s f o r m a t i o n of Sn, which induces a Mobius transformation of R™U{co} via a stereographic projection. T h e set of Mobius transformations of Sn = R™ U {oo} is called the M o b i u s g r o u p and denoted by M = M„. (8) Let U be an open subset of R n . A diffeomorphism
f(U) C M.n is said to be c o n f o r m a l if it preserves angles between smooth curves. Then there is a function k : U —• K_|_ such t h a t the Jacobian matrix of the differential of ip, Dtp(x) = \^-{x))i satisfies Dip(x) = k{x)P{x) for some P{x) 6 0(n). A Mobius transformation is a conformal transformation. On t h e other hand, when n > 3 any conformal transformation of an open subset of R" can be obtained as a Mobius transformation. It is known t h a t any Mobius transformation of R n U {oo} can be expressed as a composition of finitely many inversions in (n — l)-spheres in R n (see Definition 3.1). A Mobius transformation T of R™ U {oo} which fixes co is a similarity of R" U {oo}, namely, T(x) = \A(x) + b, for some A > 0, A G 0(n), and 6GR".

Any (k + 2)-dimensional vector subspace of R n + 1 , 1 which contains a time-like vector intersects Sn in a /c-dimensional sphere (k < n). Therefore a Mobius transformation maps a fc-sphere in Sn into another fc-sphere. T h e action of the Mobius group on the set of fc-spheres in Sn (or R n U {oo}) is transitive by Lemma 9.1.1 (7). We remark t h a t when n = 2 any Mobius transformation of S2 = R 2 U {oo} = C U {oo} can be expressed as a linear fractional transformation

The Lorentzian

z— i>

metric on Minkowski

space

139

if it is orientation preserving, and z i->- ——- if it is orientation 72 + S

jz + S

reversing, where a, 0, 7, and 5 are complex numbers with aS — fly j^ 0. The reader is referred to textbooks on hyperbolic geometry such as [Rat] for details on Mobius transformations. In what follows we will study conformal invariance, which might not be so familiar to knot theorists. We identify K3 U {00} with S 3 through a stereographic projection from a point P G Ss. This identification has the ambiguity of the choice of the point P and the affine 3-space to project on. The metric of K3 is not conformally invariant, therefore the distance between a pair of points does not have meaning in conformal geometry. On the other hand, the angle between tangent vectors is conformally invariant. Lines and affine 2-planes in M3 are regarded as circles and 2-spheres in M3 U{oo} that pass through 00. There is no reason to treat 00 as a privileged point from a conformal geometric viewpoint. We will mainly study the case when n = 3, as it corresponds to the set of spheres in S3, as will be explained in section 9.3, but the statements in what follows also hold for other n. Lemma 9.1.1 (1) Let u and v be linearly independent light-like vectors in the upper half light cone V+. Then (u,v) < 0. Therefore any pair of linearly independent light-like vectors cannot be L-orthogonal. (2) The L-orthogonal hyperplane I1- of a light-like line I is tangent to the light cone V at 1. Namely, a vector v G I1- is either space-like or v £ I. (3) The L-orthogonal hyperplane I1- of a time-like line I is space-like. (4) Let u,v e H4 = {w G M4-1 | {w,w) = - 1 , w5 > 0}. Then (u,v) < — 1 if u ^ v. (5) The L-orthogonal hyperplane I1- of a space-like line I contains a time-like vector. (6) Let V be k-dimensional vector subspace which contains a time-like vector. Then we can choose a Lorentz orthonormal basis {it 1 ,--- ,uk} of V, i.e. (u1,^) = ±d~ij, with exactly one time-like vector ul. (7) The Lorentz group acts transitively on the set of the k-dimensional vector subspaces which contain time-like vectors. (8) The Lorentz group acts transitively on the set of the k-dimensional space-like vector subspaces.

140

Preparation from conformal

geometry

Proof. (1) Assume that u = (m, • • • ,u$) and v = (vi, • • • ,vs) satisfies (u,v) > 0. Then u\Vi + • • • + U4V4 > u5v5.

(9.1)

As U5,vs > 0 this implies (uivi -\

\-u4v4)2 > u52v52.

(9.2)

On the other hand ( V + • • • + u42){v12 + • • • + vi2) > (mvi + ••• + U4V4)2.

(9.3)

As {u,u) — (v,v) = 0, (9.2) and (9.3) implies u52v52 = (ui 2 -|

hu4 2 )(^i 2 H

I-W42) > (uivi -\

h u 4 2 )(^i 2 H

h v42) - (uivi H

\-u4V4)2 > u52v52.

Therefore we have (ui 2 H

h U4W4)2

i,j = l,--- , 4 , 7 < j

which implies that there is a constant k € R such that ( M I , - - - , w 4 ) = k(vi,---

,v4).

Then 1x5 = \k\v5 as (u,it) = (i>,u) = 0 and u<s,v§ > 0. Since u and u are linearly independent, k is negative. Then by (9.1) 0 > k(v\2

+ • • • + V42) = u\V\

+ • • • U4V4 > W5U5,

which is a contradiction. (2) Assume that I is spanned by a non-zero light-like vector it = (ui, • • • , U5) and v — (v\, • • • , v$) 7^ 0 is L-orthogonal to it. Then «5 2 (^i 2 H

h V - v52) = u 5 2 ( V +

= (ui2 H

=

^2

h U42)(^i2 H

h «42) -

h i>42) - ( u i v i H

ub2vh2 h U4U4) 2

(uiVj - UjVi)2 > 0.

Then «s ^ 0 implies that (u,u) > 0, namely, v is either space-like or light-like. If v is light-like then (1) implies the conclusion.

The Lorentzian

metric on Minkowski

space

141

(3) Assume t h a t I is spanned by a time-like vector u = ( u i , • • • , 115) a n d v = (vi, • • • , v5) ^ 0 is L-orthogonal t o u. Since u\2 + • • • U42 — u 5 2 < 0, «5 7^ 0. Then

UiV\ + • • • + U4V4 — U5W5 = 0

implies v\2 + • • • + V42 7^ 0 because otherwise v = 0 . As

u52(vi2

H

> (ui2 H =

h w 4 2 - v52) = M

2 5

h u 4 2 )(wi 2 H

(V H

+ v42) -

h «4 2 ) - (wi^i H

u52v52

h W4W4)2

^ (uiVj - Ujt>j)2 > 0, i,j = l,--- ,4, i<j

(u,w) > 0. (4) Since u, v £ H 4 , 1*5,^5 > 1.

(uit;i H

h U4W4)2 < ( u i 2 H =

1- u 4 2 ) ( u i 2 H

(- w 4 2 )

(u52-l)(v52-l)

< (u5v5 - l ) 2 .

Since U5,«s > 1 this implies wi«i + • • • + U4V4 < U5V5 — 1 which means {u,v} < - 1 . (5) Assume t h a t I is spanned by a space-like vector u = ( u i , • • • , U5) G yl, i.e. (u,u) = u i 2 + • • • + W42 - W52 = 1. If M5 = 0 then P = L1contains a time-like vector ( 0 , 0 , 0 , 0 , 1 ) . If M5 7^ 0 then P = l A contains I u i , • • • , 114, — 1, which is time-like. V «5 / (6) Let v1 / 0 b e a time-like vector in V a n d let { v 1 , • • • , vk} b e any basis of V. Then we can apply Schmidt's orthogonalization t o {v1, • • • , vk}

142

Preparation

from

conformal

geometry

to obtain {it 1 , • • • ,uk} inductively as follows: v

uk=vk

u

-Ylv^u*)

u

k .

^.'"'l-v*

(fc>2),

k

uk,uk)\

Then L(v}) = — 1 and (u1,^) = ±6ij, which completes the proof as (2) of the present Lemma implies that ul (i > 2) is space-like. (7) Let V be a fc-dimensional vector subspace which contains a time-like vector. Then we can apply Schmidt's orthogonalization as in the proof of (6) to obtain a Lorentz orthonormal basis {u1,--- , u 5 } of M5 such that {u6~k, • • • , it 5 } spans V and such that it 5 is time-like. Put A = (u1 • • • u5) then A € Q = 0(4,1) and A maps the fc-dimensional vector subspace spanned by {e6_fc, • • • , e$} onto V. If A, A' £ 0(4,1) correspond to V, V then A'A-1 maps V onto V isomorphically. (8) Let V be a fc-dimensional space-like vector subspace and {u1, • • • ,uk} be its Lorentz orthonormal basis. Let v ^ 0 be any time-like vector. Put k

'^^{v,ul)ut.

v =v 1=1

As

(v,v)

= (v,v)

v is a time-like vector with (v,ul)

-

^(V,M*)2 <

0

= 0. Put u 5 = —,

^/\WM 41

Then we can

choose a Lorentz orthonormal basis {it 1 , • • • , it 5 } of M ' as in the proof of (6). This completes the proof. •

The Lorentzian

9.2

exterior

product

143

The Lorentzian exterior product

In this section we assume that n = 2. Let us generalize the vector product of two vectors in R3 in our context. Definition 9.2 ([LOl]) The Lorentzian exterior product v1 A • • • A v4 G R5 of four vectors v% = (v\, • • • , vlh) in R5 (i = 1,2,3,4) is given by v 1 A •• • A v4 = (v\,V2, v$, h,

-h)

with 1 4 v{ • • • vj

"j = ( - I ) *

M

1 ^5 • • •

-4

4 5

v

where the symbol ~ means that the j-th row vector of the (5,4)-matrix is removed. We remark that this is a generalization of the Lorentzian vector product u A v = J2,i(u x v) of u, v G R 3 , where x denotes the standard Euclidean vector product of R 3 . Lemma 9.2.1 ([LOl]) (1) The Lorentzian exterior product v1 A • • • A v4 is the uniquely determined vector u E R5 that satisfies {u,w) =det(v1v2v3v4w)

£R

(vweR5),

where vectors in R5 are considered as column vectors. (2) j ) 1 A » 2 A » 3 A w 4 / 0 if and only if vl,v2,v3, pendent.

v4 are linearly inde-

(3) (v1 Av2Av3Av4,vj) =Qforj — 1,2,3,4, namely, v1 Av2 Av3 Av4 l 2 is L-orthogonal to Span (y ,v , v3, i? 4 ). (4) The norm of v 1 A D 2 A o 3 A v4 is equal to the absolute value of the volume of the parallelepiped spanned by vl,v2,v3 and v4 associated to the Lorentzian quadratic form L:

144

Preparation from conformal

(5) Ifv1, v2, v3,v4 by these vectors is

€ TXA then the volume of the parallelepiped

| d e t ( a ; , t ; 1 , t ; 2 , v 3 , i ; 4 ) | = Kx^1 Proof. bra:

geometry

spanned

A v2 A v3 At) 4 >| = ^ / - d e t ((«*,«•?')) .

(4) T h e last equality is a consequence of a formula in linear alge-

l v \

vl\

•••

v

4

\ )

det((vi,vj))=det<

,~,2

vi

v\

<)

o

\-vl ,~,2 i r,2

,-,2

i/r — ^o — Vn — v; -Liv1 (5) Since A = {x G R 5 | L(x) (3) implies v1 A v 2 A « 3 A t i 4

Av2

Av3

Av4)

= 1}, T^yl is L-orthogonal to x. Therefore

±y/L(v1

A»2AJ)3AK4)

X

as L(a;) = 1. Since x and hence D 1 A V 2 A V 3 A W 4 are space-like, det((vl, 0 by the above formula.

v^)) < •

As a corollary of L e m m a 9.1.1 we get: L e m m a 9.2.2 1

2

(1) If v Av 1

2

Suppose v1^2 3

Av

3

4

Av

,v3 ,v4

is time-like

4

are linearly l

2

independent. 3

4

then all ofv ,v ,v ,v

are 1

2

space-like.

(2) v A v A v A v is light-like if and only if Span (t; , v , v3, v4) the tangent space of the light cone at v 1 A v 2 A v 3 A v 4 .

is

We remark t h a t D 1 A t) 2 A « 3 A t) 4 can be space-like even when all of v ,v ,v3, and v4 are space-like. In such a case, the hyperplane spanned 1 2 by v , ^ , ^ 3 , and v4 contains time-like vectors. x

2

9.3

T h e space of spheres

L e m m a 9.3.1 Let P be a hyperplane (i.e. 4-dimensional linear J4,l . Then the following three conditions are equivalent. in (1) P intersects S transversely.

subspace)

The space of spheres

145

(2) P contains a time-like vector. (3) .P-1 is space-like. Proof. The "inside" of Sfnft is the set of time-like lines, which implies the equivalence of (1) and (2). The equivalence of (2) and (3) comes from Lemma 9.1.1 (3) and (5). • Theorem 9.3.2 Let S be the set of the oriented 2-spheres in S3. Then there is a canonical bijection ip between A and S, which commutes with any conformal transformation g 6 0(4,1) modulo orientation; in particular,

Fig. 9.2

The correspondence between A and S.

Proof. A point p in A determines an oriented half line lp = Op from the origin. Let JJV = (lp)-1 be the oriented hyperplane passing through the origin that is L-orthogonal to lp. Since lp is space-like, 7Tp intersects the light cone transversely by Lemma 9.3.1, and therefore LTp intersects S3 in an oriented 2-sphere Sp. We will specify the choices of orientations in Section

Preparation from conformal

146

geometry

9.9. The map p : A 3 p ^ Ep = Si n (ip)1- e S defines the bijection from A to S. As g G 0(4,1) preserves the Lorentzian pseudo inner product, it maps the L-orthogonal complement of (p) onto that of (g{p))- Therefore
= ±g (S3 n
x) = ±g{
((p^)

a We identify A with S through this bijection ip : A —> S in what follows. Lemma 9.1.1 implies that the Lorentz group Q acts on A transitively. Proposition 9.3.3 Under this identification : (1) The 2-sphere S = S2(x,l) in Sf with center (x,y,z,w,l) and geodesical radius r corresponds to

^-l (S2r(x, l)) = ± (-?-

, —)

=

(x,l)

e A.

(2) Let p be the stereographic projection from S3 \ {(0, 0,0, —1)} to K3 = {(x, y, z, 0)}. Then the 2-sphere S = S2R{X) in R3 with center {X, Y, Z) = X and radius R corresponds to

^R'

2R

'

2R

Proof. (1) Let u = ( — , cos^-) G A. Since S 2 (x, 1) = {(y, 1) | |y| = 1, \y - x\ = 2sin ( £ ) } = {(y.l)

|y| = 1, (V,x) = l - 2 s i n 2 (-^-J = c o s r | ,

ify = ( y , i ) e S 2 O M ) t h e n y,w = ^ sin r

;

=0,

sin r

which implies u G (5^(a;, l)) 1 " = (Spanj, eS 2 (a;il) (y)J

= i i p " 1 (S 2 (ai, !))•

The 4-tuple map and the cross ratio of 4 points

(2) Suppose ip

l

(p

1

(S^(X))) = (a,b,c,d,e).

Asp

p'1 : R3 3 x = (x,y, z) ^ p~\x) = ( j ^ , \ 1 + |ai] if x = (x, y, z) e S^X) 2x

a

+

T+~W

147

Ms given by

\ ^ ^ ) z

e S3,

1 + \x\ J

then 2y

+c

Iz

2 +

T+jxj i + \x\

1 - |x| 2 _

i+|zf

_ e

'

which implies x —d+e

d + eJ

{d + e)2'

Therefore (X,Y,Z)=(-£-,-£-,-£-), \d + e d + e d + ej which implies the conclusion.

fmdR=-±-r,

\d + e\ •

It is sometimes more convenient to consider the set P 5 of unoriented spheres. Then it can be identified with the projective de Sitter space WA = A/±l. 9.4

The 4-tuple map and the cross ratio of 4 points

Four points in S3 or M3 generically determine a unique 2-sphere which passes through them. By identifying this 2-sphere with the Riemann sphere C U {oo} we can define the cross ratio of any given four ordered points in S3 or R3. We identify the 3-sphere S3 with the intersection Sf of the light cone V = {v \ L(v) = 0} and the hyperplane {« 6 K5 | u 5 = 1}. Definition 9.3 is given by

(1) The configuration space Conf„(X) of a space X

Conf^X) = {(*!,-•• ,xn)}eXn\xijtxj = X x ••• x X\

iii^j}

A,

where A is called the large diagonal. (2) ([LOl]) The set of concircular points in S±, Cc(Sf), is the subset of Conf^iS13) consisting of concircular points.

Preparation from conformal

148

geometry

T h e codimension of the large diagonal A in Xn is the same as the dimension of X, and the set of concircular points Cc(Sf) is of codimension 2 in C o n f 4 ( S 3 ) . Proposition 9.4.1 Let vl = (v\,vl,vl3,v\, 1) be four points in S\3 1 , 2 , 3 , 4 ) . Then the following three conditions are equivalent. (1) The four points v% 's are concircular l

in

Sf.

(2) The four vectors v 's are linearly dependent

as vectors in K 5 .

exterior product v1 A v2 A v3 A v4

(3) The Lorentzian

vanishes.

Proof. Observe first t h a t the vl,s in S3 C ffi4 lie in a circle C if and only l, if the v s lie on an affine 2-plane P in M4 containing C. (1) => (2): Suppose the i>"s are concircular. T h e n as v4 lies on an affine 2-plane v 1 + S p a n ( w 2 — v1, v3 — v1), we have v4 = (1—a — b)v1+av2 + bv3 for some a, b £ R, i.e. (1 — a — fyv1 + av2 + fru3 — D 4 = 0. (2) => (1): Suppose a^v1+a2V2'-{-a^v3'+a^v4 = 0 for some ( a i , 02, 03, 04) l 7^ 0. Since the fifth coordinate of the v 's is 1, 0 1 + 0 2 + 03 + 04 = 0. As (01,02,03,04) 7^ 0, one of a^'s, say, 04 is not 0. Then v4 = ai

2

3

i.e. v4 lies on an affine 2-plane which passes through

Ql + «2 + 13

u 1 , v2, and v 3 . L e m m a 9.2.1 implies t h a t the conditions (2) and (3) are equivalent.

•

T h u s the set of concircular points can be expressed as Cc(S3) = S3 fl i/ (0) if we define 1/ : (M 5 ) 4 -> M5 by u(vl ,v2,v3,v4) = v1 f\v2 /\v3 /\v4. _1

An oriented 2-sphere E(xl, x 2 , x 3 , x 4 ) t h a t passes through x1, x2, x3, and x4 is obtained as the intersection of Sf and an oriented hyperplane in M5 t h a t passes through xl,x2,x3,x4, and the origin. D e f i n i t i o n 9.4 given by

([LOl]) (1) T h e 4 - t u p l e m a p E of t h e 3 - s p h e r e is

E : Conf 4(S3)\Cc(S3) K l V '^

l>

9 (x 1 , x2, x 3 , x4) ^ V '

xl hx2 hx3 h_x^_ ^jL{xl A a;2 A x 3 A x4)

4 Then ^ ( x 1 ) £ yl = 5 is an oriented 2-sphere t h a t passes through 3 four ordered points and x . 3 4 W h e n X •) X 1 X , and x are concircular we denote by Z ^ x ^ x ^ x ^ x 4 ) any oriented 2-sphere t h a t passes through X • X> m *Xs 3 and x .

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149

(2) Let x1, x2, x3, and a;4 be four points in Sf- Letp : E\xl, x2, xA, xA) —> C U {oo} be any orientation preserving stereographic projection, where the complex plane C is oriented in the standard way. The cross ratio of four 4 ordered points (x1 ) in the 3-sphere is defined as the cross ratio , < 2\ i 3^ , \\ i 4 ^ P(x2)-Pixl) p{x3)-p{xl) {P(X ),P(X ^PiX l),p{X4) = - ^ y-rr : -j-^r —fr .

p(xz) - p(x4)

p(xA)-p(x4)

4 4 We denote it by (x2 ) or cr(x 1 ) . (Here, the order of the four complex numbers to determine the cross ratio is chosen so that the "infinitesimal cross ratio" which we will define later in Chapter 11 can be interpreted as a 2-form.) Since an orientation preserving Mobius transformation of C U {oo} is given by a linear fractional transformation which keeps the cross ratio invariant, the cross ratio of four points in the 3-sphere does not depend on the choice of an orientation preserving stereographic projection. When a;1, a:2, a;3 and x4 are concircular, their cross ratio is real and independent of the choice of £(xl,x2,x3,x4).

(3) The 4-tuple map S of R3 and the cross ratio of four ordered points in K are defined through an orientation preserving stereographic projection p:S\ - ^ K 3 . 3

Theorem 9.4.2 If we orient the 2-spheres in S = A as in Section 9.9, then the imaginary part of the cross ratio of four ordered points is always non-negative. Proof. The large diagonal A C (<S3)4 is of codimension 3, and the set of concircular points in Sf, Cc(Sf), is of codimension 2 in Conf4(S'3) = (5 3 ) 4 \ A. Therefore Conf4(S'3) \Cc(S3) is arcwise connected. As the map cr : Conf 4 (5 3 ) \Cc(Sf) —> C \ K is continuous, its image is contained either in the upper half plane {z € C | 3m z > 0} or in the lower half plane. The Example 9.1 shows that the image is contained in the upper half plane, which completes the proof as cr(Cc(S3)) CM. • We will illustrate how to orient S(x1, x2, x3, x4) and verify the above theorem on page 172.

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R e m a r k 9.4.3 (1) E(x1,x2,x3,x4) changes its orientation under a permutation of the four points. Namely, S(x\x2,x3,x4)

= s g n r • S(xT^

,xT^

,xT^

,xT^),

(9.5)

for a p e r m u t a t i o n r of { 1 , 2 , 3 , 4 } . (2) T h e cross ratios of the possible 24 orders of fixed four complex numbers z\, z2,z3, and Z4 can be written as {A, 1 — A, y , JZJ, 1 — -j, x r f } T h e sign of the imaginary part of the cross ratio {zT(2), zT(3)> ZT(I)> ZT(4)) changes in the same way as the signature of the p e r m u t a t i o n r of { 1 , 2 , 3 , 4 } . (3) Let x1, x2, x3, and x4 be non-circular points. Let us join (a;1, x2, x3, xA) and {xl,x2,x4,x3) by a p a t h 7 ( t ) (0 < t < 1) in Conf 4 (S 3 ) \ Cc(Sf) 1 2 with 7(0) = (x , x , x3, x4) and 7(1) = (x 1 , x2, x4, x3). Let at denote the uniquely determined (unoriented) 2-sphere which passes through the four points corresponding to 7(f). Then 7(0) = 7(1). Choose an orientation of <JQ at t = 0 and orient at continuously. Then geometric consideration shows t h a t if we take a suitable p a t h as 7 then the orientation of a\ at t = 1 is exactly opposite to t h a t ofCTOat t = 0. This means t h a t if we can continuously assign orientations to 2-spheres determined by four points, then the same relation as (9.5) should hold. T h e role of the 4-tuple m a p is to guarantee t h a t we can do this on the whole set of Conf 4 (5?) \ Cc(S3). Proposition 9.4.4 Let P i , P2, P3, P4 G M 3 . Then their cross ratio cr = (P2, P3', Pi, P4) can be expressed in terms of d^ = \Pi — P2WP3 — Pi\, <^3 = I P i - P 3 | | P 2 - P 4 | , and di = |Pi - P 4 | | P 2 - P 3 | as cr—

d22 + d32 T: 2

d42

2d3 . \/(d2

+ d3 + di)(-d2

+ d3 + d4)(d2 - d3 + d4)(d2 + d3 - d4) 2d32

Proof. Let Ti be the inversion in the 2-sphere with center P i and radius 1. Let P'- = T i ( P j ) (j = 2 , 3 , 4 ) . Then

Let T2 be a Mobius transformation from 17(00, P2, P3, P4) to CU{oo} which maps 00, P2, and P4 to 0 0 , 1 , and 0 respectively. Then the cross ratio cr is

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151

equal to z = T2(P^), which satisfies \z\ - I M ^ s ) - J 2 ( n ) l - -pi \z-l\-

pi - T5

\T2(P3) - T2(P2)\ _ ^ — - -

|Pi

p-JTp-

51'

_P3||p2_P4|-

The formula is then obtained by an easy calculation in trigonometry.

9.5

•

Pencils of spheres

Definition 9.5 A 2-dimensional plane P through the origin intersects the hyperbolic hypersurface A C R n + 2 in a curve or two curves. This intersection V = PnA, or the corresponding set of (n— l)-spheres in S is called a pencil of spheres. In particular, when S\ and S2 are two (n — l)-spheres which correspond to two points pi and p2 in the same component V\ of V = P (~l A, we say that V\ is a pencil generated by S\ and S2. There are three cases of the intersection of a 2-plane P and the hyperbolic hypersurface A.

(la)

(lb) Fig. 9.3

(2)

Three kinds of pencils of spheres.

(la) When the plane P is space-like (i.e. all the non-zero vectors in P are space-like), V = P f~l A is an ellipse. Since P x contains a time-like

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vector, it intersects S^ transversely in an (n — 2)-sphere P. If p € P fl A then Op C P which implies that the corresponding (n — l)-sphere
Fig. 9.4

A space-like pencil and its base sphere.

(lb) When the plane P is tangent to the light cone V, V = PV\A consists of two lines. Put I = PC\V. Then P e l 1 because otherwise P must contain a time-like vector. Hence P1- D /. On the other hand, P1- C I1- since P D I. As Z"1 is tangent to the light cone at I, the intersection P x (~l S£, = Z D S^ consists of a point. Thus the pencil is made of spheres which are all tangent to each other at the point P1- n S^ = P fl S^. Any tangent vector to this pencil is light-like. (2) When the plane P contains a time-like vector, V = P fl A consists of two non-compact hyperbolas. The corresponding set of spheres is called a Poncelet pencil. As P x is space-like, P1- fl S^ = 0. Hence a Poncelet pencil is a set of disjoint spheres. As a point p G V goes to infinity, the line Op approaches one of the asymptotic lines I C P fl V. Therefore the

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153

corresponding (n — l)-sphere
Fig. 9.5

A Poncelet pencil and its limit points.

T h e o r e m 9.5.1 Let V — P n A be a pencil of spheres generated by Si and £2 (S\ 7^ ± £ 2 ) which correspond to pi and pi in A. Let A = (pi,P2)Then the length p of the arc (one of the arcs) in the pencil V joining p\ and P2 is given by |log(A + \f\2 — 1 ) | , where the argument of the logarithm is taken between 0 and IT. More precisely, it is given according to the previous three cases as follows. ( l a ) / / the 2-plane P is space-like then — 1 < (p\,P2) < 1- The length p of the shorter arc is equal to the angle 80 (0 < #0 < "") between the two spheres Si and S2. This angle 9Q satisfies cos#o = (JP11P2) and hence the length p satisfies cosp = (pi,P2)( l b ) If the 2-plane P is tangent to the light cone then (p\,P2) = 1the length p is 0.

Then

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geometry

(2) If the 2-plane P contains a time-like vector then {puP2) > 1- The length p satisfies coshp = (jp\,P2)Proof,

(la) Put p2 = P2 — {pi,P2)Pi- Since P is space-like L(p2) > 0

which implies 1 > {p\,P2}2 > 0. Put p'2 =

2

_ . Then {pi,p2} is V (P2) a Lorentz orthonormal basis of P. The pencil V is a circle which can be parametrized by {(cos9)pi + (sin#)p 2 | 0 < 9 < 2ir}. The restriction of the Lorentzian metric to P is equal to the standard Euclidean metric. Let 6Q be the value of the parameter at p2, i.e. p2 = (cos#o)pi + (sin0o)P2- Then the length p is equal to 9Q which satisfies cos#o = (pi,P2)By Lemma 9.1.1 there is a conformal transformation g 6 0(4,1) that maps pi and p2 to (1,0,0,0,0) and (cos 9o, sin 0O, 0,0,0) in A. If we identify R4 D S3 = Sf C {x5 = 1}, the angle between g • Si and g - S 2 at a point (0,0,1,0) is equal to the angle between two 2-planes in R 4 , (Span ((1,0, 0, 0), (0,0,1,0)))" 1 and (Span ((cos 60, sin6>0,0,0), (0,0,1, 0)))^, which is equal to 9Q. (lb) First observe that (pi,P2) i^ 0 because otherwise P is spacelike. Put p2 = P2 — (pi,P2)pi- Then p2 cannot be space-like since P = Span (pi,p2) and {pi,p~2) — 0- Therefore p2 is light-like as P does not contain time-like vectors. Then L(p2) = 0 implies (pi,P2) = 1. Hence the pencil is a line {tpi + (1 — t)p2\ t £ M.} in A whose tangent vector p2 — Pi is light-like. Therefore the length p is 0. L

( 2 ) P u t p 2 = P2 — {pi,P2)pi- As p 2 is time-like l — (pi,p2) 2 < 0. Put p2 = ,

. Then {p\, pU } is a Lorentz orthonormal basis of P. The pencil V is

a hyperbola which can be parametrized by {(coshi)pi + (sinhi)p 2 | t £ R } . The restriction of the Lorentzian metric to P is a quadratic form of type (1,1). Let to be the value of the parameter at p 2 - Then the length p is equal to to which satisfies coshio = (Pi,P2)D Corollary 9.5.2 ([LOl]) (1) Si n S 2 = 0 if and only if\(S1,S2)\ > 1. (2) 3^82^® if and only if |(Si, S 2 )| < 1. Definition 9.6 Two spheres Si and S 2 are said to be nested if Si DS2 = 0 and intersecting if Si n S 2 ^ 0 and Si ^ ± S 2 . As a curve, a pencil V = {V(t)} is a "geodesic" since the geodesic curvature kg vanishes, where kg(t) is the absolute value of the tangential

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component -KTAP{t) of the second derivative V(t) in the arc-length, where TTTA : T V ^ K 4 ' 1 3 v ^ v -

(v,V(t))

• V(t) G

is the L-orthogonal projection. Therefore a pencil V(t) of the length function L:

L : {curves in A} 9 (7 : [a,b]-> A) ^

/

^\L(-y'(t))\

Tv{t)A is a critical point

dt € R.

Jo

We notice t h a t t h e infimum of the lengths of rectifiable curves joining any two points qi and q2 in A is 0 since they can be joined by a "zigzag" curve in A with light-like tangent vectors. Therefore a pencil does not achieve the minimum of the length function. W h e n n = 2, A C R 4 gives the usual theory of pencils of circles. Proposition 9.5.3 Let C\ = U\ n Sf and C2 = U-i fl S\ be two circles which intersect at {x,y}, where Ili is a Z-plane in K 5 through the origin, and let Vi= Hi fl A (i = 1,2) be a pencil of spheres whose base circle is Ci. Then the angle between V\ and V2 is the same as the angle between C\ and CiIn particular, the conformal angle 9K(X,V) is equal to the angle between two pencils whose base circles are C(x,x,y) and C(y,y,x). Proof. Let Vi (i = 1,2) be a unit tangent vector of Ci at x. Then Ili = Span (x, y, Vi). Note t h a t V\ and Vi intersect at points corresponding to the unique sphere S — Span (x, y, i>i, 1*2) H Sf t h a t contains b o t h C\ and Ci. We have only t o show t h a t t h e angle between U\ and 772, which we denote by ZTTi -772, is the same as the angle between v\ and V2, since ZV± • V2 = ZTJj- 1 • 772"1 = ZTTi • n2. x —u Let w = —

y/-Hx,y)

- and u = —

x ~\~ 11 - be two unit vectors which span

^J-2{x,y)

III H772- We remark t h a t u is time-like. Then the unit L-orthogonal vector to 77i fl 7T2 in LTi is given by Vi = Vi + (vi,u)u — (vi,w)w. Then a simple calculation implies Z77i • II2 = (^1,^2) — ( ^ l i ^ ) d

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9.6

geometry

M o d u l u s of a n a n n u l u s

D e f i n i t i o n 9.7 (Langevin [LOl]) (1) A 3-dimensional a n n u l u s (or simply, an annulus) A is a region in S3 or R 3 diffeomorphic to S2 x [0,1] bounded by two disjoint 2-spheres Si and S^- We assume t h a t the boundary 2-spheres Si and £2 are oriented as the b o u n d a r y of A . (2) Let A be an annulus with dA = S1US2 and let T : R 3 (or S3) ->• R 3 (or R 3 U {00} respectively) be a Mobius transformation which maps the two boundary spheres of A into a concentric position. Then the m o d u l u s , m(A)

— m(Si,S2)

> 0, of the annulus A is given by 771(A) = log — - ,

where R\ and R2 {R\ > R2) are the radii of the two concentric boundary spheres of T(A). (3) T h e L o r e n t z i a n m o d u l u s , 777^(A) = m i , ( 5 i , S 2 ) > 1, of an annulus A with dA = S i U £2 is given by mz,(A) = | ( 5 i , 5I2)|, where Si and S*2 are considered as points in A C R 4 ' 1 . (4) T h e (Lorentzian) modulus of a 2-dimensional annulus S1 x [0,1] can be defined similarly. R e m a r k 9.6.1 The modulus m of a 2-dimensional annulus A with dA = Ci U C2 is equal to the hyperbolic distance between two hyperbolic lines contained in C\ and C^-

O Fig. 9.6

An annulus and its concentric position.

Let A be an annulus with dA = S i U S2. Then the two b o u n d a r y spheres Si and —5*2 are nested and generate a Poncelet pencil V, where —5*2 denotes the 2-sphere with the reverse orientation.

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157

Proposition 9.6.2 ([LOl]) The modulus m{A) is an increasing function of the Lorentzian modulus m t (A) : m(A) = log (mL + ^/'uiLiA)2 - 1 J and

TOL(A)

= coshm(A).

(9.6)

Therefore Theorem 9.5.1 implies that the modulus m{A) = m(Si,S2) is equal to the length of the arc joining Si and —S2 in the Poncelet pencil generated by these two spheres. Proof. As in the proof of Theorem 9.5.1 we may assume, after a Mobius transformation, that the two corresponding points to 6*1 and — S2 in A are (1,0,0,0,0) and (coshto,sinhto,0,0, 0) for some to > 0. Then Si is an equatorial 2-sphere of S 3 = Sf and S2 is a "parallel" 2-sphere with 1

Euclidean radius r =

„

. Applying a stereographic projection to R ,

cosh to these two spheres are mapped into two concentric spheres with radii 1 and - = e'°. Therefore the modulus m(A) is equal to to• 1 — v l — r2 Remark 9.6.3 ([LOl]) (1) If the annuli Ai and A2 satisfy Ai C A2, one has m(A 2 ) > m(Ai). (2) An annulus A is uniquely determined by its modulus up to the motions of Mobius transformations. (3) Let V be a Poncelet pencil generated by Si and — S2- Then the modulus of an annulus can be computed from the cross ratio of the four intersection points of the two boundary spheres with a circle containing the two limit points of V. Lemma 9.6.4 Let II be a plane and S be a 2-sphere in M3 such that the minimum and maximum distance between TI and S are a and b (0 < a < b) respectively. Then the modulus m{E,TI) of the annulus bounded by E and LJ is given by

Proof.

We may assume that 77 is the y-z plane and E has [~—, 0,0 J

as its center. Then any inversion in a 2-sphere with center (± \/ab, 0,0) maps E and 77 to a concentric position. •

158 9.7

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C r o s s - s e p a r a t i n g a n n u l i a n d t h e m o d u l u s o f four p o i n t s

D e f i n i t i o n 9.8 (Langevin [LOl]) Let xi,X2,xs, points i n R 3 (or 5 3 ) .

and X4 be four ordered

(1) We say t h a t an annulus A is c r o s s - s e p a r a t i n g if X\ and X3 are contained in one of the connected components of E? \ IntA (or S3 \ Int A) and X2 and X4 in another component. (2) As the degenerate case when the two b o u n d a r y spheres of an annulus coincide, we say t h a t a (an oriented) 2-sphere E is c r o s s - s e p a r a t i n g in a s t r i c t s e n s e , or simply c r o s s - s e p a r a t i n g , if one of the connected components of R \ S (or S3 \ E) contains x\ and £3 and another connected component contains x-i and x\. (3) T h e m o d u l u s , m(xi,X2,X3,X4) > 0, of four ordered points xi, X2, X3, and X4 is the supremum of the moduli m{A) of cross-separating annuli A. W h e n there are no cross-separating annuli we put m(x\,X2,X3, X4) = 0. (4) A cross-separating annulus t h a t attains the modulus m{x\,X2,xz, x4) will be called the m a x i m a l c r o s s - s e p a r a t i n g a n n u l u s of t h e four point. (5) T h e L o r e n t z i a n m o d u l u s , m i ( i i , i 2 , i 3 , i 4 ) > 1, of four ordered points is defined by the Lorentzian modulus | ( 5 i , 5 2 ) | of the boundary spheres of a maximal cross-separating annulus A with dA = Si U S2. W h e n there are no cross-separating annuli we put m,L(xi,X2,X3,X4) = 1. R e m a r k 9.7.1 Any 2-sphere in a cross-separating annulus for xi,X2,xs, and X4 is also cross-separating. B u t a cross-separating 2-sphere is not necessarily contained in a maximal cross-separating annulus. L e m m a 9.7.2 (Langevin [LOl]) If A is a maximal cross-separating annulus of Xi,X2,xz, and X4, then one of its boundary sphere should contain both x\ and X3, and the other X2 and x±. Proof. We may assume, after a Mobius transformation, t h a t the boundary spheres S\ and S2 of A are concentric in M 3 , and t h a t 52 is an outer sphere. Suppose X2 is an outside point of 52. T h e n we can inflate 52 to obtain a greater sphere S'2 which does not contain either X2 or X4 inside. We can take as 5 2 a greater concentric sphere if X4 $ S2 as in Figure 9.7, or a sphere which is tangent t o 52 at x\ t h a t passes through X2 if X4 € 52 as in Figure 9.8.

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annuli and the modulus of four

points

159

Fig. 9.7 A maximal cross-separating annulus A which does not contain the four points can be inflated to produce another cross-separating annulus A' with a larger modulus; the concentric case.

Fig. 9.8

The tangential case.

Let A' be an annulus with dA = SiUS'2- Then A' is a cross-separating annulus of x\,X2,x$, and X4 with A' D A, which implies m(A') > m(A), which contradicts A being maximal. • Theorem 9.7.3 (Langevin [LOl]) (1) There are no cross-separating annuli, i.e. m = 0, rriL = 1 if and only if the Xi 's are concircular in such a way that X\ and £3 are not adjacent, i.e. when the cross ratio (x2,x3;xi,x,4) belongs to an open interval (0,1). (2) If there is a cross-separating annulus, then a maximal cross-separating annulus exists uniquely.

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(3) Let A be a maximal cross-separating annulus for (x\, X2, £3, X4) with dA = S\ U S2. Suppose we transform the annulus A by a Mobius transformation T so that the boundary spheres T(Si) and T(S2) of T(A) are concentric. Then T(xi) and T(xz) are antipodal in one of the T(Si) 's, and T(x2) and T{x^) are antipodal in the other. (4) Two boundary spheres of a maximal cross-separating annulus intersect the 2-sphere £(xi,X2,xs,X4) in right angles.

Fig. 9.9

Maximal modulus for four points.

Remark 9.7.4 (Langevin [LOl]) Being antipodal is not a conformally invariant condition. The statement (3) of the Theorem can be rephrased as follows: (3)' Let A be a maximal cross-separating annulus for (xi,X2,£3,2:4) with dA = 5 i U 5 2 . Then x\ and £3 must lie on a circle which passes through two limit points of the Poncelet pencil generated by S\ and —Si- Similarly X2 and £4 must lie on another circle with the same property. When the boundary spheres are concentric the limits points of the Poncelet pencil are the center O of the boundary spheres and 00, and circles through these limit points are straight lines through the center O. Proof. (1) We may assume that x\ = 0,£2 = z,x$ = 1, and X4 = 00 in C U {00} where z is the cross ratio (x2,x3; £1,0:4). Then there is a disc which contains X\ and £3 but not £ 2 and £4 if and only if z does not belong to (0,1). (2) Let us show the existence first. The set S\ 3 of spheres that pass ( ^ ~^' \ L through Xi and £3 corresponds to I Span(0£i, 0£3) I (~l A, which is a 2-

Cross-separating

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161

sphere in A, and hence is compact. Therefore t h e maximal Lorentzian modulus maxsjgSj 3,s2es2 4 |(<Si, Sr2)| is attained. Then Corollary 9.5.2 and Proposition 9.6.2 imply the existence of a maximal cross-separating annulus. (3) L e m m a 9.6.4 implies: L e m m a 9.7.5 Let E be a 2-sphere in M3 with center 0, and Q g R 3 be a point outside E. Then among the planes U passing through Q, the plane -/To perpendicular to the line OQ gives the maximal modulus m(E,II). Since a plane 77 passing through Q is a 2-sphere t h a t passes through Q and oo, this lemma implies (3), as, if To is an inversion in a 2-sphere whose center lies on the line OQ t h a t maps E and 77o to a concentric position as is given in the proof of L e m m a 9.6.4, then To(Q) and Xb(oo) are antipodal in To(7T 0 ). (4) is a consequence of (3). Let us come back to the proof of (2). T h e uniqueness is implied by the following characterization of the boundary circles of the maximal crossseparating annulus in the pencils with base points {x\,xz} and {£2, £4}L e m m a 9.7.6 (Langevin [LOl]) Let A be a maximal 2-dimensional annulus with dA = C\ U C2 with C\ B Xi,x$ and C% B £2,2:4- Let V\$ be a pencil of circles with base points {xi,xs} and 7^2,4 be a pencil of circles with base points {x2,x4}. Let C124 = C(xi,X2,X4) and C324 = C{x^,X2,xi) be two oriented circles in P24 (Definition 8.3). Then the pencil T'2,4 is divided into four arcs by ±Ci24 and ±C324- Then the open arc 724 between C124 and C324 corresponds to circles which can serve as boundary circles of cross-separating annuli. Then C\ or —C\ is the middle point of this open arc 724. A similar condition holds for C^. Therefore a maximal crossseparating annulus is unique. Proof. We may assume, after a suitable Mobius transformation, t h a t the boundary circles C\ and C2 are concentric about the origin O. T h e n X\ and £3 are antipodal on C\ and X2 and X4 are antipodal on C2 by (3). Therefore if we rotate the picture around the origin O by the angle ir, the circle C324 is m a p p e d onto the circle C124. Hence the angle between C and C324 is equal to the angle between C and Ci24- Since the length of an arc in the pencil is equal to the angle by Theorem 9.5.1 ( l a ) , this completes the proof. •

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Fig. 9.10 lus.

geometry

Characterization of the boundary circles of a maximal cross-separating annu-

This completes the proof of Theorem 9.7.3.

•

We remark that the Theorem can also be proved by a direct calculation. Let us give a conformal geometric interpretation of (3) and (4) of Theorem 9.7.3. We start with a dimension reduction lemma, which implies (4). Lemma 9.7.7 (Langevin [LOl]) Let Ci andCi be two disjoint circles on a 2-sphere E and g/(C\,C2) be the set of annuli whose boundary spheres intersect E in G\ and C 2 . Then the supremum of the moduli of the annuli in s^(Ci^C'i) is attained by the unique annulus AQ whose boundary spheres intersect E in the right angle. In particular, the maximal modulus of the annuli in sd{C\,C2) is equal to the modulus of the 2-dimensional annulus whose boundary circles are C\ and C 2 . Proof. Let Vi be a pencil of spheres with base circle d. Then V\ (~l Vi = ±E. Let Pi be 2-plane in R5 such that Vt = A n Pi. Let Sf £ Vi be the 2-sphere which intersects E in the right angle. Then {E, E®} C A makes an L-orthonormal basis of Pi. The pencils Vi can be parametrized by Vi = {Ei{0i) = cosBiE + smOiE?}. Since Theorem 9.5.1 (2) implies that \(£%,Z$)\ > 1

Ki^),i; 2 (0 2 ))| < Icos^cos^i + isin^sin^iKi:?,^0)! < |(r?,r2°)|,

Cross-separating

annuli and the modulus of four

points

163

where equality holds if and only if cos 9\ = cos 92 = 0. Therefore the maximal Lorentzian modulus is attained at ±17° and ±U® which are boundary spheres of AQ • • T h e above L e m m a allows us to work with 2-dimensional annuli on the 2-sphere which contains the four points Xi,x2,X3, and X4 in what follows. L e m m a 9.7.8 Let C be a circle on a 2-sphere S and x and y be two points in one of the connected components of E\C. Let si[C\ x, y) be the set of 2-dimensional annuli such that one of the boundary circles is C and another boundary circle contains both x and y. Then the supremum of the moduli of the annuli in stf(C\ x, y) is attained by the unique annulus AQ with OAQ = C U C o . If we transform this annulus AQ by a Mobius transformation T so that the boundary spheres T(C) and T{CQ) are concentric, then T(x) and T(y) are antipodal. In other words, x and y lie on a circle which passes through two limits points of the Poncelet pencil generated by C and Co • Proof. Let V be a pencil of circles with base points x and y. Let P be a 2-plane in R 4 such t h a t V = P D A. Let us identify a circle C with a point C in A as before. Let C\ be a point in (Span(C))^ L fl V and Co be a point in V which is L-orthogonal to C\. Then { C o , C i } is an L-orthonomal basis of P, and the pencil V can be parametrized by V = {C(9) — (COS9)CQ + (sin(9)Ci}. T h e n the Lorentzian modulus satisfies | ( C , C ( 0 ) ) | < \{C,C0)\, where equality holds if and only if C(9) = ± C 0 . Therefore the supremum of the moduli is attained by the unique annulus whose b o u n d a r y circles are C and CoLet P' b e a 2-plane spanned by C and Co a n d V' pencil generated by C and Co- Since the point C\ is Co, it is L-orthogonal to the limits points P' Pi 5 ^ V', which means C\, as a circle, passes through the

= P' fl A b e a Poncelet L-orthogonal to C and of the Poncelet pencil limits points. •

T h e above two lemmas imply (3) of Theorem 9.7.3 since if a 2-dimensional annulus A with dA = C\ U C2 ( C L 3 x\^x-j,,Ci 3 Xi,x$) is a maximal cross-separating annulus, then it attains the supremum of the moduli of annuli in s4{C\;xi,x±) and hence x-i and X4 must be antipodal when two boundary spheres are concentric. We close this section with a formula to express the maximal Lorentzian modulus rriLix1 ,x2,x3,x4) of the four points in terms of the cross ratio

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geometry

Proposition 9.7.9 ([LOl]) (1) The maximal Lorentzian modulus m,r,(xi,X2,X3,X4) can be expressed in terms of the cross ratio cr = cr(x2,X3m,xi,X4) as mL(xi,X2,x3,x4)

211-cif

(2) In particular, u + iv is given by

the maximal

(9.7)

{|I^|-««(T^)}

Lorentzian

modulus

of oo, 0,1, and z

2{\z-l\-Xe(z-l)}

m L (oo,0,1,2;)

(9.8)

W2

Proof. (1) We may assume, after a Mobius transformation if necessary, that the boundary spheres of the maximal cross-separating annulus are concentric about the origin. Then Theorem 9.7.3 asserts that x\ and x% are antipodal in one of the boundary spheres, and x2 and x\ are antipodal in another. We may assume that x\ = l,x3 = —1 and X2 = w,x4 = —w in C C R3 for some complex number w. Then the maximal Lorentzian modulus mi,(xi,X2,X3,X4) is equal to mL(xi,x2,X3,xi)

= -

M

since the radii of the boundary spheres are 1 and \w\. On the other hand the cross ratio is equal to cr= (x2,x3;xi,x4)

=

{w-lf 4u>

Then 1 — cr

(w + 1)2 Aw

cr 1 — cr

jw-1)2 (w + 1)2

Since \w\

$tzw2

(w + w)

w GC

we have 1 — cr

5fte

1 Iw — 1 1 — cr

w — 1

~2 I w + 1 + w + 1

V \w + l\2

The measure on the space of spheres A

165

which implies the equation. (2) The first formula implies the second since ci = (0,1; oo, z) = 1 and

= z — 1.

•

1 — cr

9.8

The measure on the space of spheres A

A choice of a point z in H n + 1 determines a metric on the sphere Sn at infinity of HT +1 by the projection on Sn of the metric of Tz{Wl+l) using the geodesic rays which start from z. Another choice of the point z determines another conformally equivalent metric on the sphere Sn. Although the sphere does not admit a measure which is invariant under the action of the Mobius group A4, the set S = A of (n — l)-spheres in Sn does. Namely, A is endowed with a (/-invariant measure dE, or more precisely, a Q+ = SO(n + 1, l)-invariant volume n + 1-form U0A- We give several equivalent definitions in what follows. (1) The Lebesgue measure \dx\ A • • • A dxn+2\ is invariant under the action of Q. Then A inherits from M™+2 a ^-invariant measure defined by dE = \ix{dx\ A ••• A dxn+2)\i where x 6 A and tx denotes the interior product (see (5) of Lemma 9.2.1). (2) This measure dE is induced by the volume (n + l)-form UJA on A associated to the Lorentz quadratic form L as dE = \UJA\Lemma 9.8.1

(1) Put n+2

CO A = 2^{—^Y xidx\ A • • • Adxi • • • A dxn+2, 1=1

where Adx^ means that Adxi is removed. Then CO A is the volume (n + 1)form on A associated to the Lorentz quadratic form L. Therefore it is invariant under the action of Q+ = SO(n + 1,1). (2) Using (xi, • • • , xn+i) as local coordinates of A+ = {{x\, • • • , xn+2) € A | xn+2 > 0}, this CO A can be expressed as CO A = Xn+2

on A+.

dx\ A • • • A dxn+i

Preparation from conformal

166

Proof.

Since xn+2 = \/xi2 + •

(2) Let x — (xi,- •• , xn+2). d

xz

<7Xj

£

geometry

-en+2

+ xn+12

- 1

( l < i < n + l).

n + 2

Then Lemma 9.2.1 (5) implies that d

d

i det (/ ^— , ^ —

det I x,

d dxi' n+l

[-n+2

d dxn+xJ \ 2

htx™+*

Xn+2

Therefore the volume form associated to the Lorentz quadratic form is expressed as 1 dx\ A • • • A dxn+i on A+.

•

Xn+2

(3) Let Q be a point in Sn = 5™ and p be a stereographic projection from Sn \ {Q} to an affine hyperplane W1 with Euclidean coordinates X — (Xi, • • • ,Xn). An [n — l)-sphere in W1 can be parametrized by its center X and its radius r. Therefore we have a bijection ip from K™+1 to a subset of A consisting of (n— l)-spheres in 5™ which do not pass through the point Q defined by V>:M++1

3(X,r)

^-1op-1(5r1(X))e^\(Span(Q))±,

where 5*™ 1(X) denotes an (n — l)-sphere in M™ with centerX and radius r, and (f : A —> S is the canonical bijection given in Theorem 9.3.2. Lemma 9.8.2

The measure dS is expressed as ip*dE =

1

\dX1 A • • • A dXn A dr\

(9.9)

•pra+l

Proof.

We prove it first in a special case when Q is the south pole and p:S"\{(0,"-,0,-l)}->Rn^{(a;i,---,:En)0)}

is the stereographic projection to the hyperplane containing the equator. Then the formula (9.4) in Proposition 9.3.3 shows that ip{X, r) € A is given

The measure on the space of spheres A

167

by iP(X,r)

'X

l + r 2 - | X | 2 l - r 2 + |X| 2

r

2r

' +1

which implies that UJA is expressed on R™ %P*UJA — ip* (

dxi A • • • A dxn+i

W+2

2r as

)=

r-rdXi A • • • A dXn A dr.

rn+l

J

Let us prove the formula (9.9) for a general case. Let p : Sn\ {Q} —>• M" be another stereographic projection, and tp : R™+1 —> A be the corresponding map. As g = p o p _ 1 : M" —>• R n is a Mobius transformation, it is enough to show that the (n + l)-form UJun+i = dx\ A • • • A dxn+\ is +

Xn + 2

invariant under an induced map 5 : R™+1 —> R" + 1 up to a sign for any Mobius transformation g. If g is a parallel transportation X \-> X + u, the induced map is given by g : (X,r) H-> ( X + u , r ) , and therefore (?*a;K„+i = ^ K n+i• If g is a homothety, then obviously g preserves WRn+i. If g is an inversion in a 2-sphere with center the origin and radius 1, then the induced map is given by ~ . iran+l -. c v

r-\ UJ. I

Therefore ]j*OJmn+i = ±UJ&n+i.

^

!

\ c

ron+1

O

(4) This measure can also be regarded as a measure on the set of totally geodesic hyperplanes of the hyperbolic space M""+1 which is invariant by the action of the hyperbolic isometries [San]. Lemma 9.8.3 ([LOl]) (1) The hyperbolic hypersurface A is unbounded and the total measure of A is infinite. (2) The set of 2-spheres which intersect a given arc of any size is unbounded in A and it has infinite volume. (3) The set T(r) of (n — \)-spheres in the unit sphere 5™ whose radii are larger than r is compact in A and hence it has finite volume. Proof. (2) Let / be a closed unit interval [0,1] in M3 and U be the set of 2spheres in M3 which intersect / . Suppose U is parametrized by (X, Y, Z, r), where (X, Y, Z) is the center of a sphere and r is its radius. Then the

Preparation from conformal

168

geometry

measure on U C M+ is — dX A dY A dZ by Lemma 9.8.2. If p < ±- then r*1 U fl {(X, y, Z, r) | r = p} contains a solid cylinder with radius p and length 1 and hence its volume is larger than np2. Therefore the volume of U is infinite. (3) Let n = 3. A 2-sphere S in Sf = V D {v\v5 = 1} has a larger radius than r if and only if it is the intersection of Sf with an affine 3plane H C M4 whose distance between the origin is smaller than \J\ — r2 . Therefore T(r) = \(a1,---

,a5)eA

l«5| 2

y/ai

(ai,--- ,a5) e A ai 2 H

+ • • • + a42

< y/l - r2 I

+ a 4 2 < l2~ J" '

Hence

Vol(^(r)) = 2/

Vol(53(/9))

_4ir2

1

( 1

^ = f ^ = ^ \ / ^ " M ^ +2

which is an increasing function of —, where S3(p) is the 3-sphere in M4 r

with radius p. 9.9

•

Orientations of 2-spheres

We orient all the 2-spheres in S = A simultaneously so that the imaginary part of the cross ratio of any given four points defined in Section 9.4 is non-negative. With this choice of orientation, the imaginary part of the infinitesimal cross ratio of any pair of points on any knot which will be given in Chapter 11 is also non-negative. The orientation argument is not necessary in the rest of this book. The reader can skip it by taking the absolute values and forgetting the signs. In fact, the map 9K • K x K -> M behaves like a map of an absolute valued function as we will see in Example 10.2. The orientation of a connected orientable manifold M can be given by the choice of one of the two equivalence classes of ordered bases of a tangent space TpM at a point p in M, where two ordered bases of TpM are said to be equivalent if and only if the determinant of the transition matrix is positive. A set of ordered basis is called positive when its equivalence class

Orientations

of 2-spheres

169

coincides with the orientation of M. We denote an ordered basis of TPM by («i,--- ,vn). Let us start with the definition of the orientation of an intersection of oriented subspaces in an oriented ambient space. Let Pi = {x | X5 = 1} be a level hypersurface in M5. We assume that P1 = M4 is oriented in the standard way. Let us identify S3 with S3 = V fl Pi, where V is the light cone. Definition 9.9 (1) Let p G A and D\p = (Op)1- be its L-orthogonal complement used in Theorem 9.3.2. The orientation of Hp is given as follows. An ordered basis {v\,vi,v$,v±) of TxIIp at a point x £ IIp is positive if and only if (p,Vi,V2,v^,V4) is a positive ordered basis of TXK5. (2) We orient S3 as the boundary of P 4 C M4 with the homological convention, namely, an ordered basis (^1,^2,^3) of TXS3 at a point x in S3 is positive if and only if (v\,V2,vz,x) is a negative ordered basis of TXM4, i.e. (x, vi, U2, ^3) is a positive ordered basis of TXK4. This orientation of S"3 induces that of Sf C Pi = M4. (3) The intersection Ep of the oriented 3-sphere S3 and an oriented hyperplane TLp is given as follows. Let x 6 Sp- An ordered basis (i>i,i>2) of TxEp is positive if and only if there are u 6 TXS3 and wi,W2 G TxIIp such that (u,v\,V2) is a positive basis of TxSf, (vi,V2,w1,W2) is a positive basis of TXIIP, and that (u,v\,V2,w\,W2) is a positive basis of TXK5. We remark that the homological convention to orient Sn~1 as the boundary of Bn C K" is opposite to the geometric convention if and only if n is even. Example 9.1

Xi

Let 0 0 0

, X2

=

W

0 0 0

, x3 =

1 0 0

W

, x4 =

0 1 0

u

Then the 4-tuple map is given by x\ A X2 A Xz A X4 ^ ( ^ 1 , ^2, ^3,^4)

\jL[x\

= (0,0,0,-1,0) € A

A X2 A X 3 A X4)

Let us denote this vector by p € A and put i7 p = S\ n (Op) -1 . Then i7p = {(^1, ^2, ?3,0,1) I ^1 + £f + Cl = 1} is a 2-sphere which passes through

Preparation from conformal

170

geometry

the four a%-'s. Let us take

f°\ 1

(°\0 u =p

0 -1

GTXlSf,

Vl

=

0 0

/ ^2

Vo7

Vo /

/!\ 0 101 =

0 0

w

0

\ 0 -1 0

e^i;,

Vo/

/ °0 \ , w2 =

0 0

eTXlnp.

Iv - 1 /

Then ( f i , ^ ) is a positive ordered basis of TXlSp. Therefore a stereographic projection n from "the north pole" X4 is orientation preserving. Since it maps X\,X2,X^,XA to 1, — l,i,oo, cr(xi,x2,x3,x4)

— ( - l , i ; l,oo) = 1 + i.

Thus S r a c ^ a ; ! , ^ , ^ , ^ ) > 0. Let us give an explicit formula for the orientation of a 2-sphere
£ = 5f n 77 = sf n (77 n Pi), where Pi = {x | £5 = 1}. Let i be a point in 17. Since TxSj» = (Span (a, es))"1-, Tx(77nPi) = (Span(^e5))± we have T^Z1 © Span (x, 1/, e 5 ) = K 5 . Therefore det(vi,V2,x,v,e5) ^ 0 for any basis {t>i,t>2} of T^Z1, and the orientation of TXS can be given by the sign of this determinant. Since in the previous example det(vi,V2,xi,p,B5) > 0 for a positive ordered basis (^1,^2) of TXlSp, we have:

Orientations

of 2-spheres

171

Proposition 9.9.1 Let v e A, E = (p(y), and x G E. Then an ordered basis (^1,^2) ofTxE is positive if and only if det(vi,v2,x,u,e5)

> 0.

Let us then give a more convenient expression of the orientation of a 2-sphere which passes through any given non-concircular four points. Proposition 9.9.2 Let xi,X2,Xs, and X4 be non-concircular points in S3, and E = E(xi,X2,xs,X4) be the oriented 2-sphere given in Definition 9.4. Then E contains two oriented circles, C134 = C(xi,X3,Xi) and C124 = C(x\,X2,X4) (see Definition 8.3). Let 1/134 (cci) and ^124(2:1) be the positive unit tangent vectors at x\ to C{x\,xz,Xi) and C(xi,X2,X4) respectively. Then the ordered pair of basis (1^134(xi), 1^124(^1)) gives the positive orientation of the tangent space TXlE(xi,X2,X3,Xi) of E at x\.

Fig. 9.11

The orientation of E =

E(X\,X2,XZ,XA).

Proof. There is an orientation preserving conformal transformation T which maps E to Cu{oo} such that T(x{) = 0,T(xz) = 1, and T(xi) = 00. Let z = T(x2) then z is equal to the cross ratio (2:2, £35x1, £4), and hence Sm z > 0 by Theorem 9.4.2. Therefore an ordered basis ( 0 1 , Oz) of T 0 C is positive. Since T maps C(xi, £3, £4) to the oriented real axis C(0,1, 00), and C'(xi,X2,Xi) to an oriented straight line C(0, 2,0c), T* (1/134(2:1)) = k\ OX and T*(1/124(2:1)) = &2 Oz for some fci,A;2 > 0, which completes the proof. •

172

Preparation from conformal

geometry

Let xi,X2,a:3, and X4 be non-concircular points in M3. Let us see that the cross ratio (x2, x3; x1, x4) is in the upper half plane {z £ C | 9m z > 0} using the same notations as in the above Proposition. Let 77 be an oriented 2-plane through £1, £3, and X4 and let p : S —> 77 be an orientation preserving stereographic projection.

77 ^

Fig. 9.12 An orietation preserving stereographic projection p from 17 to 77. The images of p are denoted by the same symbols as their preimages.

Let us take any identification 77 = C. Consider a Mobius transformation T of CU{00} that maps x\,xs, and X4 to 0,1, and 00 respectively. Then x-i 4 is mapped to the cross ratio cv = (x ) . Then C134 is mapped into the real axis with the standard orientation, and C124 into the line through the origin and X2 whose orientation is given by Oa^-

•

Fig. 9.13

> c i:

The cross ratio in the upper half plane.

Since the orientation of C is given so that (01, Oa^) is a positive basis of TQC, X2 = cr must be in the upper half plane.

Orientations

of 2-spheres

173

Let us finally consider the degenerate case when x-i approaches x\ and X4 approaches X3. Definition 9.10 Let vx and vy be unit vectors in tangent spaces of M3 or S3 at x and y respectively. Suppose x,vx,y, and vy are not concircular, namely, there is no circle C passing through x and y which is tangent to vx and vy at x and y. Then they uniquely determine a 2-sphere £(x; vx,y; vy) that passes through x and y and that is tangent to vx and vy at x and y. We will call it a twice tangent sphere. We assign an orientation to S(x; vx,y;vv) as the limit of E(x,x + £\vx,y,y + £2Vy) (£i,£2 > 0) as £\ and £2 decrease to +0. Then C(x,y,y + £2Vy) approaches an oriented circle C(x,y;vy) passing through x which is tangent to vy at y whose orientation is given by vy at y-

Let vy(x) denote the unit tangent vector to C(x,y;vy) at x. We call vy(x) the unit vector vy conformally transported to x, and call the correspondence vy 1—>• vy(x) the unit conformal transportation of a unit vector from y to x as in Definition 3.3. Then: Proposition 9.9.3 The ordered pair (vy(x),vx) gives the positive orientation of the tangent space TxS(x;vx,y;vy), where E(x;vx,y;vy) is oriented as the limit of S(x,x + £ivx,y,y + £2Vy) (£1,^2 > 0) as £1 and £2 decrease to +0.

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Chapter 10

The space of non-trivial spheres of a knot

10.1

Non-trivial spheres of a knot

Any set of four (ordered) non-concircular points determines the unique (oriented) 2-sphere which passes through them. In this chapter we study the set of spheres determined by four points on a given knot K. This set in A is related with the geometry of the knot K. Definition 10.1 (Langevin [LOl]) (1) An oriented 2-sphere E is called a non-trivial s p h e r e for a knot K if E intersects K at more than or equal to four points, where the number of intersection points in K n E is counted with multiplicity. (2) An oriented 2-sphere E with E is called a non-trivial s p h e r e in t h e strict sense for a knot K if each connected component of M3 \ K (S3 \ K) contains at least 2 connected components of K \ E. (3) We denote the set of the non-trivial spheres (or the non-trivial spheres in the strict sense) by S(K) C A (or respectively, S{K) C A). R e m a r k 10.1.1 A 2-sphere E is non-trivial in the strict sense for a knot K if and only if we can find four points x\,X2,X2,, and X4 on K in this cyclic order such that E is a cross-separating 2-sphere in the strict sense for xi,X2,x$, and X4. A non-trivial sphere in the strict sense generically intersects K at more than or equal to four transversal points. L e m m a 10.1.2 The set S{K) of non-trivial spheres for a knot K is a non-empty compact ^-dimensional subspace of A when K is not a circle. When K is a circle S(K) is a pencil with base circle K and hence is a 1-dimensional compact subspace of A. 175

176

The space of non-trivial

spheres of a knot

Proof. Suppose K is not a circle. Lemma 13.4.2 implies that S{K) is non-empty and is 4-dimensional. Let us fix a metric on the ambient space S 3 , which corresponds to fixing a stereographic projection from S3 onto M3 and consider knots as knots in M3 with the standard metric. Recall that A(K) = \3(K) =

min

r(a(xi,X2,x3,xi)),

where r(a(xi, X2, x$, X4)) denotes the radius of the smallest 2-sphere that passes through Xi,X2,X3,x± (see Definition 8.6). As a(xi,X2,X3,x4) is a non-trivial sphere for K, O(K) > 0 is the minimum of the radii of the non-trivial spheres. Therefore S(K) C A is bounded. The closedness of S(K) C A is obvious. • Suppose K is not a circle. The boundary of S{K) consists of spheres which are tangent to the knot K, i.e. <% = {S(x1,x2,x3,x4)

I xk e K, Xi = Xj (i ^ j)}.

This boundary is not smooth. In particular, it has "corners" which are the transverse intersections of two folds ^ / s of the set of the tangent spheres. There are two types. One is the intersection of 2?^ and =3^/, where {i,j,k,l} — {1,2,3,4}, and the other is the intersection of ^ij and £?ik where j ^ k. In the first case we get the set of twice tangent spheres, the spheres which are tangent to the knot in two distinct points. A "corner" of the first type meets another "corner" in an osculating sphere, a sphere of type S(x, x, x, x). The closure of the boundary dS{K) and the closure of its corner may have other singularities. In this chapter we identify R4 with an affine horizontal hyperplane in M5 of level 1, M4 = {{xir-,x5)\xB = 1}, and S3 with Sf = VnRf, where V is the light cone. We assume that a knot K lies in S3 and is parametrized by f(t) =

(fi(t),f2(t),f3(t)J4(t),l)eS3.

We assume that the derivative / ' never vanishes and that the knot K = f{Sl) is oriented via / . (The assumption of orientation of a knot K is not necessary. The reverse orientation will give the same results.) In the following two chapters we use the following notation: x = / ( s ) , x + Ax = f(s + As), y = /(<), and y + Ay = f(t + At).

The 4-tuple map for a knot

10.2

177

The 4-tuple map for a knot

In this section we define a map that assigns an oriented 2-sphere that passes through a given 4-tuple of points on a knot, and then a map that assigns tangent spheres as degenerate cases in the next section. Definition 10,2 ([LOl]) (1) The set of (distinct) concircular points of a knot K, Cc(K), is given by 3

Cc(K) = i {tut2,t3,U)

e Conf 4 (S' 1

i.e. Cc(K) = (/ x / x / x

C = CK(ti,t2,h,t4) : a circle\ C9/(tl),/(*2),/(*3),/(t4) J'

f)-\Cc(Sf)).

(2) We call the above circle CK(ti,t2,t3,t4) (tut2,t3,t4) in Cc(K).

the base circle of

As a circle in Sf is an intersection of an affine 2-plane II in M.f with Sf, n iTS\

f iJ. , , + \

= \(h,t2,t3,t4) [

^

r

,nu

€ Canf^S 1 )

3

77 : an affine 2-plane in M? 1

Spany^),/^),/^),/^)) is of dimension 3

and therefore Proposition 9.4.1 implies that Cc(K) = {(
Is Cc(K) non-empty for any trivial knot Kl

Moreover we will show later that Cc(K) is generically of dimension 2. Intuitively, this is also indicated by the following observation: Let Kp = Is (K) be an inverted closed knot in a sphere with center p € R 3 \ K. Then G. Kuperberg's theorem implies that there is a line Lp such that Ip n Kp, and hence Ip(Lp) n K, contains 4 points. This means

The space of non-trivial

178

spheres of a knot

t h a t for any point p e M3 \ K there is a circle Cp t h a t intersects K in at least 4 points. This implies t h a t , by considering t h e normal plane t o Cp at p, t h e set of such circles, a n d hence Cc{K) has dimension 2. D e f i n i t i o n 1 0 . 3 ([LOl]) (1) T h e 4 - t u p l e m a p for a k n o t K, ipK, which is a m a p which assigns an oriented 2-sphere t h a t passes through /(£*) (i = 1,2,3,4) t o a 4-tuple (ti,t2,t3,t4:) e Conf 4 (5' 1 ) \Cc(K), is defined by ^

= r o ( / x / x / x

/ ) : C o n f 4 ( 5 1 ) \Cc{K)

3 {tut2MM)

/(*i)A/(*2)A/(*3)A/(*4)

-»• EA^S.

^(/(ti)A/(i2)A/(i3)A/(t4))

(2) T h e u n o r i e n t e d 4 - t u p l e m a p is defined by ipK = TY O t/jK, where n : A -> ¥A = A/ ± 1 is t h e projection. Proposition 10.2.1 ([LOl]) The set of concircular points of a knot, Cc{K), is of dimension greater than or equal to 2 if it is not an empty set, in particular if K is a non-trivial knot. Proof.

Put vf = !//(*!, h, h, tA) = f{h) A f(t2) A f(t3) A f{U).

Assume t h a t (t\,t2,£3,£4) S Cc(K). Then Proposition 9.4.1 implies t h a t , considered as vectors in M 5 , f{t\), f(t2), f{t3), a n d / ( £ 4 ) are linearly dependent, whereas it is easy t o see t h a t three of them, say, f(t\), f(t2), a n d f(t3) are linearly independent. Suppose

f(t4) = af(t1) +

bf(t2)+cf(t3).

Then

^=-a/(ti)A/(t2)A/(t3)A/'(t1) ^

^-=-bf{tl)^f{t2)^f{h)^fl{t2)

^=-c/(i d

-^~=

014

1

)A/(t

2

)A/(i

3

)A/'(i

3

)

f(h)Af(t2)Af(t3)Af'(t4).

Generalization

Since f{ti),f(t2),

of the 4-tuple map to the diagonal

179

and /(is) are linearly independent, S

P

a n ( ! ^ , | ^ \ \ oil

ot2

ot3

otA I

has dimension at most 2. Therefore the kernel of di/f, which is equal to the tangent space of Cc(K), has dimension at least 2. • 10.3

Generalization of the 4-tuple map to the diagonal

The set of concircular points and the 4-tuple map can be naturally extended to a subset of the diagonal A in the following way. Let us stratify the diagonal A C (S1)4 by putting A(2'2) = {g • (a, 3, i, i) e (S1)4 | s + i, g G 6 4 } , A( 4 ) = { ( S ) S , S ) S ) ) e ( 5 1 ) 4 } , A( 2 - 1 ' 1 )= {g-(S,s,t,u)e(S1)4\ A ^

s^t^u^s,ge&4}

= {g-(s,s,s,t)e(S1)4\s^t,geei}

, ,

where 64 is the symmetric group. We do not use the "full" compactification of the configuration space Conf4(S'1) as is given in [BT] or [FM], where one takes into account the way how points come together. Therefore, the 4-tuple map can only be extended to a map to FA = A/ ± 1, since even if there is a limit lim

ipK(s,s +As,t,t

+At)

e A,

As,At-y+0

for instance, it changes its orientation if we take the limit in the other way: lim

ipK(s,s-As,t,t

+ At) = -

As,At^+0

10.3.1

lim

ibK(s,s + As,t,t

+ At).

As,At-++0

Twice tangent

spheres

Let us consider the limit of ipK(ti,t2,t3,ti)

as i 2 approaches t\ + 0 and f f ( t ~i

i 4 approaches i 3 + 0. Put x = / ( i i ) , vx = ,,,,*{,, V = f(h),

and vy =

( 3 . Then, in general, 4>K(ti,t2,t3,t4) approaches a twice tangent l/'(*3j| sphere £K(X,V) of a knot K at x and y, which is an oriented 2-sphere that is tangent to the knot K both at x and at y (see Definitions 8.2 and

The space of non-trivial

180

spheres of a knot

9.10). It is uniquely determined unless there is a circle C which is tangent to K at both x and y. When the twice tangent sphere Ex(x,y) is uniquely determined, its orientation is given as the limit of the orientation of E(x, x + Ax, y,y + Ay) as Ax and Ay converge to 0, where we assume that the cyclic order of these four points satisfies x -< x + Ax -< y -< y + Ay. Namely, by Proposition 9.9.3: Proposition 10.3.1 The ordered basis (vy(x),vx) entation of the tangent space TxEx(x,y). Let S^2,2^(K)

gives the positive ori-

denote the set of the twice tangent spheres.

Let us give a formula to assign a twice tangent sphere to a pair of points on a knot. As we specify the way how points come together in this case, there is no ambiguity of the sign. Definition 10.4 knot K by

(1) Define the set of (2, 2)-concircular points of a

3

C{K' '(s,s,t,t) : a circle which is tangent to K both at f(s) and at /(£)

g-(s,s,t,t)€AW

2

Cc^ \K)

ge&4

(2) We call the above circle CK'

(s, s,t,t) the base circle of g-(s, s, t,t) 6

Cc^(K)(ge&4). As in the case of Cc(K), Cc^2\K)

=

'g-(s,a,t,t)e

A<2>2)

g£64

3

n : an affine 2-plane in Rf

n3f(s),f'(s),f(t)j'(t)

where f'(s) and /'(£) are considered as tangent vectors in Tf^Sf Tf^Sf respectively. Therefore t] E A(2 2) CcW (K) = l\ g - ( S ' S 'g *' ' € 64

Span ( / ( s ) / , ( s ) / ( f } f { t ) } ' ' ' is of dimension 3

and

1.J

Since the dimension of Span ( /'(s), /''(s), /(t), /'(t)) is greater than or equal

Generalization

of the J^-twple map to the diagonal

181

to 3, Proposition 9.4.1 implies that >(s,s,M)eA(2'2) 1

9&&4

/(*) A/'(*) A/(*) A/'(t) = 0 (10.2)

Put 77 = Span (f(s)J'(s),f(t), /'(*)>• Then the base circle of g • (s,s,t,t) is given by CK' '(s,s,t,t) = S3 H II, and hence it is the base circle of a pencil V = A n II1- of twice tangent spheres at f(s) and /(£). Taylor's expansion formula implies /(s)A/(S')A/(i)A/(*') = (s' - s)(t' - i) • /(s) A f'(s) A /(t) A /'(i) + higher order terms. Definition 10.5 ([LOl]) (1) Define the twice tangent sphere map for a knot K, 4>{K'2) : A<2'2) \ Cc^2\K) -*• A S 5, by yAL(/(s) A/'(s) A / ( i ) A / ' ( i ) ) ' f2 21

where g G 64. We denote the right hand side simply by ipK' (s,t). We remark that ipK' (s,t) = ipK' (t,s). (2) The unoriented twice tangent sphere map is given by ipK = | 2 21

^ ° WK'

1

wnere

Remark 10.3.2

"" '• A —> FA is the projection. (1) If / (s) A f'(s) A f(t) A /'(*) + 0 then lim

ipK(t1,t2,t3,t4)

=

ipK(s,s,t,t)

(titete,t4,)^rg-(s,s,t,t)

no matter how points come together. (2) Generically Cc^2'2^ (K) has dimension 0 since g • (s, s, t, t) belongs to 22 CS- ' \K) if and only if /'(£) is a multiple of vx{y), the unit tangent vector at x conformally transported to y (see Definition 3.3). We remark that CC(2>2\K) has dimension 1 when if is a (p, g)-torus knot on the Clifford 3 torus i n S c C 2 which is an orbit of an ^ - a c t i o n defined by 5-1

3 e 2«*

^

(C2

D 53 9

^;W) ^

(e2mptz, e2*iqtw) £ S3 C C 2 ) .

Problem 10.2 Is it true that if K is non-trivial then the twice tangent sphere map ipK' ' is not injective, namely, there is a sphere S that is tangent to K at more than or equal to three points?

The space of non-trivial

182

spheres of a knot

Let us see the behavior of the twice tangent sphere map near Cc^2'2^ (K) in the next example. Example 10.1 Let p and q be relatively prime odd numbers. Let r be a (p, q) torus knot on a Clifford torus T 1 C Sf : -v/2"

{

\

T(S) =

l

(

l

•

l

l

•

1

—=- cos ps, —==• smps, —==• cos qs, —=- smqs, 1

Vv^

Then (0,0,7T,7r) e expansion implies

V2

CC( 2 ' 2 )(T)

V2

V2

since r(0) A r'(0) A

T(TT)

r(0) A T ' ( 0 ) A T(TT + <) A T'(TT + t) = t- r(0) A r'(0)

(p2 - q2^

V2

A

T'(TT)

= 0. Taylor's

A T(TT) A T"(TT) +

(07q,07-p,0)

0(t 2 )

+ O(t2)

for small Itl. Therefore

^>(o,*

+

t)

' (o,^L=,o,-

for small |i|. Thus the twice tangent sphere ip)- ' (0,n + t) G A = S has a limit as t decreases to 0 only as an unoriented 2-sphere in FA. Example 10.2 The formula (3.7) implies that the conformal angle OK(X, y) satisfies cos 9K(X, y) = —(vx,vy)+2(vx,w)(vy,w), where w = —. Therein ~ y\ fore, the conformal angle 6T(s) — 0 T ( T ( O ) , T ( S ) ) of the (p,q) torus knot r above, which is equal to QT(r(t),r(t + s)) for any t, is given by 1 f 2 (p sin ps + q sin qs 2 cos 0T(s) = —x < — p cos ps — q cos qs £> + q { I — cos ps — cos gs The conformal angle 6T(s) of the (3,5) torus knot is illustrated in Figure 10.1. This example shows that a map 9K '• K x K \ A —> [0, ir] is continuous, but not necessarily of class C 1 . It behaves like an absolute value function.

Generalization

Fig. 10.1

10.3.2

Tangent

of the ^-tuple map to the diagonal

183

The conformal angle 0T(s) of the (3,5) torus knot.

spheres

Put ^CK' ' (s, s, t,u) : a circle g- (s,s,t,u) G A^2,1'1) tangent to K at f(s) ge&4 passing through f(t) and f(u Then g- (s,s,t,u) e A*2'1-1' g e&4

f(S)Af'(s)Af(t)Af(u)=0

We call the above circle C- K. ('2 , 1 ', 1 ) ,(s,s,t,u) the base circle of g • which is the base circle of the corresponding pencil of spheres. Define ~i>K = ^ , u ) : A'2-1-1) \ Cc^'l'l\K) -»• FA *£ PS by ^K{9-

(s,s,t,u))

= TT

< f(s)Af'(s)Af(t)Af(u) K^/L(f(s)Af(S)Af(t)Af(u))

where g G ©4 and n : A —> PAis the projection.

\ ) '

(s,s,t,u),

The space of non-trivial

184

spheres of a knot

Put Cc^

(K) =

I

9 • (s,s,s,i)

G A (3,1) The osculating circle of K at passes through f(t) 5GS4

f(s)

Y

Then

Cc^(K)

3 1 g- (s,s,s,t) G A* ' fir G 6 4

-{

/(a) A f'(s) A f"(t) A f{t)

"}

We call the osculating circle at /(a) which passes through /(£) the base circle of g • (a, s, s, t), which is the base circle of the corresponding pencil of spheres, and denote it by CK' (s,s,s,t). Define ^K = ^ l ) : A ^ 1 ) \ Cc^l\K) -> PA ^ PS by T

,

,

Q Q

,ss

(

f(s)Af(s)Af"(t)Af(t)

WL(f(s)Af(S)Af'(t)Af(t)) where g G 64 and 7r : yl —> PA is the projection. We remark that if f(s) A f'(s) A f"(s) A f(t) ^ 0 then lim

ipK(t1,t2,t3,t4)

=

ipK(s,s,s,u)

{tl,t2,t3,U)-+g-(8,S,8,u)

no matter how points come together. 10.3.3

Osculating

spheres

A twice tangent sphere degenerates to an osculating sphere as the limit when two tangent points approach each other as we saw in Proposition 8.3.1. Definition 10.6 of a knot K by Cc{i\K)

(1) Define the set of osculating concircular points

= j ( a , a , a , a ) GA<4>

The osculating circle of K at /(a) is tangent to K to the fourth order

(2) When (a, a, a, a) G Cc^(K) we call the osculating circle at f(s) the base circle of (a, a, a, a) and denote it by C(K\s, a, a, a).

Generalization

of the 4~tuple map to the diagonal

185

Taylor's expansion formula implies /(a) A /'(a) A f{t) A f\t)

= ±(t

- s)4f(s)

A /'(a) A f"(s) A f"(s)

+ higher order terms. Then Proposition 8.3.1 implies: Proposition 10.3.3 (1) The order of tangency of the osculating circle at /(a) is exactly 3 if and only if f(s) A /'(a) A / " ( s ) A f"'{s) £ 0. Therefore C z^{K)

= { ( a , s, a, s) e A ' 4 ' |/(a) A / ' ( a ) A / " ( a ) A / " ' ( * ) = 0 }

(2) When f(s) A f'(s) A f"(s) A /'"(s) ^ 0 it is space-like, and therefore the osculating sphere SK (f(s)) is uniquely determined by r(4)(/(fi))

_

f(s) A f(s) A f"(8)_A y/L{f(S)Af'(s)Af"(8)Af'"(s))

f'"(s)_

Proof. Put vf\s) = /(a) A /'(a) A /"(a) A /'"(a). (1) Suppose the osculating circle C is tangent to the knot at f(s) to the fourth order. Then, for some parametrization of C — C(t), f^(s) = CW(s) for i = 0,1,2,3. Since C'"(s) belongs to Span(C"(a), C"(a)), / ' " e Span(/'(s), /"(s)> which implies f(s) A f'(s) A /"(a) A /'"(a) = 0. Suppose vf\s) = 0. Then f (s), f (s), f" (s), and /'"(a) are linearly dependent as vectors in K5. Then Lemma B.l implies f'"(s) belongs to Span(/'(a),/"(a)). We may assume, after a Mobius transformation, that f(s) = (1,0,0,0,1), f'(s) = (0, a, 0,0,0) for some a ^ 0, and /"(a) belongs to xy-plane. Since (f(t)J(t)) = 1, we have ( / , / " ) + ( / ' , / ' } = 0 and ( / , / ' " ) + 3 = 0. Therefore /"(a) and /'"(a) is given by /"(a) = (-a 2 ,5,0,0,0) and f'"(s) = (-3ab,c, 0,0,0) for some 6 and c. Then if we put g(t) = a(t - s) + | ( t - a) 2 + ^ ^ - ( t - *) 3 then C{t) = (cos(g(t)),sm(g(t)),0,0,1) satisfies C « ( s ) = / ^ ( a ) for i = 0,1,2,3, and hence the osculating circle C is tangent to the knot at /(a) to the fourth order. (2) Lemma 9.1.1 (3) shows that v\ (a) is not a time-like vector. If it is a light-like vector then 9.1.1 (2) implies that v\ (a) = kf(s)

for some

The space of non-trivial

186

k € M. Since L(f{t),f(t))

spheres of a knot

= 0 for any t e S1 4

L(f(t),f"(t))

= - L ( / ' ( t ) , /'(*)) = £ ( / / W ) 2 7^ 0

for any

t E S1.

i=i

On the other hand as (i/f, f"(s)) tion.

= 0, we have k — 0, which is a contradicD

Definition 10.7 ([LOl]) (1) Define the osculating sphere map for a knot K, ^ : A<4' \ Cc^{K) ->• yl = S, by

^>(,, ,,,,,) = /(«)A/'(')A/»Ar(,) _ ^ VL(/( S )A/'( S )A/"( S )A/'"( S )) (2) The unoriented osculating sphere map is given by ipK = woipK. We remark that if f(s) A f'(s) A f"(s) A f"(s) lim

^ 0 then

^^(^1,^2,^3,^4) = tpK(s,s,s,s)

£ PA

no matter how points come together.

10.4

Lower semi-continuity of the radii of non-trivial spheres

Definition 10.8

Put

~Cc{K) = Cc(K) U CcW (K) U Cc^1^

(K) U C^ 3 - 1 ' (K) U Cc^ (K)

and call it the set of generalized concircular points and define the extended 4-tuple map, %\>K : (S1)4 \ Cc(K) —> FA, in the obvious way. Proposition 10.4.1 Let K = f{S1) be a knot inM.3 and fe£r(erft;(£i, £2, £3, £4)) be the radius of the smallest sphere that passes through f(t\), f(t2),f{tz) and ffa), where the points are counted with multiplicity (see Definition 8.5). Then the map (5 1 ) 4 3 {t\,t2,t$, £4) ^ r(&K(^1,^2,^3, ^4)) G R is lower semi-continuous. Proof. Fix a stereographic projection S3 ->• K3 U {00}. Let U D (S1)4 \ Cc(K) be the maximal subset of the closure of (S1)4 \ Cc(K) which makes the following diagram commutative for some continuous function ipK:

Lower semi-continuity

of the radii of non-trivial

U

4 (S^XCciK)^

spheres

187

-**-,TA

]>¥A

where t is the inclusion. Suppose t = {ti,t2,t$,ti) $ U and t' = (t^t^t^t^) is close to t. If t' ElA then ipK (t) G A is in some neighborhood of the pencil V whose base circle is the base circle of t, C/f(t). Therefore r(crK{t')) > r(Cx(t)) — e = r{(7K(t)) — e for some small e > 0. If t' 0 U then the pencil V whose base circle is the base circle of t' is close to the pencil V'. Since both of the pencils are compact, r(ax(t')) > r( 0. D

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Chapter 11

The infinitesimal cross ratio

In this chapter we introduce an infinitesimal interpretation of the 2-form — in the integrand of E^ in terms of the cross ratio from a conformal \x - yY

geometric viewpoint. We show that we can pull back a complex valued 2form on K x K \ A using twice tangent spheres.

11.1

The infinitesimal cross ratio of the complex plane

Let us start by giving a complex valued 2-form on C x C \ A = R2 x R 2 \ A. Suppose w,w + Aw,z, and z + Az are four complex numbers with \Aw\, \Az\
. (w + Aw) — w z—w + Az) — (w + Aw) - {z + Az) ' z - (z + Az)

AwAz (w - z)2 '

Definition 11.1 The infinitesimal cross ratio of the complex plane is a complex valued 2-form UJCT on C x C \ A given by dw A dz (w — z)7 Lemma 11.1.1 ([LOl]) Both the real part 5Rewcr and the imaginary part 9m wcr of the infinitesimal cross ratio of the complex plane are exact as real 2-forms on R2 x R2 \ A. 189

190

Proof.

The infinitesimal

cross ratio

Let w = u + iv kind 2: = x + iy (u,v,x,y

€ 'R).

Then

2 2 { ( « - - z ) - (i> — y) }(du Adx —dv A dy) 2 2 2 {(« , — x) + (v — J/) }

$teu)cr

2(u ; — a:)(i>— j/) (du A dy + dv A dx) {(u- - x)2 + (v — y)2}2 If we put ^n

dw Rew—z

'•=

(u — x)du + (v -y)dv 2 (u — x) + (v -y) - 2

i

then Keojc r = dAsR. As for the imaginary part, o ( « — x)(v - y)(du Adx — dv A dy) 2 2 2 { ( « - - x) + (v — y) }

9 m ujcr -

,{(« - z )

2

-

(v — y)2}(du

A dy + dv A, dx)

{(u — x)2 + (v — y)2}2 If we put dw w—z

(u — x)dv — (v — y)du (u — xy + (v — yy

then 3mw c r = d\%.

D

Since the infinitesimal cross ratio cucr is kept unchanged under the replacements w i->- —, z H-> —, it can be considered as a 2-form on C x C \ A, or a meromorphic 2-form on C x C, where C denotes the Riemann sphere CU{oo}. As we mentioned before, any Mobius transformation of C can be expressed aszH> ——- if it is orientation preserving, and z i-> ^ — - if it is 7^ + 5

72 -t- 5

orientation reversing. Since the replacement a) 4 i , z 4 I maps ujcr to its complex conjugate, we have: Lemma 11.1.2 ([LOl]) (1) The infinitesimal cross ratio uCT is invariant under the diagonal action TxT:CxC3(w,z)^

(Tw, Tz) e C x C

of an orientation preserving Mobius transformation

T.

(2) It is mapped to its complex conjugate by the diagonal action of an orientation reversing Mobius transformation: (T x T)*uiCT = cucr.

The real part as the canonical symplectic form of T* S 2

191

Corollary 11.1.3 The real part $ltuCT (or the imaginary part 3mw c r ) of the infinitesimal cross ratio is invariant (or invariant up to sign, respectively) under any diagonal action of Mobius transformations onR2 x R 2 \ A . We give an alternative proof using real coordinates. Let I be the inversion in the circle with radius 1 and center the origin. Then (IxI)*\?R

= XSi +

jdlog(\u\2),

(I x /)*AQ = — Acj — darctan I — ), VuJ which implies (/ x iy^teto^ = 5Rewcr and (I x J)*Srati; cr = 3mw c r . We will see in the next section that the real part of the infinitesimal cross ratio 5Rewcr can be interpreted as the canonical symplectic form of the cotangent bundle of the 2-sphere (Proposition 11.2.8). Then Lemma 11.2.6 and Theorem 11.2.7 show that 5Rewcr is exact and is invariant under the diagonal action of a Mobius transformation. This infinitesimal cross ratio wa has been used in various fields such as algebraic geometry, symplectic topology, conformal geometry, and physics.

11.2

The real part as the canonical symplectic form of T*S2

In this section we study the canonical symplectic form U>T*sn of the cotangent bundle T*Sn of an n-sphere, which is diffeomorphic to the configuration space Sn x Sn\A. Then we show that the real part of the infinitesimal cross ratio of the complex plane can be interpreted as the pull-back of the canonical symplectic form UT,s2 of T*S2 = S2 x S2 \ A through any stereographic projection S 2 = CU {oo}. Definition 11.2 Let M be an n-dimensional manifold and T*M be its cotangent bundle. Let (<7i,--- ,qn) be a local coordinate system of M and (pi, • • • ,pn) be a local coordinate system of fibers of T*M associated with the basis {dqi,--- ,dqn}, namely, the coordinates of a point (x,v) G T*M are equal to (qi,--- ,9n,Pi)"' - >Pn) ii v = ^pidqi. Then (gi, • • • , qn,Pi, • • • ,pn) serves as a local coordinate system of T*M. The canonical symplectic form Uo of the cotangent bundle T*M is a

192

The infinitesimal

cross

ratio

globally defined non-vanishing 2-form which is locally expressed by u

=

°

^2d1i

hdpi.

D e f i n i t i o n 1 1 . 3 Let T(T*M) be a tangent bundle of T*M. Let (x, v) E T*M. T h e projection n : T*M -> M induces dn : T(x,v)T*M -> TXM. Define a 1-form 9 of T*M by (6(x, v))(w)

= v(dn(w))

£ R

we

T{x,v)T*M,

This 1-form 8 is called the tautological form of

(x, v) e T*XM.

T*M.

T h e n one can verify t h a t 6 can be locally expressed by 8 = Y^Pi^liTherefore the canonical symplectic form U0 of T*M is exact with U>o = —dO. Since 9 is defined without using local coordinates, UJ0 is independent of the choice of the local coordinates of M, which implies the invariance of the canonical symplectic forms under diffeomorphisms between manifolds: L e m m a 11.2.1 Let f* : T*M -)> T*M be a diffeomorphism induced by a diffeomorphism f : M —)• M. Then the canonical symplectic forms correspond via f*, namely, (f*)*(u)o) = COo, where Uo and uJo are the canonical symplectic forms of M and M respectively. Proof. We give an alternative proof using local coordinates to show t h a t the fundamental 1-forms of T*M and T*M correspond via / * . Let (§i, • • • , qn) be a local coordinate system of M, and (qi, • ,q„,Pi,--,Pn) be the corresponding local coordinate system of T* M. W h e n f(q1,- • • ,qn) = (q1} • • • ,qn) we put % = ^ ( g 1 ; •• ,qn) a n d % = liiQi,---

A

i J

Then

,9qi

f \\ f \ \

/ : \ V

'Sqf

,Qn)- P u t A

8 dqt

f*dqi

(\\ = *A

f:\

\ {r)*Pi

A

P

V: J

\ I J

Since (/*)*% = qi(qi,---

(

dqi

,qn) (

:

\

(f*ydq.,

J

\ d(

which implies {f*)*{Y.Pi li)

=

HPidqi-

(

') d%

\

'•

)

•

The real part as the canonical symplectic form of T* S2

Lemma 11.2.2

193

([LOl]) There is a canonical bijection i / ) : 5 n x S n \ A - > T*Sn.

Proof. Let LTX be the n-dimensional hyperplane in Rn+1 passing through the origin that is perpendicular to a; 6 5™, and px • Sn\ {x} —>• IIX be the stereographic projection. We identify IIX with TxSn. Take an orthonormal bas {v\, • • • , vn} of nx = TxSn. Suppose y € Sn\ {x} is expressed as V — 2/l«l H

1" VnVn + Vn+lX of M n + 1 . Then

with respect to the orthonormal basis {vi, • • • ,vn,x} Px{y) = •:

; Vn+l

Vi

1-

Vn+l

Put

^(y) = r J ^ — «!* + ••• + - ^ «

n

* e T X

(n.i)

1 - 2/n+l 1 ~ Vn+l where {t>;*} is the dual basis of T*Sn. Then the map 4>x : Sn\{x} -> T^S"1 does not depend on the choice of the orthonormal basis {vi} of Hx = TxSn. Thus the map V : 5 " x Sn \ A 9 (x, y) - • (x, Mv))

e T*S™

makes the canonical bijection. Lemma 11.2.3

•

([LOl]) Let (xlf • • • ,xn ) be a local coordinate system of

Ut+i = {(xu- • • > xn+l)

eSnc

M n + 1 | xn+1 > 0}.

It determines the associated local coordinate system o/7r _ 1 ([/^ + 1 ) C T*Sn as in Definition 11.2. The canonical bijection ij; : Sn x Sn \ A ->• T*Sn is expressed with respect to this local coordinates as ( 4>{x,y) = [xir--,xn, \

VX-V-^X1 Xn+1

1

1-x-y

yn-*mXn\

,-•-,

,

Xn+l

1-x-y

€T*Sn

J

on U^+1 x Sn\A , where xy denotes the standard Euclidean inner product. Namely, the map tpx : Sn\ {x} —> T*Sn in the proof of the previous lemma

The infinitesimal

194

cross ratio

is given by „

yi-1b±LXi

^(y) = E

i

/L

—' i=l

Xn+1

dx

>-

1 —x • y a

The exactly same formula applies to

Un+1 = {(xi, • • • ,xn+1)

eSnc

R n + 1 | x n + 1 < 0}.

Proof. Let {v\,- • • ,vn} be an orthonormal basis of TIX =TxSn, {ei, • • • , en+i} be the standard orthonormal basis of M n + 1 . Put

and

A = {vx-'-Vn) £ M ( n + l , n ) , A = (t>i •••vnx) e 0(n+l), and

B

d dx\

( ' 0

d dxT{

\

0 G M(n+l,n)

X\ Xn + 1

Xn Xn+1

/

where vectors in Rn+1 are considered as column vectors. Let us calculate the matrix for the change of basis of T*Sn from {v*} to {dxi}. Since / « i \

ei

= 'A

\xj

>> e ™+i,

we have ei

,en+i.

/«i\

w

The real part as the canonical symplectic form of T* S

We have

fv1\

ei

0X1

t

B\

| = lBA

:

d

ven+l

\ dx„ I

\ x )

As

l

BA-

i —^-1

0

x

\

n+l

Xn+l,

)

o\ l

BA : > 0/

/ — \

Vi

9xi

l

BA

d \ dxn /

and so = \Vn

fBA)

/

\dxn/

Suppose y e Sn \ {x} is expressed as y = v\V\ -\

h ynvn + i/n+ix.

Then Vi

\

(

2/i

= A v2/n+l,

KVTI+I,

which implies (m •••Vn)

=

(yi---yn+i)A.

195

196

The infinitesimal

As yn+1 = x-y,by

cross ratio

(11.1) 4>x(y) is given by

V'x(y)

l - x y

(m

• • • yn)

' dx\ * t

l - x y

{y1---yn+1)A\ BA)

Since E = AlA = (Ax) [ ; ) =AtA

A XBA) = AtAB

= B-

+ xlx,

xCxB) = B.

Therefore ( ipx(y)

l - x y

{yi---y„+i)

0^

1

0

' dxi

1

\ - —

VJ n

a

n

n

Remark 11.2.4 The canonical bijection ip : S x S \A-+ T*S is not consistent with Mobius transformations. Namely, suppose g G 0(n+l, 1) acts on Sn as a Mobius transformation, g : Sn —>• Sn. Then we have the diagonal action gxg:

SnxSn\

A3 (x,y)^(g-x,g-y)GSnxSn\

and a pull-back map g* : T*Sn

—} T*Sn. But the diagram

Snx5n\A

i>

gxg[ n

A

n

1 kr -^T*S n

S xS \A

i>

is not commutative in general: ip o (g x g) ^ (g*)

J

o^

The real part as the canonical symplectic form of T* S2

197

This is because the zero vector in T£Sn corresponds to a pair of antipodal points (x, —x) in Sn x Sn\ A, and the choice of antipodal points is not preserved by Mobius transformations. Lemma 11.2.5 ([LOl]) The pull-back ui = tp*UJo of the canonical symplectic form U0 ofT*Sn by i/> : Sn x Sn \ A -> T*Sn is given by

1 —x • y

I

\

1—x • y I

_ YIi=l dxi A dVi , (Y17=l Vidxi) A CLn=l 1 —x •y (1 — x • y)2

Proof.

Xjdyj)

As U0 = —dJ^QidPi x

y% f—' i=i

% -dxj

1— X • 1 - »

on t/TJ"+1 x Sn \ A by Lemma 11.2.3 (2). Since — Y^=i xid%i = on Sn, it implies the formula.

xn+idxn+i •

Lemma 11.2.6 ([LOl]) Let p : Sn \ {(0, • • • , 0,1)} -> IT be the stereographic projection. Put Pn = p-1

x

j T 1 : R" x W1 \ A -> 5 " x 5 n \ A.

TTien £/ie pull-back coKn = Pn*uj = Pn*tp*UJo is given by

2

Proof.

l*-y|2

/

£ da;,—77^A dy,

- 2 -

^

|a;-y|

Let Pn(x,y)

2

= (X,Y)

V

l^-yl2

( E f e - 2/i)^») A (H(xj - 2/j)%.?\ |a; - y| 4 be expressed as

P n (x 1 ,--- , x n ; yi,--- ,y„) = (A"i,--- , X „ + i ; Fi,--- , Y n + i ) . Then

* {H^iXidvA

T,? (xi-yi)dyi

= =1 = p^ i Tl-X-Y V T ) I = ^ . T\x-y\,,r

+ diogdyp +1).

The infinitesimal

198

cross ratio

Therefore Pn*u = Pn*d

2d

l-X-Y

H?=i(xj 2

yi)dVi\

\x~y\

J'

a T h e o r e m 11.2.7 ([LOl]) The 2-form UR" is invariant under any diagonal action of the Mobius group M. on W1 x W1 \ A defined by g- (x,y) =

{g-x,g-y),

where g E M and (x, y ) £ l " x K " \ A. Proof. It is obvious that Wi» is invariant under the diagonal action of multiplication by a scalar (x,y) — i >• (cx,cy), or of addition of a vector (a;, y) i-» (x + v, y + v). The previous Lemma implies that Win is invariant under the diagonal action of the orthogonal group 0(n). Therefore it suffices to show the invariance under the diagonal action I x I o£ the inversion / in the (n — l)-sphere with radius 1 whose center is the origin. Let (/ x I)(x,y) = (X,Y) be expressed as ( / x / ) ( > ! , • • • ,xn;y1,---,yn)

= (Xlt---

,Xn+1;

Ylr--

,Yn+1).

Then

Therefore T\*nj(YJ{xi-Yi)dYi\ xl) 2d ^ l x _ y ] 2

(T-*T\*<,

- I T . ,

(I x / ) u^

-(I

=

0j{T,(xj-yi)dyi\

) =M^

{x_y?

j

Ws".

We give another proof of the invariance. Let Si be the n-sphere in Mn+1 with center 0 and radius 1, and S/^ be the n-sphere in E n + 1 with center (0, • • • ,0,1) and radius \[2. Let 1^ denote the inversions of M n + 1 U {oo} in an n-sphere S in M.n+1. Then as p — Isr1

P-

(I(X))

= / ^ ( / £ i ( a . ) ) =It

{El)

(lE^{x))=I^{p-\x))

The real part as the canonical symplectic form of T* S2

199

for x £ W1. (This equality can be verified using elementary geometry.) Therefore the following diagram l " x l " \ A —^—>• Sn x Sn \ A Ixl R n x M™ \ A

I«n X 7»n n

> S x Sn \ A P 1

n

commutes. Lemma 11.2.3 implies that the following diagram Sn xSn\A -—> T*Sn I I]Rn X 7Rr,

Sn x Sn \ A

I ( i l R n * ) - 1 = IRn *

y T*Sn

v commutes. As Lemma 11.2.1 implies that (IVLn*)*u)0 = W0 the following commutative diagram l n x R n \ A — ^ — > Sn x Sn \ A — - — > T*Sn I X l[ J/K" X IRn j/R„* n n l"xK"\A >S xS \ A > T*Sn Pn rp gives the conclusion. • Proposition 11.2.8 (Folklore) The real part of the infinitesimal cross ratio ujcr is equal to minus one half of the pull-back of the canonical symplectic form of the cotangent bundle T*S2: „ Proof.

/ dw Adz\

1

1_„,„

The left hand side is equal to

{QEI ~ Vi)2 - (x2 - V2)2}(dx1 A dyi - dx2 A dy2) {{xi -yi)2 + (x2 - J / 2 ) 2 } 2

+2

(xi - yi)(x2 - y2){dxx A dy2 + dx2 A dyi) {(xi - 2 / i ) 2 + {x2 - J/2)2}2

dx\ A dyi + dx2 A dy2 {x\ -y\)2 + {x2 - j/2) 2 p{(^i ~ Vi)dxi + (3:2 - y2)dx2} A {(xi - yi)cfc/i + (x2 2

{(xi-yi)

which is equal to the right hand side.

+

y2)dy2)

2 2

(x2-y2) }

•

200

The infinitesimal

cross ratio

T h u s t h e real p a r t of t h e infinitesimal cross ratio has a n a t u r a l generalization t o higher dimensions, whereas t h e imaginary part does not, because of the following Proposition which t h e author was informed of by a personal communication. T h e o r e m 11.2.9 (Y. Kanda) Let u be a 2-form satisfies the following two conditions: (i) u> is invariant

on Sn x Sn \ A that

under the diagonal action of the Mobius

group.

(ii) LO is invariant up to a multiplication by ± 1 under the action involution (x, y) \-> (y, x) of Sn x Sn\A.

of the

Then LO is uniquely determined up to a multiplication by a constant if n > 3. Therefore u> must be a multiple of the pull-back of the canonical symplectic n form of T* S . When n = 2 there are essentially two such ui's, which correspond to the real and the imaginary parts of the infinitesimal cross ratio. Proof. Since Sn x Sn \ A is homogeneous space of t h e Mobius group, it suffices t o consider u> restricted t o a point X = (x,y) and t h e action of its isotropy group Hx- Let us take a pair of antipodal points X = (x, —x). Then

T(x,-x)(Sn

xSn\A)

= TxSn 8

T_xSn.

T h e first factor and t h e second factor of t h e right hand side can be identified through t h e antipodal m a p , and so, we can p u t

T{x,-x){Sn

xSn\A)

= V®V,

V^W1.

T h e induced action of t h e isotropy group H^x _ x j on t h e tangent space is given by R+ x SO{n), where R+ 3 c acts by c-{u,v)

= {cu,c~1v)

(11.2)

and SO(n) acts by t h e diagonal action g-(u,v)

= (gu,gv).

(11.3)

T h e proof is then reduced t o linear algebra as follows. Let A be an antisymmetric bilinear form of R n x R n t h a t satisfies t h e following two conditions. (i) A is invariant under t h e action of K+ x SO{n) given by (11.2) and (11.3).

The infinitesimal

cross ratio for a knot

201

(ii) A is invariant up to a multiplication by ± 1 by the involution (x,y) >-> (y,x) of W1 x R " . Then it can be easily checked that A is expressed as =tuGq-tpGv

A({u,v),(p,q)) for some n-matrix G that satisfies l

BGB = G

V

B e SO(n).

If n > 3 then G is a constant times the identity. When n = 2, then

also satisfies the condition, which corresponds to the imaginary part of the infinitesimal cross ratio. •

11.3

The infinitesimal cross ratio for a knot

As we mentioned at the beginning of the previous chapter, we use the following notation: K = f{Sl),

x = / ( s ) , x+Ax = f(s+As),

y = f(t), zndy+Ay

=

f(t+At).

We assume that the derivative / ' never vanishes and that the knot K = f{Sl) is oriented via / . (The assumption of orientation of a knot K is not necessary. The reverse orientation will give the same results.) As is stated in Section 9.4, we can define the cross ratio (x + Ax, y; x, y + Ay) of the four ordered points x,x + Ax,y,y + Ay using the oriented 2sphere S(x, x + Ax, y,y + Ay) that passes through them which is uniquely unless the four points are concircular. Let II be the plane which contains C(x, x + Ax, y) with an identification with the complex plane C, and let p : S(x, x + Ax,y,y

+ Ay) -> 77 U {oo} = C U {oo}

be the orientation preserving stereographic projection. (When f(s), f(t), and /'(<) are concircular, we can take 77 as ZJ(x,x + Ax,y, y+Ay)

f'(s), and

The infinitesimal

202

Fig. 11.1

U(x,x

+ Ax,y,y

cross ratio

+ Ay) and its orientation.

the identity map of H as p.) Let x = p{x), x+ Ax= p(x + Ax), y = p(y), and y+ Ay= p(y + Ay) denote the corresponding complex numbers. S(x,x

+ Ax, y,y + Ay)

..... - - - ..

K

C(x, x + Ax, y) Fig. 11.2 The twice tangent sphere and the infinitesimal cross ratio. In the case of the above figure, the stereographic projection from the south pole gives an orientation preserving map from S(x,x + Ax,y,y + Ay) to C.

Then the cross ratio (x + Ax, y;x,y + Ay) is given by {x + Ax) — x {x + Ax)-{y

+ Ay)'

y—x y - (y + Ay)

Ax Ay (x - y)2 '

Let us take the limit as \Ax\ and \Ay\ goes to 0. Then the sphere E(x,x + Ax,y,y + Ay) becomes the twice tangent sphere Sx(x,y), and

The infinitesimal

cross ratio for a knot

203

the circle C(x, x + Ax, y) becomes the tangent circle C(x, x, y). Let vx{y) be the unit tangent vector of C(x,x,y) at y (see Definition 3.3). As is stated in Remark 3.4.1, The angle between vx(y) and f'(t) is equal to the conformal angle 0K(x,y) (0 < 9pc(x,y) < TT), which is the angle between two tangent circles C(x,x,y) and C(y,y,x) (see Definition 3.2). Our convention of the orientation of the twice tangent sphere SK (x, y) given in Section 9.9 implies that the angle from vx(y) to /'(£) with respect to this orientation is non-negative. As x,x + Ax and y are in ZV(x,2/) D 77, they are invariant by the stereographic projection p. Since | Ay | = \Ay\ as illustrated in Figure 11.3 we have:

south pole Fig. 11.3 The circle as the intersection of E(x,x + Ax,y,y + Ay) and the 2-plane containing the center of E(x,x + Ax,y,y + Ay), its south pole, and y + Ay.

Theorem 11.3.1 ([LOl]) The cross ratio (x+Ax, y; x, y+Ay) has the absolute value -———|i and the argument 0K{x,y) modulo higher order terms \x-y\2

of\Ax\\Ay\. Definition 11.4 (Langevin [LOl]) The infinitesimal cross ratio QCR = Qcn(x,y) of a knot K at (x,y) is the cross ratio of four points (x + dx, y;x,y + dy). It is a complex-valued 2-form on K x K \ A given by nCR(x,y)

=

ei8«^

dxdy

(11.4)

The infinitesimal

204

cross ratio

Thus the conformal angle 8K can be interpreted as the argument of the infinitesimal cross ratio. A stereographic projection from a sphere is a restriction of a Mobius transformation of R3 U {00}. Hence the infinitesimal cross ratio (or its complex conjugate class) is the unique invariant of dx and dy under the action of the orientation preserving Mobius group M+ (or respectively, the action of the Mobius group A4). Since the absolute value of the 2-form x" v\ on K x K \ A is the \x

absolute value of flcR and the angle 0K(x,y) to the argument of (2CR, the cosine formula

-yy

(0 <

6K(X,V)

<

TT)

is equal

implies the infinitesimal cross ratio formula

Ei2)(K)=[[ J

{\nCR\-mnCR}.

(11.5)

JKxK\A

This formula applies to closed and open knots in K3 and in S3. Since the infinitesimal cross ratio QCR is invariant modulo complex (2) conjugate under Mobius transformations, El ' is Mobius invariant. As a corollary of Proposition 9.7.9 we have: Corollary 11.3.2 ([LOl]) The integrand of E(?\K) can be expressed in terms of the infinitesimal maximal (Lorentzian) modulus as \&CR\ - $ttftcR = mL(x,x

+ dx,y,y + dy) - 1 m(x,x + dx,y,y + dy) : sinh 2 2

Proof.

When the cross ratio cr satisfies \cr\ <§C 1 then (9.7) implies rriL ~ 1 + (\cr\ — 5Re cr) + higher order terms of cr.

We close this section with the following claim which we will use later. Lemma 11.3.3 ([LOl]) The conformal angle of a closed knot K in R3 is identically 0 if and only if K is a circle. Proof. Fix a point x G K and consider the inverted open knot at x, Kx. Since 6x(x,y) = 0 for any y 6 K, the tangent vector of Kx at y is equal

From the cosine formula to the original definition of E^2'

205

to — f'(s) for any y G Kx. Therefore Kx is the straight line, which means that K is a circle. • 11.4

From the cosine formula to the original definition of

In Section 11.2 we showed that the real part of the infinitesimal cross ratio is equal to minus one half of the pull-back of the canonical symplectic form of T* S2. As a corollary, we show that the infinitesimal cross ratio formula (11.5) implies the original definition (3.5). Now let us reformulate the infinitesimal cross ratio of a knot K. Let

(x0,y0) =

(f(s0),f(t0))eKxK\A.

Let SK(xQ,yo) be the oriented twice tangent sphere given in Definitions 8.2 and 10.5. Let T0 be a Mobius transformation of E 3 U {oo} that maps ^K(xo,yo) to K2 x {0} C M3 preserving the orientations. Consider — (w — zY as a complex valued 2-form on M 3 x l 3 \ A = {({w,x3),(z,y3))

e ( C x i ) x ( C x R ) \ A} .

Then, as a 2-form, the infinitesimal cross ratio £2CR satisfies ^CR(XO,VO)

= (T0 x T0)* I -

— )

(x0,yo).

P r o p o s i t i o n 11.4.1 ([LOl]) Let K = /(S1) be a knot. Then the real part of the infinitesimal cross ratio QCR defined in Definition 11.4 is equal to minus one half of the pull-back of the canonical symplectic form of the cotangent bundle T*S3: dxdy (J I (III

_

pj — UtiicR COS0K~\x — — y \ A = %tlfiCR

where

LKXK\A

_

1I

.

— R3\KxK\A = --Z-LU -~^^V.AKXK\\

I

:^xif\A-}I

_

1I

L — = —— Z -T^( KxK\A)*P3*1p3*Uo,

x K 3 \ A i s the inclusion map.

Proof. We show that their pull-backs (/x/)*Ke QCR and coincide at any given (so,
x f)*uR3

= - y ((To o / ) x (To o

f))*uR3.

-\(fxf)*UJR3

206

The infinitesimal

cross ratio

On the other hand, since {(T0ofydx3)(s0,t0)=

(J-poofh}

(so),

((To o /)*dj/ 3 )(so, *o) = ( ^ ( T o o / ) 3 ) (to) we have 1

2{(To°f)x(T0of)yuR3(s0,t0)

( ( T 0 o / ) x ( T 0 o / ) ) * ( - ^ = i ^ 2A ^ \x-y\ [Y?i=l(xi - Vi) A dxi)

+2

~

=

A

( E L l ( ^ - Vi) A % ) \

)

\hw

M

((T0of)x{T0of)y(^l^Adyt \

\x — y\

x

( E L i ( * ~ Vi) Adxi) A ( E i = i ( x ; - 2/i) A eh/*) \ +2^ r ^ ^ ^ (so, to) \x-y\4 I = (/ x /)*(T 0 x T0ym

(?™*i2)

( s °^°) = (/

x

/)*^«cfl(so,*o). D

Let ATg (A; 5 1 ) be the complement of the -—-tubular neighborhood of v2

the diagonal A of S1 x S1:

iV£c(A; S 1 ) = {(s, t) e [0,1] x [0,1] | e < \s - t\ < 1 - e} for 0 < £
xS1\t-s

= ±e (mod Z)} .

Eo

207

-energy for links

Then 7isixsi\A

= limn / /

\\x-y\2

~ U x fT^cR

(ur^Jim

On the other hand, as

by Proposition 11.4.1 and Lemma 11.2.6, we have

by Stokes' theorem. As (f(s)-f(t),f'(t))

= (s-t)

+

0(\s-t\3),

we have

^ e +O^)^

(

^

£

+ 0(£3)^

dt + J0 e^+0(e^ J7!1 e2 + 0 ( e 4 e + 0(e ) 2

2

4

+ 0(e).

£

Therefore we restored the original definition (3.5):

E?\K) = 11.5

Vn.(EW{K)-iy

n(2)

EQ -energy for links

Let us generalize Eo = E^ — 4 for links. If we try to apply the formula (2.15) to a pair of different components of a link L, the counter term is not well-defined as the arc-length d^x, y)

The infinitesimal

208

cross ratio

is not defined. But in t h a t case we do not need any counter terms as the integral of the main t e r m does not blow up. \x ~y\2

In [FHW], Freedman, He, and Wang defined the total energy of a link L = Kx U • • • U Kn by

TE[Fliw](L) = J2E(Ki) 4 E i

where E(Ki,

E

^ - ^

(1L6)

i=£j

Kj) for different components is given by

Jx€Kt

JyeKj

\x

V\

Namely, the counter t e r m is removed for different components. It was suggested in [KS1'] t h a t it would be more n a t u r a l to count the cross t e r m E(Ki,Kj) twice as much as in (11.6). D e f i n i t i o n 11.5 L is defined by

([KS1']) T h e c o n f o r m a l (1 — cos 6) e n e r g y for a link

/ / .LxL

l-cos0L(x,y) r—"^ dxdy. x \

(11.7)

V\

We show t h a t our cross ratio formula (11.5) naturally generalizes t o links by the same formula. This formulation is free from the ambiguity of a constant due to the choice of a counter term which is necessary in the case of knots. It also gives a n a t u r a l interpretation of counter terms. T h e o r e m 11.5.1 ([L02]) (1) Let L = Kx U • • • U Kn be an n-component link. Then the conformal (1 — cos9) energy of L can be expressed as Ei2){L)=

/ / (\QCR\-^QCR) LxL

where f2cn{x,y) is the infinitesimal way as in the case of knots.

cross ratio which is defined in the same

(2) The integral of the real part of the infinitesimal cross terms of different components vanishes:

KiXKj

(lU

cross-ratio

in the

Ea

-energy for links

209

Therefore the conformal (1 — cos 6)-energy of links satisfies dxdy _^y y ^ i \JxeKiJy t n , •JyeKj i+j

i

Proof. that

X

\

V\

(2) In case of a knot Lemmas 11.2.6 and Proposition 11.4.1 imply

^.znCR{x,y) for x ^ y, where and

LKXK\A

= - — (LKxK\A)*{2dr))

: ^ x i ( \ A - > R 3 x IR3 \ A is the inclusion map,

=

T,l=1(xi -yi)dyj \x-y\2

is a 1-form on R3 x M3 \ A. In case of a 2-component link Ki U Kj (i ^ j), the argument is parallel to show that Ke flCR = -{1-KiX.KjYidri), where iKtxKj • K-i x Kj —> M3 x M3 \ A is the inclusion map. Then Stokes' theorem implies

[[

KznCR = -ff

J JKiXKj

d((LKiXKjyr,)

J

JKiXKj

J

Jd^KiXKj)

= ~

(iK.xK.YV

Let us study the cross terms. Definition 11.6 Put

([L02]) Let L = Kx U • • • U Kn be a link in K3.

Ei2)(Ki,KJ)= [

f

(\nCR\-mnCR)=

[

JxGKi Jy€Kj

f

Jx£Kt

\nCR\.

Jy&Kj

The mutual conformal (1 — cos0) energy of L is given by

E^-\L)

= YiE^\Ki,Ki) = W

/

^

-

(11.9)

The infinitesimal

210

cross ratio

Theorem 11.5.2 ([FHW]) The conformal (1 — cos (9)-energy of a link L = K\ l)K2 bounds the average crossing number (see Definition A.5) from above:

Therefore it bounds the Unking number from above: ^2)(K1,^2)>47r|Lic(X1,^2)|. Proof.

This is a consequence of Theorems 12.5.1 and 12.5.2.

•

Example 11.1 Let us consider a family of Hopf links obtained as pairs of fibers of the Hopf fibration 7r : S3 —> S2, which is given by 7 r : C 2 D S 3 9 (21,22) >-)• {2^ez1z2,2^mz1z2,\z1\2

-\z2\2)

£ S2 C K3

as it is the composition of a map C \ {0} 2 D S 3 3 (21,22) >-»• — e C P 1 ^ C U {00} 22

and a stereographic projection ^

.

,

,

25Rew

29mw

|w| 2 - 1 ,

n..

Let Pj = (0, 0,1) and P 2 = (2r\/l - r 2 , 0, 2r 2 - l ) , and L r = Kx U AT2 (0 < r < 1) be a Hopf link with Ky = T r " 1 ^ ) and K2 = ir-l{P2). Then K\ and AT2 can be parametrized as f(s) = (e",0) e A"i, and

5 (t)

= ( r e " , ^ 1 - r 2 e") e X 2 .

Then |s(i)-/(s)|2 = 2-2rcos(£-s).

EQ

AsEi2){K1)

= Ei2)(K2)

-energy for links

211

= 0,

1 „r2U, x „(2\.mut,r , f v f -E(2){Lv r) = E{2)>mut{Lr) = f" f 2 ' Jo Jo Jo Jo :4TT

/ '0

2+

dsdt 2cos(t-s)

^-— 1 — r cos 2£

(£ = tan 77) 2

2TT'

4

*L (T^TRl + r)r]

2

A/1

- r2

4^2 IP1-P2I Therefore we get Proposition 11.5.3 ([KS1']) The mutual (1 — cos9) energy of a Hopf link which is obtained as a pair of fibers of the Hopf fibration over a pair of points Pi and Pi in S2 is equal to An2 times the ordinary electrostatic energy of a pair of unit charges on Pi and P2 in S2 C K 3 . We conjecture that the infimum of Eo {K\,K2) of non-splittable links K\U K2 is attained by the extreme Hopf link obtained as a pair of fibers of the Hopf fibration over a pair of antipodal points on S2, when EQ {K\, K2) = 2ir2. This is a weaker conjecture. See Theorem 12.5.1 and Conjecture 12.2.

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Chapter 12

T h e conformal sin energy -EUnfl

In the previous section we introduced the infinitesimal cross ratio ficR, which is the unique Mobius invariant of a pair of infinitesimal curve segments dx and dy, and showed that E^ can be expressed in terms of the absolute value and the real part of flcR- In this chapter we can use the imaginary part of QCR-

12.1

The projection of the inverted open knot

Definition 12.1 ([KS1'; Lin]) The conformal sin© energy of a knot K, Esino(K), is defined by ff

sin 6

J JKXK\A

K(x,y)dxdy

\x-y\z

Then it can be expressed in terms of the infinitesimal cross ratio as

Esin6(K) = f f J

%mnCR, JKxK\A

hence we also call it the imaginary cross ratio energy ([LOl]) . Proposition 12.1.1 Let vx denote the unit tangent vector to K at x, and v~x(y) be its image by the conformal transportation from x to y (see 213

The conformal sin energy Esln $

214

Definition 3.4). Since

= ^vx(y)

6K{X,V)

Esin9(K)

=

• vy and \vx(y)\ =

\vx(y) x

=,

vy\dxdy,

J JKxK

where x denotes the vector product. This functional Esing is well-defined since the conformal angle OK is of the order of \x — y\2 near the diagonal by the following Lemma. Lemma 12.1.2 9x(x,y) satisfies

([LOl]) If a knot is of class C 4 , the conformal angle

eK{x,y)=

V

fcT +fc 6

\x-y\*

+

0{\x-y\%

near the diagonal, where k is the curvature, r is the torsion of the knot at x, and k' is the differential of the curvature k with respect to the arc-length parametrization. Thus if we put 9K(X,X) = 0 we get a continuous, but not necessarily of class C1 map 6K • K x K -^ [0, n] (see Example 10.2). Proof. Assume that the and let x = f(s) and y = Then (3.7) implies that ported to y (see Definition

knot K = /(S"1) is parametrized by arc-length, f(t). the unit tangent vector at x conformally trans3.3) is given by

I/W-/WI7 1/(0-/Ml Put s = 0. If we take the Frenet frame at x then Bouquet's formula implies that f(t) is expressed as

(f

/(*)

\ k ~2~

k kr o

3

0(t4).

The geometric meaning of Es{n g

215

Since /'(O) = (0, 0,1), vx(y) is given by . (,, __ (/i 2 W ^ h2{t) x[y) ~

h\t),Vx{t)h{t),2h{t)h{t)) \f(tW

(t2 - 7-^t4 + 0{t5),kt3 + ~tA + 0{t5), ^-t4 + 0{t5))

t2{i-~t'2

+ o(t*))

On the other hand + o(i 3 ), %t2 + o(* 3 ))

( i _ #t* + 0(t*),kt + U fi(T\

f {

_V

2

2

' ~

2

J_

1 + 0(*3)

Hence sin^Kr = !/'(*) x ^(j/)! = t 2 ( l - ^ + 0(*3))(l + 0«3)) \/jfc2T2 + /c'2

2

3

t2 + 0 ( i 3 ) .

• 12.2

The geometric meaning of Es-lng

Let us give a geometric interpretation of Eslag using the inverted open knots. Put \r lis \ Vsmg{K;x)=

f/ JK

s

™9Kd—. y \x-y\2

Let Ix, Kx — IX(K), and y = Ix(y) be as in Section 3.4. Let TIX be the normal plane to K at x, n = —f'(s) be the normal vector to IIX, and TTX : M3 —> TIX be the orthogonal projection. Since Ix*{vx(y)) = n the conformal angle 9x(x,y) n and the tangent vector to Kx at y. Hence Vs-mg(K;x)=

/

^ir

= /

JK \x-yr

is the length of the projection irx(Kx) C iX

JKX

sin 6 K\dy\

216

The conformal sin energy

Eslng

"=-/'(«) Fig. 12.1

The projection of the inverted open knot.

Thus the imaginary cross ratio energy is equal to the total length of the projection of the inverted open knots:

Esin9{K)=

f

Length(irx(h(K)))dx.

JK

12.3

Self-repulsiveness of Es\a g

We show that Es-mg is self-repulsive with respect to the C2-topology.

Self-repulsiveness

Theorem 12.3.1

([LOl]) Fix 5 (0 < S < \)

217

and K (K > 2TT). Put

(1) The length of if is 1. (2) The curvature of K is not greater than K. K : aknot (3) ^x, y € K such that dK{x,y) = 5 and |x — y\ < d,

JtrK(d)=JtrK(5,d)

for0
of Es-ln g

Then lim I inf Esine(K) d-n-o \KeJtrK(d)

1 = oo. )

In particular, ES[ng is self-repulsive with respect to the C2-topology. Lemma 12.3.2

Let 2\

5

do = min <

J_ 1

•l + x / l + 5 ^ >.

100' 5 '

Let d < do. Suppose K = / ( 5 1 ) E J^K{d) is parametrized by arc-length and satisfies that |/(0) — f(S)\ = d. If 5d < t < yd then at least one of Vsin6(K; f(±t)),Vsine(Kis qreater than — • 100

f(S

±t))

. (J + f

We show that the Lemma implies the Theorem. Proof of Theorem 12.3.1. Since lOd < Vd < \

f([-Vd,-5d})uf([5d,Vd])uf([6

+ 5d,5 +

Vd})\Jf([S-Vd,5-5d})

is a disjoint union of curve segments of K where Sl is regarded as R modulo Z. Since VSin$(K; y) > 0 for any y G K7 rVd

Esin$(K)>

/

{Vsing(K; f(-t))

+

Vsin6(K-J(t))

J5d

+Vsine(K; Vd 1

>

L

dt

f(5 - t)) + Vsin6(K; f(S + t))} dt 1

mJTl 'm\^(1

+

-\-

7s-ks(il

The conformal sin energy Esin g

218

which blows up as d decreases to 0, which completes the proof of Theorem 12.3.1. Proof of Lemma that

12.3.2. Assume there is a t with 5d < t < \fd such

Vsin9(K; f(±t)), Vsine(K- f(5 ± *)) > - L . _ 1 _ . Put x± = / ( ± t ) . Let Rt = l/\f"(t)\ be the radius of curvature at x+, and C(x+,x+,x+), Kx+ = IX+(K), IIX+, and nx+ be as before. Define x'+ by C(x+,x+,x+) Let x+ = Ix+(x'+).

CMIX+

={x+,x'+}.

Then

i

i

400 ' d+t

Fig. 12.2 The osculating circle, x+,x'.,

and x.\-.

Since Vsing(K;x+)

= Length(irx+(KX+))

< — • -^-j

irx+ (Kx+) lies inside the circle on IIx+ with center x+ and radius l/(200(
Self-repulsiveness

of Es[n g

219

we have . 1 +

, +l

1 1 2Rt ~ 2

1 1 1 - 2 200 d+Vd

1 ~ 400

1 d+t

Therefore TTX+(KX+) lies inside the circle F on IIX+ with center x+ and radius 3/(4O0(d + t)). This means that Kx+ lies inside the cylinder D2 x R, where dD2 = r and the direction of K. is / ' (t). If we apply the inversion

Ix+ again, this implies that K lies outside the "degenerate open solid torus" (by which we mean the interior of a torus of revolution of a circle around a tangent line) Nt whose meridian disc has radius (400(d+i))/3, as illustrated in Figure 12.3.

Fig. 12.3 The circle F and the degenerate solid torus.

The same argument applies to —t to conclude that K lies outside the "degenerate open solid torus" AT_t. Since

| / ( t ) - / ( - * ) l < 2 i « ^ p ( d + <) /([—£,£]) i s contained in a bounded component of M3 \ (Nt U N-t), which we denote by D, as illustrated in Figure 12.4. Sublemma 12.3.3

f(6) is contained in D.

We show that the Sublemma implies Lemma 12.3.2. If f(8) is contained in D, then the knot K must have double points at x+ and x_, which is a contradiction, which completes the proof of Lemma 12.3.2. •

The conformal sin energy Esln g

Fig. 12.4

The domain D illustrated as a 2-dimensional figure.

Proof of Sublemma 1 2 . 3 . 3 . Since | / ( 0 ) — f(5)\ = d, which is much smaller t h a n the radius ( 4 0 0 ( d + t ) ) / 3 of the meridian discs of the degenerate solid tori Nt and iV_ t , it suffices to show t h a t |/(0)-/(±i)|>d to prove Sublemma 12.3.3. First we remark t h a t

t < Vd <

< 200K

For 0 < s < t

^(/'(0),/'(*)) = (/'(0),/"(*))<«, hence (f'(0),f'(s))
K

> d.

1/(0) - g{t)\ > J (l - ns)ds = t (l which completes the proof of Sublemma 12.3.3.

^t)

D

^sin 9

an

d the average crossing

number

221

We conjecture that Theorem 12.3.1 holds without any assumption of the boundedness of the curvature, namely: Conjecture 12.1

12.4

We conjecture that Esing is self-repulsive.

Es-mg and the average crossing number

We show that Es[ng can detect the unknot. Example 12.1 Let KQ be a circle. Then 6^o(x,y) Therefore Esing(K0) = 0.

— 0 for any

We remark that Lemma 11.3.3 implies that Esing(K) K = KB. Proposition A.5.1 implies:

(x,y).

= 0 if and only if

Theorem 12.4.1 ([Lin]) The imaginary cross ratio energy bounds the average crossing number (see Definition A.5) from above as Esin6(K)>4ir(c)(K) for any knot K. Corollary 12.4.2

12.5

If K is a non-trivial knot then Esing(K)

> 12ir.

-Esin0 for links

Definition 12.2 Let L = Kx U • • • U Kn be a link. (1) The conformal s i n # energy, or the imaginary cross ratio energy, of L is defined by TP

,T\

If

o n

ff

sin 6

L(x,y)dxdy

(2) Put EfABg(Ki,Kj)=

[

[

xeKi JyeKj

SmnCR=

f JxeKi

sm.9r,(x,y)dxdy \x-y\2 JyeKj f

The conformal sin energy Es{n g

222

The mutual conformal sin 9 energy, or the mutual imaginary cross ratio energy, of L is defined by

<j

sin 0 j_l{x,y)dxdy eK, \x-y\2

i<3

Since

,K )=f

E?\KUK2)= 2

I

>[

fI

\nCR

[

SmnCR

=

Esine(K1,K2)

yeK2

Theorem 12.5.1 ([L02]) The mutual conformal (1 — cos9)-energy for links bounds the mutual conformal sin 6-energy from above: Ei2)(Ki,K2)

>

E^eiKuK,).

The argument in the proof of Proposition A.5.1 works for links because we did not assume that both x and y belong to the same connected component. Therefore we get: Theorem 12.5.2 ([Lin]) The imaginary cross ratio energy of a link L = K\ U K2 bounds the average crossing number from above: ^ine(^l,^2)>47T(c)(^1,^2).

Therefore it bounds the linking number from above: Esine(KuK2)>4Tr\Lk(KuK2)\. Since the infimum of the mutual imaginary cross ratio energy for a 2-component link is 0, this proposition implies that E™£Q can detect the trivial link. The estimate of the above Proposition 12.5.2 does not seem to be sharp. Definition 12.3 A link L = K1 U • • • U Kn in R3 is said to be splittable if there are B\,-Bn C M3 which are mutually disjoint and which are diffeomorphic to 3-balls such that Bi D Ki (1 < i < n). Conjecture 12.2 We conjecture that the infimum of Esine{K\, K2) of non-splittable links K\ U K2 is attained by the extreme Hopf link obtained as a pair of fibers of the Hopf fibration over a pair of antipodal points on S 2 , whenEsin0(KuK2) = 2ir2.

Chapter 13

Measure of non-trivial spheres

In this chapter we study the volume of the set S(K) spheres of a knot K. 13.1

of the non-trivial

Non-trivial spheres, tangent spheres and twice tangent spheres

Generically a sphere intersects a knot K in 0, 2 or an even number of points. The set of spheres that intersect any given knot is unbounded in A as it contains arbitrarily "small" spheres, and hence its volume is infinite as we saw in Lemma 9.8.3. In order to have a finite valued functional we should get rid of the contribution of the set of "small spheres" which intersect K in at most 2 points. Let us consider the space S{K) of non-trivial spheres that intersect K in at least four points (see Definition 10.1). Lemma 10.1.2 implies that S(K) is a non-empty compact 4-dimensional subspace of A if K is not a circle. It has therefore a finite non-zero volume. If K is a circle then S{K) is a pencil with base circle K which is a 1-dimensional compact subspace of A and hence its volume is 0. Let Vol (S(K)) for a knot K.

denote the volume of the set S(K) of non-trivial spheres

The boundary dS(K) has two types of "corners", one of which is the set of twice tangent spheres S^2'2^ (K) as was explained in Chapter 10. Definition 13.1 ([LOl]) Let Atwt(K) denote the area of the set of the twice tangent spheres of K. 223

S{2'2\K)

224

Measure of non-trivial

spheres

It is invariant under Mobius transformations. L e m m a 13.1.1 ([LOl]) (1) The set of the twice tangent spheres forms a space-like surface in A. Namely, the restriction of the Lorentz form to the tangent space at any regular point of this surface is positive definite. Therefore its area Atwt(K) is positive. (2) The area Atwt(K)

is not

self-repulsive.

Proof. (1) Let u(s,t) = f(s)Af'(s)Af(t)Af'(t) and ^(s,t) = ^'2\s,t) be the twice tangent sphere m a p given by xjj = v • (is, v)~^ G A. Then

where tps = — and vs = — . A s i / S = f(s) A f"(s) A f(t) A f'(t) b o t h v and vs are space-like unless vs = 0. Therefore (ips,ips) > 0 unless vs = kv for some k £ M. (2) Let Kn (n > 3) be a knot with length 1 lying on a fixed sphere SQ which has a pair of points G Kn with 8K„{pn,qn) = 1/2 and \pn — Qn\ = 1/n. Then S^2,2\Kn) = {So} and hence Atwt(Kn) does not blow up as n goes to oo. • R e m a r k 13.1.2 Unlike E^ or EB[ng, Atwt(K) cannot be expressed as an integral over K2 of a simple function of the infinitesimal cross ratio QCR which does not contain its derivatives. This is because the local contribution of a neighborhood of a pair of points on K to Atwt(K) depends on the curvature, whereas flcn does not.

P r o b l e m 13.1

Can Atwt(K)

be expressed in terms of the infinitesimal

cross-ratio and its derivatives — —— —— ) etc. ? 9s \\f(s) - /(t)j 2 13.2

T h e v o l u m e of t h e s e t of t h e n o n - t r i v i a l s p h e r e s

Unlike E1^ or Esin0 the measure Yo\(S(K)) of the set of the non-trivial spheres cannot be expressed as a double integral over K x K of a function of the infinitesimal cross ratio since it does not take into account the number of intersection points of K with spheres. In order to get such an integral formula, we have to count each sphere E with multiplicity, which is the

The volume of the set of the non-trivial spheres

225

number of pairs of points in S fl K where the algebraic intersection number has the same sign. Definition 13.2 ([LOl]) (1) Langevin's measure of spheres of a knot K, EmntB(K), is defined by Emnts{K)

= / JS(K) IS(K)

„C nK(S) = „ U22dS, U i , where wneie nu = = nxyzj)

non-trivial

B(rnif)

where the number of the intersection points §(£ fl K) is counted with multiplicity, and dS denotes the ^-invariant measure of A given in section 9.8. (2) Another related functional is the 4-tuple measure of non-trivial spheres of a knot K which is defined as the measure of the image of the 4-tuple map ipK of the knot K with multiplicity:

E£L(K)=

I JConf4(Sl)\Cc{K)

WK*UJA\=

[

2nC4dS,

JS(K)

where U>A is C?+-invariant volume 4-form of A. This definition of -Ejnnts i s based on a suggestion by J. Cantarella to the author when he gave a talk on Emnts(K). We study Emnts(K) in what follows since it can be expressed as a double integral over K x K of a function of the infinitesimal cross ratio as will be shown in Theorem 13.3.1, whereas E^J^^K) has no such expression. Example 13.1

Let KQ be a circle in S3. Then Vol (5 (if 0 )) = Emats(K0)

since S(K0)

= E{2ts(K0)

=0

is an empty set.

Proposition 13.2.1 ([LOl]) These three junctionals, Vo\{S(K)), and .Emnts are Mobius invariant.

Emnts,

Proof. Let g G Q be a Mobius transformation. Then g maps a nontrivial sphere E of K to a non-trivial sphere g • £ of g • K keeping the number of the intersection points. Therefore S(g • K) = g(S(K)) and ng.f((g-£) = riK(S)- Since the measure d£ is ^-invariant, i.e. g*dS = dS, we have Emnts(g • K) = Emnts(K) etc. • As we mentioned before, the measure Vol (S(K)) of the set S(K) of the non-trivial spheres is finite for any smooth knot K since S(K) is bounded in A. On the other hand, it is not easy to control the number of the

Measure of non-trivial

226

spheres

intersection points of K and a sphere E. We will show later at the end of the next section that the measure of non-trivial spheres is finite for a smooth knot using formula 13.1. Problem 13.2

Is -E"mnts finite for any smooth knot?

We conjecture that for any smooth knot K there exists a natural number n such that the measure of the set of spheres that intersect K in at least n points is 0, and hence the answer to the above question is affirmative. Remark 13.2.2 These three functional, Vol(S(K)), are different in general: Vol(S(K))

< Emnts(K)

<

Emnts,

and -B^nts

E^nts(K).

In fact, if an oriented 2-sphere E has exactly 6 transversal intersection points with K, then E is counted 1, 3 = 3C2, and 15 = QC^ times respectively in these functionals.

13.3

The measure of non-trivial spheres in terms of the infinitesimal cross ratio

Definition 13.3 Assume that a knot K is oriented. Let p € K (~l E be a transversal intersection point. We say that E intersects K at p positively if the orientation of K is "outward" at p with respect to the orientation of E, namely, if the algebraic intersection number K • E is equal to +1 at p. We call p a positive intersection point. Then the measure of non-trivial spheres satisfies -Emnts (K) = / n'C2W, JS(K)

where n' is the number of positive intersection points of K and E. We remark that the above definition does not depend on the choice of the orientation of a knot K. Let S+(K;x,x';y,y') denote the set of cross-separating spheres E — 3 dB of x,x',y, and y' such that B3 contains x and y. Generically these spheres intersect xx' and yy' positively, where xx' denotes the arc of K

The measure of non-trivial spheres in terms of the infinitesimal cross ratio

227

joining x and x' with the natural orientation derived from that of K. Put \ntK\x>y)l

Vol {S+ {K;x,x + Ax;y,y

1 =, \Ax\,\A i™\^o ,

+ Ay))

\Ax\\Ay\

y

Then the measure of non-trivial spheres of a knot K is expressed as Emnts{K)=

'Y\Vo\{S+{K\Xi,xi+l;Xj,Xj+]))

lim max \xi — Xi-tx (—>0 —

\nt(K]{x,y)\dxdy.

-^ 1

J

Proposition 13.3.1 pressed as Emnts(K)

JKXK\A (K\A

([LOl]) The measure of non-trivial spheres is ex-

= ^- f f 4

(smeK - 9K coSeK)dxdy

(13.1)

\x ~ V\

J J KxK

where OK = 0K{x,y) is the conformal angle. The proof decomposes into the following two lemmas. Lemma 13.3.2 ([LOl]) Let C+(K;x,x';y,y') denote the set of cross2 separating circles C — dD ofx, x', y, and y' on the sphere S = S(x, x', y, y') such that D2 C £ contains x and y. Then we have the dimension reduction formula: Vol (S+(K-x,x';y,y'))

= ~Vo\

(C+(K; x, x'; y, y')) .

Proof. Since Vol (S+(K; x, x'\ y, y')) is conformally invariant, we may assume that x,x',y, and y' are in R2 = {(X,Y,0)} C ffi3 by a stereographic projection (S3, S(x,x',y,y')) —¥ (M3 U {oo},E 2 U {oo}). Suppose a sphere (and a circle) is parametrized by the pair of the coordinates of its center and its radius. Let the space of the parameters of S+(K;x,x';y,y') and C+(K; x, x'\ y, y') be denoted by CTR(4) (x, x', y, y') and CTR<-3) (x, x', y, y') respectively. Namely : CTR^(x,x',y,yi) v y

yj

= \(X,Y,r) eR3++ " C G W ^ J , ^ with j \x ' ' center (X, Y) and radius r J

CTR^(x,x',y,y') v w

y ;

= {(X,Y,Z,r)eRX \v ' ' '

= {(X,Y,Z,Vr2

+ Z2) \(X,Y,r)

^ ^ ( ^ x' ; 2 / y ') with j + center (X, Y, Z) and radius r J e CTR{3\x,x',y,y'),

Z e t } .

Measure of non-trivial

228

spheres

Then \ol(S+(K;x,x';y,y'))

= f

4 CTR(4Hx.x'.y.v') >(x,x',y,y') JcTR(

f

r

U

(r2 +

\J-oo

CTR(3)(x,x',y,y')

2

.dZ

(T—

JcTR^(x,x',y,y')

•K

^dXdYdZdr

dXdYdr

Z2)2

3dXdYdr

r1

Vol(C+(K;x,x';y,y')).

'

•

Lemma 13.3.3 ([LOl]) Let z be the cross ratio {x1 ,y;x,y') and 9 be its argument. If \z\
= (sin<9 — 6>cos6>)|^| + o(|2rj).

Proof of Theorem 13.3.1. We show that the above two Lemmas imply the Proposition. By Lemma 13.3.2 Vol (S+ {K; x,x + Ax; y,y + Ay)) = |-Vol (C+(K; x,x + Ax; y,y + Ay)). Since the cross ratio (x + Ax, y;x,y + Ay) has the absolute value ,

If

\x-y\2

and the argument 6K, Lemma 13.3.3 implies Vol {C+{K; x,x + Ax; y,y + Ay)) , • „ „ „ JArM/tyl (\Ax\\Ay\\ = (sm 9K - 9K cos 9K) L - l L j I l + o(^J) . Therefore l (4)/ ^ | _ sin 8K - 9K cos 9K \ntK{x,y)\^_^2 which completes the proof of Theorem 13.3.1.

•

Proof of Lemma 13.3.3. We assume that the argument 9 satisfies 0 < 9 < -7T. A similar calculation works when -|- < 9 < n. Let z = a + ib (a, b > 0). We may assume that x = (0,0), x' = (a,b), y = (1,0), and y' = oo,

The measure of non-trivial

spheres in terms of the infinitesimal

cross ratio

229

by a suitable orientation preserving stereographic projection from S S(x,x',y,y') to R2 U {ex.}. Then CTR{3)(x,x';y,y') is given by

l

CTR^(x,x']y,y )=\(X,Y,r)G

J>3

X2 + Y2 < r2 (X-l)2+Y2 r2

Let M(r0) be the intersection of CTR}3'(x,x';y,y') {r = ro}. Then by Lemma 9.8.2 Vol (C+(K;x,x';y,y'))

and the level surface

f°° 1 = / -^ Area(JV(r))dr.

Let Ci(r) = dDi(r) denote the circles defined by C\(r) = {{X,Y)\X2 + Y2=r2}, 2 C 2 (r) = {{X,Y)\(X-1) + Y2=r2}, 2 C3(r) = {(X, Y)\(Xa) + (Y - b)2 = r 2 } . Then M(r) = IntDi(r) n IntD 2 (r) n (K2 \ D3(r)). Since A/"(r) = 0 when 9 — 0 we assume 9 > 0 and hence b > 0 in what follows. We remark that II — z\ M(r) is non-empty if and only if r sin 9 > . Let Pij(r) (i ^ j) be one of the two intersection points of Ci{r) f~l Cj(r) with smaller y-coordinate as is shown in Figure 13.1. Then they are vertices of a curved triangle Af(r). Let {£ij(r), rjij(r)) be the coordinate of Pij{r) with respect to the frame (ei,e'2) where e\ = (cos 9K, sin OR) and e 2 = (—sin 9K, cos 9K)Let Qi2(r) be the intersection point of a half line starting from P\2(f) in the e[ direction and C3. Let V(r) C M2 be a domain swept out by the subarc of C$(r) between Q\2{r) and P2z{r) by a parallel transportation in the —e[ direction by \z\, M$r) be a curved triangle Qi2{T)Pi2{r)P2zir), and M2(r) be a subset bounded by Ci(r) and the half line starting from Pi3(r) in the —e\ direction as in Figure 13.2. Then M(r) = V(r) U M2(r) U Mi(r). Direct calculation shows that 1

/

r3

Area (M^r)) dr = O (\z\2) = o(\z$

(i = 1,2).

(13.2)

Measure of non-trivial

230

Fig. 13.1

Ci(r), Pij(r),

spheres

and M(r).

Since AreaP(r) = |«|(r/i 2 (r) - rji3(r)), we have

Vol(C+(K;x,x';y,y'))

=

Vu{r)

-Vi3{r)

dr + o(\z\).

As

rii2(r) - r?i3(r) = \ / r 2 - —

— - cosO^r2

- —

Non-trivial

annuli and the modulus of a knot

231

C,

Pair)

Ci

-c2

Pi2(r) Mi(r) Fig. 13.2

Af(r) and M i ( r ) .

we have 7712(r) " W O

sin#

1 2 sin t

r

a

dr

\

2

cos 9

dr + o(|z|)

sin 6

1 — usin# — cos^v 1 — u 2 J du + o(\z\) Jo

1 2r

= sin 0 — 0cos9 + o(\z\), which completes the proof of Lemma 13.3.3.

D

Remark 13.3.4 (1) Lemma 12.1.2 implies that the integrand (sin 9K — 2 9 K COS9K)I\% — y\ of the measure of non-trivial spheres is 0 at the diagonal. Therefore, the measure of non-trivial spheres is finite for any knot of class C4. (2) There are no apparent inequalities among three numerators of the integrands, 1 — cos0, sm9, and sin$ — #cos nints-

(3) Lemma 11.3.3 implies that the measure of non-trivial spheres of K, and hence E^l^K), is equal to 0 if and only if K is a circle. 13.4

Non-trivial annuli and the modulus of a knot

Suppose a non-trivial sphere £ for a knot K intersects K transversally in at least four points. Thickening £ we get an annulus which is crossed by at least four subarcs of K.

232

Measure of non-trivial

spheres

Definition 13.4 (Langevin [LOl]) Let K be a knot and A be an annulus with dA = Si US2. (1) A strand (of K in A) is a subarc of K that is contained in the annulus A and whose end points are on dA. A non-crossing strand is a strand whose two end points lie on the same boundary sphere of A. (2) A cross-strand is a strand of K with one end point contained in Si and another in S2. A minimal cross-strand is a cross-strand whose intersection with the interior of A is connected. (3) An annulus A is called a non-trivial annulus for a knot K if there are at least four cross-strands with disjoint interiors. Remark 13.4.1 (1) An annulus A is a non-trivial annulus for a knot K if and only if there are four points xi,x2,x%, and X4 on K in this cyclic order such that A is a cross-separating annulus for them. (2) An annulus A is a non-trivial annulus for K if and only if each connected component of R 3 \ IntA (or S 3 \ IntA) contains at least two connected components of K \ [K PI IntA). (3) Any 2-sphere in a non-trivial annulus for a knot K is a non-trivial sphere (in the strict sense) for K, but a non-trivial sphere for K is not necessarily contained in a maximal non-trivial annulus for K. Lemma 13.4.2 If K is not a circle then there is a non-trivial annulus, and hence a non-trivial sphere (in the strict sense) for a knot K. Proof. Suppose there are no non-trivial annuli for a knot K. Take any three points xi, x2, and X3 on K in this cyclic order. Let £4 be a point in the subarc of K between X3 and xi which does not contain x2. Then Theorem 9.7.3 (1) implies that X4 must lie on a subarc of the circle C(xi,X2,xz) between x% and xi which does not contain x2- By taking 23,^4, and Xi the same proposition then implies that the subarc of K between xi and £3 which does not contain X4 is contained in the subarc of C(x3,X4,xi) = C(xi,X2,xs) between x\ and £3 which does not contain X4. Therefore K = C(xi,x2,x3). O Definition 13.5 (Langevin [LOl]) (1) The modulus of a knot K, is the supremum of the modulus of a non-trivial annulus for K.

m(K),

(2) A non-trivial annulus for a knot K is called maximal if it attains the modulus of the knot K.

Non-trivial

annuli and the modulus of a knot

233

non-crossing strand

Fig. 13.3 Non-trivial annulus.

Proposition 13.4.3 (Langevin [LOl]) There exists a maximal non-trivial annulus for any knot K which is not a circle. Proof. Let O(K) > 0 be the infimum of the radii of the non-trivial spheres (see Definition 8.6). Suppose A is a non-trivial annulus with dA = S\ US2. Then each boundary sphere Si is a non-trivial sphere for K, and hence its radius is not smaller than O(K). Therefore the set of the pairs of boundary spheres of the non-trivial annuli for K is a non-empty compact subset of A x A. Hence the supremum of the absolute values of the Lorentzian moduli is attained. • Let us now look for properties of maximal non-trivial annuli. Theorem 13.4.4 (Langevin [LOl]) Let A be a maximal non-trivial annulus for K with dA = SiUS^. Applying a suitable Mobius transformation, we may assume that Si and S2 are concentric spheres in K 3 . (1) The knot K is contained in A. (2) K C] Si (or K f] S2) consists of two points where K is tangent to Si (or S2, respectively). (3) The two tangent points of K n Si (or K n S2) are antipodal. Proof. (1) Suppose K is not contained in A. Let X2 be a point of K which is not contained in A. Then we can choose three other points, xi,X3, and X4, of K such that the cyclic order of xi,X2,xa, and X\ coincides with the

234

Measure of non-trivial

spheres

orientation of K, and that A is a cross-separating annulus for xi,X2,x^, and X4. Since any cross-separating annulus for 2:1,2:2, £3, and X4 is a nontrivial annulus for K7 A must be the maximal cross-separating annulus for Xi,X2,a;3, and X4. Then Lemma 9.7.2 implies that each Xi should belong to one of the boundary spheres of A, which is a contradiction.

Fig. 13.4 If a maximal non-trivial annulus A for K does not contain K then A can be inflated to produce another non-trivial annulus A' for K with a larger modulus.

(2) and (3). The statement (1) implies that the intersection K U 5, consists of tangent points. Suppose x\,X2,x$,X4 are four points on K in this cyclic order which satisfies x\,x$ £ S\ and 2:2,3:4 £ S%- Then Theorem 9.7.3 (4) implies that x\ and X3 are antipodal in Si and Xi and X4 are antipodal in 52. Suppose there is another tangent point x5 £ K U 5i (2:5 ^ £1,0:3). We may assume without loss of generality that £5 is contained in an arc in K joining X2 and X4 which contains X\. Then x5,£2,0:3,2:4 are four points on K in this cyclic order and A is a cross-separating annulus for them. Then A must be a maximal cross-separating annulus for 2:5,2:2,2:3, X4 and hence 2:5 and X3 must be antipodal in 5i by Theorem 9.7.3, which is a contradiction. Therefore there are only two tangent points in each boundary sphere of

A.

•

Self-repulsiveness

13.5

spheres

235

S e l f - r e p u l s i v e n e s s o f t h e m e a s u r e of n o n - t r i v i a l s p h e r e s

T h e o r e m 13.5.1 repulsive. Proof.

of the measure of non-trivial

([LOl]) The measure

Fix 8 (0 < 8 < \).

of non-trivial

spheres

is self-

For 0 < d < 8 put

3

j(r(d) p(d) =

K : a knot with length 1{K) inf K€^(d)

x, y E K such t h a t ' i) dK(x,y) = 5l(K), ii)\x-y\
{Emnts(K)}

We remark t h a t p is a monotonic decreasing function of d because d\ < d
C Jtifid-i), and t h a t p(d) = 0 for d >

because J(f{d) IT

contains a circle. Assume t h a t there is a positive constant M' such t h a t supp(d)

lim

p(d)

M'.

d>0

There is a positive constant M = M(M') such t h a t if the measure of nontrivial spheres of K is smaller t h a n 1.1M' then the ratio of the radii of the two spheres of any concentric non-trivial annulus for K is smaller t h a n M. Let Kd be a knot with length 1 in Jff(d) whose measure of non-trivial spheres is smaller t h a n 1.1M'. Let i , j / b e a pair of points on Kd such t h a t the shorter arc-length between t h e m is 8 and t h a t do = \x — y\ < d. (We use the axiom of choice here.) T h e knot K is divided into two arcs, 71 and 72, by x and y. P u t •y

a = m m •; sup

y

, sup w WS72

Let Sr denote the 2-sphere with center forms a non-trivial annulus, a < '

-

and radius r. Since ( E a , Ed^ j and hence a < — .

2

}

-

2

Let Kd be the _

part of Kd which is contained in SMA . The length I {Kd) of Kd is greater t h a n or equal to 8 since EMA contains at least one of the arcs 71 and 72. T h e n by Poincare's formula ([San] page 259, see also pages 111, 277) we

236

Measure of non-trivial

spheres

have f $(g(2r) D Kd)dg = °

3

JG

^

l Q

° • Area(£ r ) • l{Kd),

V2U\

where 0 „ denotes the volume of the n-dimensional unit sphere, G denotes the group of the orientation preserving motions of M3, which is isomorphic to the semidirect product of R 3 and 5*0(3), and dg denotes the kinematic density ([San] page 256). We remark that On =

, in particular, O 0 = 2, Oi = 2TT, 0 2 = 4?r, 0 3 = 2ir , • • • .

+1

We have dg = dP A dK[P], where P e l 3 , dP is the volume element of M3, and dK^p] is the kinematic density of the group of special rotations around P, which is isomorphic to 50(3). Since

/ •

dK[P] = 0 2 0 i

Poincare's formula implies /

%(£r(X, Y, Z) n Kd) dXdYdZ

= 27rr2 (Kd) ,

JEr(X,Y,Z)nKd^ip

where Sr(X,Y,Z)

denotes the sphere with radius r and center

Suppose —
Then the measure of non-trivial spheres of Kd

= f 0 Js{Kd)

^

-M4

(X,Y,Z).

+ r

l

^

™

^

(x+y\ 2 )

•\

r4

dXdYdZdr

2

I )

where D\{c) denotes the 3-ball with center c and radius R. Thus Md

Emnts(Kd)>~J^

r 2

3 \

l-hnr l(Kd)-jn(^+rj

\dr

Md

> i 2**

1

jTT(Md)3

1

The measure of non-trivial

spheres for non-trivial

knots

237

which blows up as d decreases to 0, which contradicts the assumption that limd^+0 p(d) < M < oo. •

13.6

The measure of non-trivial spheres for non-trivial knots

Theorem 13.6.1 (Langevin[LOl]) Th ere exists a constant ao > 0 such that if a knot K is a non-trivial knot, then its modulus is larger than or equal to ao. We may imagine a very thin knitted curve contained in an annulus of small modulus as a possible counter example to the above theorem.

Fig. 13.5

A knitted curve.

We assume that a knot K has a very small modulus and control the behavior of K at any scale to show that it is a trivial knot. The outline of the proof is as follows. We construct inductively a sequence of ^-tubular neighborhoods Ni of trivial polygonal knots -Tj which contain K so that Si decreases exponentially. We show that there is a natural number k such that K cannot be knotted in Nk and hence is unknotted. Proof. We start with a knot K of modulus m = m(K) and choose a stereographic projection to M3 such that the two boundary spheres of a maximal annulus A are concentric about the origin.

238

Measure of non-trivial

spheres

Recall t h a t the modulus m(K) is larger t h a n or equal to the modulus m(xi,X2,X3, £4) for any four points on the knot chosen in t h a t cyclic order, since any cross-separating annulus of (xi, X2, £3, X4) is a non-trivial annulus for K. Fixing three points £1,2:2, £3 defines a function pXl,X2tX3 by: Pxi,x2,x3(%) = m(xi,X2,X3,x). It is a continuous function, the zero level of which is the arc 7 of the circle through (2:1, 2:2,2:3) which does not contain the point X2- Moreover, given a small value e of p X l ) X 2 ) X 3 , if the sphere S2 is endowed with a metric such t h a t the length of the circle through x 1, X2, £3 is of the order of 1, then the distance from the level pXltX2,x3 = e to the circle 7 satisfies: c • Length(7) < d(x , 7 ) < C • L e n g t h ^ ) for all x e {pXuX2tx3 = e}Let us suppose t h a t the curve K has a small modulus less t h a n 1 0 Q 0 0 , and choose a stereographic projection of S"3 on R 3 such t h a t the two boundary spheres Si and £2 of a maximal annulus A for K are concentric about the origin (the north pole of the projection is one of the limit points of the pencil generated by S\ and 62, the tangent point of S3 and R 3 , which we choose as origin of M 3 , is the other limit point). Finally, compose the stereographic projection with a similarity to transform the inner sphere into the unit sphere centered at the origin. L e m m a 13.6.2 The knot K is contained in a thin tubular neighborhood of a geodesic circle i"i of the middle sphere Sm {defined by the intersection of the ray containing \{o~i + 02) and A, where a\ and 02 are the points in A corresponding to the spheres Si and S2) of the annulus A. Proof. We know t h a t the knot K is tangent to Si in two antipodal points xi and X3. We choose X2 on the intersection of the knot with the "equatorial" plane II associated to the north and south poles xi and X3. Let 71 be a subarc of K with end points x\ and X3 which contains X2, and 72 be the closure of its complement, K \ 71. Let X4 be one of the points in the intersection of 72 with 77. P u t C123 = C(xi,X2,xs) and C134 = C(xi,X3,X4)Then 72 has to stay in a neighborhood of C123, and 71 in a neighborhood of C134. T h e two circles, C123 and C134, are close to each other, and close to a great circle Z\ of the sphere Sm, which completes the proof. •

The measure of non-trivial

Fig. 13.6

spheres for non-trivial

knots

239

The knot is trapped in a neighborhood of a circle.

Consider the t u b e of radius (Si around J i containing K, and geodesic discs D\,...D]li of this t u b e , normal to i~i through equidistant points of i"i, such t h a t the length of those arc is roughly 100 times as large as the diameter 5\ of the t u b e . Chose a point xf of K in each disc D\. Joining consecutive (for the cyclic order) points xf and xf+1 by a small arc of the circle defined by the two consecutive points xf and xf+1 and one of the almost antipodal points to xf: x\, t h a t we will call 7?, we get an unknotted polygon r<2 with all the angles almost flat (the tangent of those angle is at most if? < 1.5 • arctan(yig) < -^). Using the same idea as for the proof of L e m m a 13.6.2, we deduce t h a t the subarc of the knot joining xf to xf+1 has to stay in a neighborhood of the small arc of the circle 7? of "diameter" of the order of <5i • (length of 7?). T h e knot is now confined in a thin neighborhood of i~2 • Let us call the diameter of this t u b e 82, which is also of the order of 5i • (length of7j 2 ). Consider a sequence of consecutive normal geodesic discs D\...D\ in the t u b e of radius 2 • 62- We choose t h e m to be roughly 100 • 62 apart. We can define t h e m with no ambiguity, except in a neighborhood of the vertices of i~2. T h e choice of the discs near the vertices of F2 has to be made with more care: they should be roughly at a distance of 5OS2 from the vertices of i~2- Again we choose a point xf in each normal disc Df. Then we get a new unknotted polygon i~3. T h e knot K is now contained in a thin tubular neighborhood of TV T h e diameters 8{ of the successive tubes

Measure of non-trivial

240

spheres

become smaller exponentially. The only obstruction left for an inductive construction of unknotted polygons inscribed in K is the angles of the polygons. We need to guarantee they stay flat enough. The angles 9f of T^ a r e bounded by
Fig. 13.7

The successive polygonal approximations of a knot.

That is also true for the angles at all the vertices of the successive polygons except for the ones which are adjacent to some vertex of the previous polygon. A priori the angle cannot grow more than linearly with the index of the polygon. In fact 8 • if 2 bounds all the angles of all the 7^. If the angle 9 at a vertex of i"fc_i is larger than 5 • ip2, then our construction n

will replace it by two angles of 7^ smaller than \- 3^2 > therefore never reaching 8 • ip2 • Having obtained a polygon J \ such that the diameter 8k of its tubular neighborhood containing the knot K is smaller than -A- -A(K), where A (AT) is the minimum global radius of curvature of K given in page 124. The same argument as in Theorem 8.5.6 implies that the knot K is trivial, which completes the proof of Theorem 13.6.1. • Theorem 13.6.1 implies that Emnts(K)

can distinguish the unknot. Namely,

Corollary 13.6.3 (Langevin [LOl]) There is a positive constant c > 0 such that if K is a non-trivial knot then Emnts(K), Vol (<S(AT)) > c.

Measure of non-trivial

spheres for links

241

Proof. Let K be a non-trivial knot. Let A with dA = SiU S2 be nontrivial annulus for K with m(A) = m > 0. Let S C A be a set of 2-spheres contained in A. Then S can be obtained as the intersection of two half light cones with vertices Si and 52- One can compute the volume of S using the measure given in Lemma 9.8.2 to get Vol(5) = l . { 2 1 o g ( c o B h f ) - l

+

I

- i

w

}

>

which is an increasing function of m. Since S(K) D S and Theorem 13.6.1 implies m > ao > 0 the proof is completed. • The modulus and the measure of non-trivial spheres probably do not behave the same way for connected sums of knots. Let m([K]) (or Emnts([K})) denote the infimum of the modulus (or the measure of non-trivial spheres respectively) of a knot that belongs to an isotopy class [K]. Problem 13.3

Do we have m([Ki$K2}) = max{m([A- 1 ]), m([K2})}7

Problem 13.4 (1) Does the measure of non-trivial spheres bound the average crossing number (see Definition A.5) from above? (2) (A weaker question is that:) Is the number of the knot types [K] with EmntB([K}) < b finite for any real number b? 13.7

Measure of non-trivial spheres for links

Definition 13.6 (Langevin) (1) An annulus A is called a non-trivial annulus for a 2-component link L — Ki U K2 if each Ki has at least two cross-strands with disjoint interiors, i.e. if and only if there are x, x' € Ki and y, y' G K2 such that A is a cross-separating annulus for x, x', y, and y'. (2) The modulus of a 2-component link L, m(L), is the supremum of the modulus of a non-trivial annulus for L. (3) A non-trivial annulus for a 2-component link L is called maximal if it attains m(L). Theorem 13.7.1 ([Moni]) (1) There exists a maximal non-trivial annulus for any 2-component link L.

242

Measure of non-trivial

spheres

(2) The modulus of a non-splittable link is bounded below by the modulus of the extreme Hopf link LQ obtained as a pair of fibers of the Hopf fibration over a pair of antipodal points on S2. (3) / / the modulus of a non-splittable link L of class C1 is equal to that of the extreme Hopf link LQ, then L = LQ. As in the case of knots, this claim provides a lower bound for the measure of non-trivial spheres for a non-splittable link. Proof. We only prove (1) and (2). Send a point x in K\ to oo by a Mobius transformation T. Then there is a pair of points T(y),T(y') in T(K2) such that the chord joining T(y) and T(y') intersects T(Ki) in a point T(x'), because otherwise the link L should be splittable. Applying T to the line T(y)T(y'), we get a circle C which contains x, y, x', and y' in this cyclic order. Then simple calculation shows that one of the moduli of maximal cross-separating annulus for x,x',y,y', and that for x,x',y',y is not smaller than the modulus of the extremal Hopf link, which corresponds to the case when x,x',y, and y1 are equally located along C. •

Appendix A

Generalization of the Gauss formula for the linking number

A.l

The Gauss formula for the linking number

Let L = K1 U K2 be a 2-component link in M3 with Kx = / ( S 1 ) and K2 = g{Sl). The linking number Lk(Ki,K2) can be defined as the intersection number of K\ and F2, where F2 is any Seifert surface of K2, i.e. an oriented surface whose boundary is K2. In other words, Lk(Ki,K2) is the sum of the signatures ± 1 of the intersection points of K\ and F2. The linking number has several definitions which are equivalent up to sign (see [Ro] for example). We use the definition by the degree as we are interested in the Gauss formula. Definition A . l

Define the map g : T2 —> S2 by

Fig. A.l

The Gauss map.

243

244

Generalization

of the Gauss formula for the linking

number

Then the l i n k i n g n u m b e r Lk{K\, K2) of K\ and K2 is the degree deg - ( / ( s ) , g(t)) € Conf2(M 3 ) into the configuration space and a homotopy equivalence Conf2(K 3 ) 3 (x,y) >->• Z~LV\ € >S2. In terms of the homology group, the degree deg (pL is given by ipL * ([T 2 ]) = ( d e g v ? J - [ S 2 ] , where [T 2 ] <E H2(T2;Z) and [S 2 ] 6 H2(S2;Z) denote the fun2 2 damental classes of T and 5 . In terms of de R h a m cohomology group ff^fl, the degree deg ]) = ( d e g ^ J • [UT2] e

H2DR(T2),

where LOsi = — (xidx2

A dxs, + x2dx^

A dx\ + x^dxi

A cfa^)

and CJT2 = ds A dt are the unit volume elements of S2 and T 2 . Lemma A.1.1 mula:

The degree degcpL

GI(f,g)=

f

JT2

is given by the following

ipL*uS2

irt

Jrt(f(.W(0./(.)-^))^,i \f(s)-g(t)\3

47ryy5ixSi where f(s),g(t) product in R 3 . Proof. T

(s,t)T

(AJ)

are considered as column vectors and x denotes the vector

P u t w = w(s,t)

2

G a u s s for-

= f(s)

- g(t).

Then

Vh{s,t)

= ^-

If £ , |

e

then

Ws2

^

'.as' a*y ~~s' v^* \8sJ ' ^L* V^

- 4 " ( l ) - ( l > - ^ (A-2) where we orient S"2 by the inner normal vector for the sign convention. Since 9L

*

l

d \ _ f ds) \w\

(f',w)w M3

f d \ _ ' ^ * \dti L

9' \w\

(g',w)w

The writhe and the self-linking

the right hand side of (A.2) is given by PL

'

S2 =

det(f',g',w)J

number

, '

\w\d

245

— , which implies that

,u

D

—H^—

This proof shows that 4irGI(f, g) is equal to the area with sign of the image p{T2). In other words, the degipL is equal to the times y{T2) covers S2 homologically. Let us introduce another definition of the linking number. If v G S2 is a regular value of
^2

sgmpL„(s,t).

(s,t)e
Let nv : K3 —>• 77 be the orthogonal projection in the v direction to a 2-plane 77. Then (s,t) S yjT1(t') if and only if irv(g(t)) is an over crossing point of TTv(f(s)) in the link diagram irv(L) C 77. Furthermore, sgnpLif(s,t) — +1 if and only if this crossing point is a + crossing point. Therefore the linking number is given from a link diagram by deg
2_. tt{+crossing points} — t({—crossing points}, K2 over Kt

(A.3)

where the sum is take over all the crossing points of K2 over K\. Problem A . l Are there any conformally invariant closed 2-form w = LO(KI,K2) for a 2-component link L = K\\J K2 such that

Lk(Kl,K2) = ± [to JT

for any torus T which is a boundary of a tubular neighborhood of K{*. Here, the conformal invariance means that T*(W(T{K1),T(K2)))=UJ{K1,K2)

for any Mobius transformation T.

A.2

The writhe and the self-linking number

Let us consider the case when g = f in (A.l).

Generalization

246

Definition A.2

of the Gauss formula for the linking

(1) The w r i t h e of a knot / is given by

and is denoted by

Wr(f).

(2) T h e d i r e c t i o n w r i t h e Wrv(K) given by Wrv(f)

number

of a knot K in the direction v is

= 2_] tt{+crossing points} — )j{—crossing points},

where the sum is take over all the crossing points in the knot diagram

The writhe is well-defined since the numerator of the integrand is of order \s — i| 4 near the diagonal. Theorem A.2.1 vanishes.

([Cal; Ca2; Ca3; Pol]) Suppose the curvature of f never det(f f" f") Let r be the torsion of f (which is given by r = — y f 'n '

when the knot is parametrized q u n

S1{f) = then Sl(f)

1 ff

by arc-length).

Put

det(f'(s)J'(t)J(s)-f(t))

^JJs^

1 f

\m-ntw

is integer valued and satisfies

the

^t+-jsiris)ds following.

(1) Let np denote the unit principal normal vector field along the Then Sl(f) = Lk(f, f + enp) for sufficiently small e.

knot.

(2) Let nb denote the unit binormal vector field along the knot. Sl(f) = Lk(f, f + enb) for sufficiently small e.

Then

(3) Let H be such a homotopy of f that the curvature meanwhile. Then Sl(f) is kept invariant if there are no Otherwise Sl(f) changes by ± 2 whenever a self-intersection Definition A.3

We call Sl(f)

never vanishes self-intersections. appears.

the self-linking n u m b e r of / .

Remark A.2.2 T h e self linking number is not a knot invariant because SI changes its value by ± 1 every time we add a kink to a knot. T h e reader is referred to [BT; Po2; W h l ; Wh2; Wh3] for the self linking number and its generalizations.

The total twist

Fig. A.2

A.3

247

A kink.

The total twist

Instead of using Frenet frame one can use a moving frame t, e2, e3 of mutually perpendicular unit vectors along a knot K to generalize Theorem A.2.1 as follows. Let ei be the unit tangent vector field of K, e2 be a unit normal vector field along K, and let e3 = e.\ x e2. Definition A.4

([BW]) (1) The total twist of e 2 along K is given by Tw(K;e2)

= ±- f 27r

(de2,e3).

JK

(2) Its reduction modulo Z is called the total twist mod Z of K and denoted by fw(K): Tw(K) = ±- f (de2,e3)

^ JK

(modZ).

Lemma A.3.1 ([BW]) The total twist mod Z of K does not depend on the choice of vector field e2. Proof. Then

Let a2 be another unit normal vector field and 03 = e.\ x a2. a2 — cos 9e2 + sin 9e3 03 = — sin 9e2 + cos 9e3

for some function 9 on K, which implies (da2, 0,3) = (de2, e3) + d9. Hence — / (da2,a3) 2TT JK

- — / (de 2 ,e 3 ) = — / dO e Z. 2n JK 2n JK

248

Generalization

of the Gauss formula for the linking

number

Remark A.3.2 Suppose K has nowhere vanishing curvature. If we take the principal normal vector field as e 2 then (de 2 ,e3) = r(s)ds, where r is the torsion and s is the arc-length parameter. Therefore

Tw(K;np) = i - j T{s)ds. Theorem A.3.3 twist satisfy

([Whl]) The linking number, the writhe and the total

Lk(K, K + ee2) = Wr(K) + Tw(K; e 2 ) for small e. This formula plays an important role in applications to biology, in particular, DNA, and high polymer science. Proposition A.3.4

([BW]) Let I be an inversion in a 2-sphere. Then Tw(I(K))

=

-fw(K).

Proof. Let P be the center of the sphere of the inversion 7". Let us use e.i = /*(ej)/1/„,(e^)| as a unit vector field along I(K) (i = 1,2,3). Let h(x) denote an oriented line through x whose tangent vector is equal to ei(x). Then ei(I(x)) is the unit tangent vector to the image circle I(li(x)) at I(x). Therefore (x-

P, ej)

e f (/(*)) = et(x) - 2\_ ';J(x \x-P[

t

- P).

(Note that — e.i(I(x)) is equal to the image of ej(x) by the unit conformal transportation from x to I(x) (see Definition 3.3) since any circle which passes through x and I{x) is mapped to itself by the inversion I.) Direct calculation shows (de.2,ez) = (de2,e3). Since e.\ x e 2 = — e.3 — / (de 2 ,ei x e 2 ) = - — / (de 2 ,e 3 ). 27r 27r J UK) JK

n

Average crossing

A.4

number

249

Average crossing number

Freedman and He [FH] defined the average crossing number by replacing the integrand of the Gauss formula (A.l) by its absolute value:

m.g)-L[[

Bffl&M^

(A.4)

Since its integrand is the absolute value of the Jacobian det(d(pL), An{c)(f,g) is equal to the area without sign of the image ipL(T2), which is the same as

f

Jves2

$(pL-Hv))uS2,

where $('fiL~1(v)) denotes the cardinality of the pre-image
-4*JJslxS1

|/(«)-/(t)|3

1 47T

ves2

We remark that in the case of knots, (c) does not give the half but the whole of the average of the numbers of the crossing points of the knot diagrams as the directions of the projection vary over S2. It is because each crossing point in a knot diagram nv(h(S1)) is counted exactly once in §(tpfj~l(v)). Definition A.5 ([FH]) (1) (c)(/, g) is called the average crossing number of a link f{Sl) U ^(S 1 ). (2) (c)(f) is called the average crossing number of a knot / : S1 —> R 3 . The average crossing number of an open knot / : M —>• M3 can be defined similarly.

250

Generalization

of the Gauss formula for the linking

number

T h e average crossing number of a knot K is obviously greater t h a n or equal to the minimum crossing number of its knot type [K]. Therefore if ( c ) ( / ) < 3 then / ( S 1 ) is unknotted. Freedman, He, and Wang showed in [FHW] t h a t the number of the knot types whose minimum crossing number are not greater t h a n n is bounded by 2 • (24) n . Therefore any upper bound for a knot energy functional e induces the finiteness of knot types if e bounds the average crossing number from above. Remark A.4.1 T h e average crossing number is not Mobius invariant. Let 7 be a curve on a 2-plane 77 and T be a Mobius transformation which maps 7 to a curve on a 2-sphere S. Then the average crossing number of 7 is 0, whereas the average crossing number of T o 7 is not necessarily 0.

A.5

T h e conformal angle and t h e Gauss integral

Let L = K\ U K-2 be a 2-component link in R 3 with K\ 1

K2 = g^S ).

Let x = f(s),

— f(Sl)

y = g(t), vx = j £ ^ j , vy = ™ | ,

and

and vx(y)

be the image of vx by the unit conformal transportation from x to y (see Definition 3.3). P u t w =

—. Suppose 9L(X,y)

^ 0,n.

T h e n the twice

\x-V\

tangent sphere SL(x,y) is uniquely determined. Let ip(x,y) be the angle between w and the outer normal vector n to Er,(x,y) at y. Since the orientation of TyEi,(x,y) is given by the ordered basis (vx(y),vy) and the angle between vx(y) and vy is 6i,(x,y), {™,vx(y) As vx(y)

= 2(vx,w)w (w,vx

x vy) = cosip(x,y) s i n 9 L ( x , y ) .

- vx x vy) =

-cosip(x,y)sm9L(x,y).

W h e n 8L(X, y) = 0 or n then define ip(x, y) to be -|- since we can take the plane which contains the circle C(x,x,y) = C(x,y,y), which also contains the chord joining x and y. Then (w,vx

x o v ) = 0 = - cos ip(x,y) sin

0L(x,y).

Proposition A.5.1 (Langevin[LOl]) Let ip = ip(x,y) be the angle between the chord joining x and y and the outer normal vector to the twice tangent sphere 27^(x, y) at y. Then the Gauss integral hence the linking

251

The conformal angle and the Gauss integral

number of a link and the writhe of a knot, and the average crossing number is given by }_ f f GI{KuK2) = - ~ / 4TT JKl JK2 i\m TS\ 1 f f (c){K1,K2) = -r / J 4TT JKl JK2

cosip(x,y)smO (x,y) ^°^;2yl„L L^y)dxdy, \X - y \ \cosip(x,y)\sm6L(x,y) ; rdxdy, 2 2 \X - y\

which also holds in the case of a knot K\ = K2 • Let us give another formulae of the Gauss integral in terms of angles. Let nx = rjx(x,y) (or r\y = rjy(x,y)) (0 < r]x,r)y < ir) be the angle between w and vx (or between w and vy respectively). Let nxy = nxy(x,y) be the angle between the plane Sp&n(w,vx), which contains C(x,x, y), and the plane Span (u;, v y ), which contains C(x,y,y), with a sign convention so that (w,vx x vy) = sin r\x sin r\y sin nxy. Then njtTs is \ 1 f f GI(K1,K2) = T- I / 4TT JKl JK2 {c){KuK2)

=— 4TT JK%

/ JK2

smr]xsmriysmr]xy y ; ^ -dxdy, 2 \X - y\ : \X

^ - y\2

y

-±dxdy,

(A.5)

which also hold in the case of a knot K\ = K2. Since 2r(C(x,x,y)) smnx = \x — y\ Wr{K)

= -L [[ smnxy(x,y) !6TT J JKXK r{C(x, x, y))r(C(x, y, y))

([GM]). Since

r(C(x,x,y)

< - 7 ^ = RKK) ~ A(K)

when the knot is normalized to have total length 1, the above formula implies

whose coefficient can be improved to:

252

Generalization

Theorem A.5.2 C < — such that

of the Gauss formula for the linking

number

([Bu2; KS1'; CKS1; BS3; CDGl]) There is a constant

~ 4

(c)(K)

A.6

Mobius invariance of the writhe

Although the integrand of the Gauss formula (A.l) is not Mobius invariant, its integral is Mobius invariant up to sign. Let us show it for the writhe first. Theorem A.6.1 ([BW]) Let I be an inversion in a 2-sphere. Then Wr(I(K)) = —Wr(K). Therefore, if T is a Mobius transformation of R3 U {oo}, then Wr(T(K)) = Wr(K) if T is orientation preserving, and Wr(T(K)) = —Wr(K) ifT is orientation reversing. Proof. Let Wr(K) denote the reduction of the writhe Wr(K) modulo Z. The reduction of the formula in Theorem A.3.3 Lk(K, K + ee2) = Wr(K) + Tw(K; e2) modulo Z implies 0 = Wr(K) +

fw(K).

Then, by Proposition A.3.4, Wr(K) + Wr(I(K))

= -(fw(K)

+ fw{I{K)))

= 0,

hence Wr(K) + Wi{I{K)) is an integer. We show that it is 0 by a deformation argument. Let P and r be the center and the radius of the sphere of the inversion / respectively. Let v be a unit vector such that P(t) = P + tv (t > 0) does not meet the knot K. Let It be the inversion in the 2-sphere with center P(t) and radius r + t. As Wr(K) + Wr(It(K)) is continuous in t and integer valued, it is constant. Since lim^oo It is the reflection about a plane and hence Wr(K) + Wr^^K)) = 0, we have Wr(K) + Wr(I0(K)) = 0. • The Mobius invariance of the linking number up to sign can be shown similarly.

The circular Gauss map and the inverted open knots

253

Problem A.2 Show the Mobius invariance of writhe up to sign directly, i.e. without using the invariance of the total twist.

A.7

The circular Gauss map and the inverted open knots

Let C(y,y,x) and vy(x) be as in Subsection 3.6. We remark that when y approaches x then C(y,y,x) approaches the osculating circle at x, which will be denoted by C(x,x,x), and vy(x) approaches the tangent vector at x, which will be denoted by vx(x). Definition A.6 defined by

([LOl]) The circular Gauss map
e S2.

We remark that
([LOl]) The degree of' ip°K vanishes for any knot K.

Proof. Any knot K can be deformed continuously by an ambient isotopy to a very "thin position" so that K is contained in xy-plane except for some 'bridges' which are contained in {0 < z < e} as illustrated in Figure A.3. Then the pre-image of the vector (0,0,1) G S2 by (p°K consists of pairs of points near the crossing points. Suppose a knot satisfies

'/(o = e.o,«)

(--i-
(i-}<«
>)=(M-I,O) for a small e (0 < e < -|-). Put L 8' 8 J

L2

8' 2

8J

254

Generalization

of the Gauss formula for the linking

Fig. A.3

number

A knot in "thin position'

The pre-image (
Fig. A.4

(^)-1((0,0,l)).

• Let us introduce two non-trivial functionals derived from ip°K. Definition A.7 ([LOl]) (1) The absolute circular self-linking number a la Gauss, \csl\, is defined by

\csi\(K)= [ \{
\Jvv(x) =
The circular Gauss map and the inverted open knots

255

on S2. T h e t o t a l v a r i a t i o n of t h e c o n f o r m a l l y t r a n s p o r t e d u n i t t a n g e n t v e c t o r s , Itv, is defined by Itv(K) By definition Itv{K)

> \csl\(K)

= /

lK(x)dx.

for any knot K.

Let us give geometric interpretation of Itv using t h e inverted open knot. Let Ix, Kx, and y be as in Subsection 3.6. Then Ix(C(y,y,x)) is the (oriented) tangent line to Kx at y. Let Vy(y) be the unit tangent vector to Ix(C{y,y,x)) at y. Then -vy(x) = Ix*(vy(x)) = vy(y). Therefore lK(x) is equal to the length of the curve {Jye^ ^V (&) ^ S2- Thus Itv(K) can be considered as the "total variation of the tangent vectors to the inverted open knots". If T : K 3 U {00} —¥ M3 U {00} is a Mobius transformation t h a t fixes {00}, then T|R3 is a similarity. Therefore we have: Proposition A.7.3 ([LOl]) The similarity class of the inverted open knot Kx is a conformal invariant of K. Namely, IT^(T(K)) is similar to Kx for any Mobius transformation T. Therefore the functional Itv is a Mobius invariant functional. Although Itv is Mobius invariant, it cannot be expressed in terms of the infinitesimal cross ratio like E^ and Es[n0, because Itv is determined by up to the second order derivatives of the knot. Theorem A.7.4

Neither

\csl\ nor Itv is an energy functional

for

knots.

Proof. It will suffice to show t h a t Itv is not an energy functional because hv > \csl\ for any knot. We show t h a t the contribution to Itv of a pair of straight line segments of a fixed length with the closest distance e does not blow u p even if e decreases to 0. Let T\ = {x(s)\a < s < b} and i~2 = {y(t)W < t < b'} be straight line segments with mmX£rlty€r2 \x ~ y\ = e- Let us consider and w = * y . Then vx and vy are I3- y\ constant (with respect to x s 7 \ and y g i ^ ) - Fix t and put j(s) = vx{y), where vx (y) is the image of the unit conformal transportation of vx from x to y (see Definition 3.3). T h e n as j(s) = 2(vx,w)w — vx - M A U J V P u t vx = f'(s),

dj

vy = f'(t),

(

2 v

dw \

Ts= { -Ts)

w

+

.

. dw

2{v w)

- Ts'

Generalization

256

of the Gauss formula for the linking

number

hence d'y dw < 2y/2 ds ds Therefore ,6

dj ds < 2s/2 \ ds Ja

irAy) =

dw ds < ds

2V2TT,

and hence rb'

Itv\nur2

= /

rb dt

lrAy) -

J a'

/

lr2(x)ds

< 2V2-K(b' — a! + b — a)

Ja

D

which is independent of e. We show t h a t they can detect the unknot. Example A.l therefore Itv(K0) Theorem A.7.5

Let K0 be a circle. Then vy(x) = \csl\(K0) = 0. ([LOl]) If K is a non-trivial

= f'(s)

for any y and

knot then Itv(K)

> TT.

Proof. T h e unit tangent vector Vy(y) of the inverted open knot Kx is asymptotic to — vx(x) as y goes to 00. If the angle between Vy(y) and —v x (x) is smaller t h a n y for any y then Kx is unknotted, which contradicts the assumption. Therefore IK(%) > TT for any x G K. D Conjecture A.3 There is a positive constant C such t h a t if A' is a nontrivial knot then IcsZKif) > C.

Appendix B

The 3-tuple map to the set of circles in S3

B.l

The set of unoriented circles in S3

We showed in Theorem 9.3.2 that the set of the oriented 2-spheres in S3 can be identified with the hyperbolic hypersurface A in R 4 ' 1 . Let us study the set of circles. The set C of unoriented circles C in S3 can be identified with the set i^sp of space-like 2-planes P through the origin. The bijection is given by assigning to P the base circle C = P1- f~l S3^ of the pencil V — P n A. This set 5ssp is a subspace of the Grassmannian manifold Gr 2,3-

B.2

The 3-tuple map to the set of circles

We defined in Definition 10.3 the 4-tuple map for a knot K which assigns an unoriented 2-sphere that passes through 4 points on K. (3)

Definition B . l The 3-tuple map ipK is a map which assigns an unoriented circle that passes through 3 points / ( t i ) , /(*2) 5 and /(is) on a knot K. Let / : S1 —»• S3 = S3 C M4'1 be a knot. We can assume without a loss of generality that S 1 = [0,Z]/~ and \f'{t)\~l. Lemma B . l (1) Under the identification C = ^Sp C Gr2,3 the 3-tuple map ipK is given for (^1,^2,^3) G Conf3(S'1) by ^)(ti,*2,*3) = (Span(/(t1))/(t2),/(t3)»-L, 257

The 3-tuple map to the set of circles in S 3

258

and when ti = t2 7^ £3 by ^(tutuh) and if t\=t2

= (Span(/(t1))/'(i1),/(i3)))±,

= £3 then ^(t,t,t)

= (Span < / ( t ) , / ' ( * ) , / " ( i ) » X .

(2) The 3-tuple map ipK : (S1)3 —> ^sp

C Gr2i3 is continuous.

Proof. (1) We show first that the three vectors in the right hand sides of the formulae are linearly independent. It is clear in the first two cases. Assume af(t) + bf'(t) + cf"(t) = 0. Since ( / , / ) = 0 we have ( / , / ' ) = 0 and (/, /"} = - ( / ' , / ' ) = - 1 . Therefore (/(<), af(t) + bf'(t) + cf"{t)) = 0 implies c = 0. Since ( / ' , / ' ) = 1 we have ( / ' , / " ) = 0. Therefore (f'(t),af(t) + bf'(t) + cf"{t)) = 0 implies 6 = 0. Thus a = b = c = 0, which implies that /(£), f'(t), and f"(t) are linearly independent. Then Lemma 9.1.1 implies that the right hand sides of the formulae are space-like. (2) Let V®2 denote the set of orthonormal 3-frames in R5 with the standard Euclidean metric. By using Taylor's expansion and Gram-Schmidt's orthogonalization we can define a continuous map from (S1)3 to V°2- As ipK is the composition of continuous maps (S1)3 —> V3°2 -^ it is also continuous.

Gr 3 , 2 ^

Gr 2>3 , •

Appendix C

Conformal moduli of a solid torus

The modulus of a 2-dimensional annulus S1 x D1 can be naturally generalized to that of a 3-dimensional annulus S2 X Dl as in Definition 9.7. It can be also generalized to a solid torus S1 x D2 in the following way, which works for 3-dimensional annulus as well.

C.l

Modulus of a 2-dimensional annulus revisited

Let A be a 2-dimensional annulus in M2 with concentric boundary circles C\ and Ci about the origin with radii a and b (a < b). Then its modulus and Lorentzian modulus is given by m(A) = log —. Put m FH (A) = inf f f JA

\Vf\2dv,

where / : A —» S1 = K/Z is any degree 1 C1-function, and du is the standard area 2-form of K2. Then mFH(A) =— log —. Put m*(A) = inf / a JA

\Vg\2dv,

where g : A —> [0,1] is any C^-function such that g = 0 on C\ and g = 1 on C 2 . Then m*(A) = 27r/log - which is attained by 5 {x) Therefore, mFH(A) =— m(A) and mFH(A)m*(A) 2-7T

259

=1.

logN

"

loga

log b — log a

Conformal moduli of a solid torus

260

C.2

Conformal moduli of a solid torus

Let T be a solid torus with some Riemannian metric. Definition C.l

(1) ([Ge]) The conformal modulus of T is given by

I pds L )

/93dvol

TOi(T) =

inf

sup

'7

where p is any non-negative densidy function on T and 7 is any degree 1 curve in T. (2) ([FH]) The conformal modulus of Freedman and He of T is given by m{T)=

inf f |V/| 3 dvol, / JT

where / : T —• R/Z is any degree 1 C1-function. A fat solid torus has a large modulus, while a very thin one has a modulus close to 0. Theorem C.l Definition C.2

([FH]) We have m(T) >

mi(T).

([FH]) (1) Define m*(T) = inf f S3d vol, s JT

where 5 : T —> K_|_ is any density function which satisfies 52da>

1, /is. for any singular surface S representing the generator of H2 (T, dT; Z). Here, a singular surface means a union of 2-manifolds S and a smooth mapping (S,dS) —>• (T,dT). The metric and the area element da on 5 is given by pullback. (2) Define m*{T) = inf / ItulTrfvol, w JT where w is any closed 2-form representing the generator of H2(T, dT; Z).

Conformal moduli of a solid torus

T h e o r e m C.2

([FH]) (1) ml(T)2m(T)

261

= m*(T)2m(T)

= 1.

(2) Suppose f : T —>• R/Z is i/ie extremal function realizing m(T). Then 5 = \df\/m{T)^ is the unique extremal density realizing m\{T), and w = * (\df\df/m(T)) is the unique etremal 2-form realizing m*(T). T h e o r e m C.3

([FH]) For any solid torus T of knot type [K] we have m(T) < A(asympt.c(K)) 2

where asympt.c(K)

is the asymptotic crossing number which is given below.

Definition C.3 ([FH]) Let T be a tubular neighborhood of a knot K, and p, q be positive integers. A 2-component link {L\, L2) is called a degree (p,q) satellite link of K if (Li,L2) can be (simultaneously) isotoped to a pair of curves (L'1;.L2) C T with degree(L'1) = p and degree(L 2 ) = q. Let c + (Li,L 2 ) denote the minimum number of over-crossings of L\ over L2 among all planar link diagrams of (Li,L 2 ). Let c+(K) be the minimum of c + (Li,L 2 ) over all degree (p,q) satellite links (Li,L2) of K. The asymptotic crossing number of K is given by asympt.c(K)

c+JK) = liminf-^ P,9->OO

pq

c+JK) = inf - ^ . P,9>I

pq

The equivalence of two definitions can be shown using a smartly chosen fc-fold planar parallel diagram (Li,L2) of a degree (p,q) satellite link (Li,L 2 ) that is a degree (kp,kq) satellite link with c+(Li,L2) — k2c+(Li,L2). If we take two parallel copies of a minimal knot diagram of if as (L\, L 2 ) then c + (Li, L 2 ) < C(K), where c(K) denotes the minimum crossing number of a knot K. Therefore asympt.c(K) < c(K). Conjecture C.l

([FH]) asympt.c(K)

= c(K).

This page is intentionally left blank

Appendix D

Kirchhoff elastica

In order to produce beautiful knots one might ask what actually happens if one forms a knot in a piece of elastic wire. According t o the BernoulliEuler theory the bending energy of an elastic rod is proportional to the total squared curvature K ' 2 ' . A Critical point of K<2> is called an e l a s t i c a . Langer and Singer showed Theorem D.l ([La-Sil; La-Si2; La-Si3]) (1) / / / is a closed planar elastica then f is equal to the circle or a figure eight which is unique up to similarity or a multiple cover of one of these two. (2) The knot types of non-planar with p > 2q.

closed elasticae are (p,q)-torus

(3) The only stable closed elastica in R 3 is the singly-covered

knots

circle.

T. Kawakubo ([Kawakl]) takes the twisting energy into consideration. He studies his functional on the space of framed knots, where the framing is not necessarily Z-valued. He obtained a local solution of its Euler-Lagrange equation and showed t h a t the non-trivial knot types of critical points of his functional are again torus knots. See also [Kol; Ko2; Ko3]. These functionals are not self-repulsive as they can be expressed as an integral of a locally defined quantity.

263

This page is intentionally left blank

Appendix E

Open problems and dreams

We end with open problems and conjectures, some of t h e m are vague, just a kind of dream. Problem E.l (See Remark 3.5.2) Hatcher showed t h a t the space of the 3 trivial knots in S deformation retracts onto the space of great circles in S3 ([Ha]). Are there any knot energy functional E such t h a t the circles are the only critical (or, local minimum) points of E in the trivial knot type? P r o b l e m E.2 W h a t is the topological (homotopy) type of the space of the knots t h a t belongs t o a given knot t y p e [K\l Problem E.3 Can we get any topological informations from critical points (or infimums etc.) of any geometric functional on the space of knots? For example: (1) The infimum of the total curvature of a knot type is equal to 27r times its bridge index. (2) T h e non-planar critical embeddings for the total squared curvature functional are torus knots ([La-Sil-3]). (3) There are critical embeddings for E1-2^ in prime knot types, but it is conjectured t h a t there are no critical embeddings in composite knot types. Conjecture E.4 Let E be a knot energy functional. T h e n it takes its minimum value for non-trivial knots at a trefoil, in particular a (2,3)-torus knot on a Clifford torus in S3 when E is defined on the space of knots in S3. 265

266

Open problems and dreams

If, furthermore, E is defined for links, it takes its minimum value for non-splittable links at a Hopf link, in particular the one obtained as the inverse image of antipodal two points on S2 by the Hopf fibration map when E is defined on the space of links in 5 3 . Problem E.5 Find any relations between any knot energy functional and any (algebraic) knot invariant. Find more relations between knot energy functionals. Problem E.6 Define a measure on the space of knots and integrate a knot energy functional E or expiE on a knot type to define a new knot invariant. Problem E.7 Let £ be a knot energy functional. Are there any functional E for knotted tori (or, for surfaces in general) such that E(K) = lim

E(dNe(K)),

e—s-0

where N£(K) is the gr-tubular neighborhood of K? Problem E.8 By a framing of a knot K we mean a (unit) normal vector field v to K. Let £ b e a knot energy functional. Are there any functional E' for framed knots (K, v) (or 2-component links) such that E{K) can be expressed in terms of E'(K, K + ev) (for some framing v)l Problem E.9 Can we define a knot energy functional E by taking spaces X and Y, a differential form (or a current) ui on Y, then defining a map Y for each knot / , and finally putting

E(f)=

f

Jx

Problem E.10 (K. Ono) A knot K in Sz = dB4 is called a slice knot if there is a disc D in BA such that 3D = K. Let E be a Mobius invariant knot energy functional. Can E(K) be expressed in terms of D when K is a slice knot dDl Problem E . l l als.

Here are some candidates for new knot energy function-

Conjecture 3.1 ([KS1']) (1) There are no minimizers in composite knot types because all the tangles will pull tight.

Open problems and dreams

267

(2) Let [Ki], [K2] be knot types. Then

Ei2\[K4K2}) = M2)([#i]) + Ei2\[K2}). Conjecture E.12 In general, let E be a Mobius invariant knot energy functional which takes 0 at circles like Es-lng or Emnts. Then

(1) There is an E minimizer in a prime knot type. (2) There are no E minimizers in composite knot types because all the tangles will pull tight. (3) Let [K{\, [K2] be knot types. Then E([K^K2})

= E^K,))

+ E([K2}).

Conjecture 3.2 ([KK]) fror r0)Pi9 where r0 is obtained by (3.21) is a stable local minimum for E1-2' if and only if p = 2 or q = 2. Problem 4.1 (1) Does the map Ah : R>o —> R>o given by R 3 f e h 4 A r e a { ( s , t ) e S1 x S1 : \h(s) - h(t)\ > b} G M determine the knot h up to a motion of R 3 ? (2) Put a = 2 for example. Does the map Bh = e2'* : M>o —> K>o given by R > 0 9 P ^ e 2 ' p (/i) e R determine the knot ft up to a motion of R 3 ? Conjecture 5.1 If a > 3 then ft.«/ satisfies the following conditions: (1) hho is independent of a. (2) /i!» attains the local minimum value of the total squared curvature functional among [K], where [K] is a closure of [K] in the set of immersions with respect to the C2-topology.

(3) h£> e[Ki\[K\(2)

Conjecture 7.1 There are Es3

.

.

.

(2)

minimizers for any knot type. Since J5L (2)

is not Mobius invariant any more the number of Es3 minimizers for any knot type is finite. Problem 7.2 Do we have e^{K) > e%?(K) for any knot K in H 3 ? It is not straightforward as the arclength also decreases if we use the hyperbolic metric for the ambient space.

Open problems and dreams

268

C o n j e c t u r e 7 . 3 There are no E^' minimizers in any knot type. Any knot will converge t o a point if it evolves so as t o decrease its value of .Ejjjp . Suppose / „ is a convergent sequence in a prime knot type [K] with limn^oo Ey{fn) = Ey{[K]). T h e n n o / „ converges t o an E^3' minimizer 4 after rescaling where n : R —> M3 is t h e orthogonal projection. C o n j e c t u r e 7.4 Among knots with length I in H 3 , circles with length / minimize e^ {ap > 2). If this conjecture is ture, then there is an e ^ p minimizer for any knot type if t h e index ap > 2. C o n j e c t u r e 7.5 (1) As for t h e scale invariant energy, there are no e^, p s minimizers in any knot type. Any knot will converge t o a point if it evolves so as t o decrease its value of e^s. Suppose / „ is a convergent sequence in a knot type [K] with liiri n _ i . 0O e ^ P s ( / „ ) = e^3ps(K]). Then 7r o / „ converges P to an e^3 minimizer if exists after rescaling where n : M4 —>• R 3 is t h e orthogonal projection. (2) As for t h e unit quantity energy, there are no e ^ p minimizers in any knot type. Any knot will diverge t o infinity if it evolves so as t o decrease its value of e j ^ p . Suppose / „ is a diverging sequence in a knot type [K] with limra-joo e j ^ p ( / „ ) = eJjpP ([K]). If ap > 3 then there are singular knots on the limit sphere S^ as t h e limit of / „ . P r o b l e m 8.1 Are Uhp{K)

(j = 1,2,3,4) self-repulsive if p > j?

C o n j e c t u r e 8.2 We conjecture t h a t s d _ 1 bounds the distortion from above, and hence, sd~ is self-repulsive. P r o b l e m 1 0 . 1 Is Cc(K) non-empty for any trivial knot Kl P r o b l e m 1 0 . 2 Is it true t h a t if K is non-trivial then t h e twice tangent sphere m a p tpR is n ° t injective, namely, there is a sphere S t h a t is tangent t o K at more t h a n or equal t o three points? C o n j e c t u r e 1 2 . 1 We conjecture t h a t Es\ng is self-repulsive. C o n j e c t u r e 12.2 We conjecture t h a t t h e infimum of Es\ng{Ki, K2) of nonsplittable links K\ U Ki is attained by t h e extreme Hopf link obtained as a pair of fibers of t h e Hopf fibration over a pair of antipodal points on S2, when Esing(Ki,K2)

= 2TT2.

Open problems and dreams

Problem 13.1 Can Atwt(K)

be expressed in terms of the infinitesimal d /

cross-ratio and its derivatives — —— ds

Problem 13.2 Is E^lta Problem E.13

269

eie

\

etc. ?

\\f(s)-f(t)\i)

finite for any smooth knot?

Is there any integral geometric interpretation of Mobius (2)

invariant knot energies such as Eo and Es-mg like -Emnts, i.e. are there any functions F on the set of non-trivial spheres S(K) such that Ei2) = [

F(S)dS

etc.?

JS(K)

Problem 13.3 Do we have m([K^K2})

= maxima]),

m([K2})}?

Problem 13.4 (1) Does the measure of non-trivial spheres bound the average crossing number (see Definition A.5) from above? (2) (A weaker question is that:) Is the number of the knot types [K] with Emnta([K]) = for a 2-component link L = K\ U K2 such that

CJ(KI,K2)

Lk{K1,K2) = ± f u JT

for any torus T which is a boundary of a tubular neighborhood of K{1 Here, the conformal invariance means that T*{uJ(T{Kx),T{K2)))=u(KuK2) for any Mobius transformation T. Problem A.2 Show the Mobius invariance of writhe up to sign directly, i.e. without using the invariance of the total twist. Conjecture A.3 There is a positive constant C such that if K is a nontrivial knot then \csl\{K) > C. Conjecture C.1([FH]) asympt.c(K)

= c(K).

This page is intentionally left blank

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differen-

Index

a-energy, 20, 28 a-energy polynomial, 79 D(K), 124 e-self avoiding voltage, 10 vx{y), 40, 41

4-tuple map extended —, 186 of S 3 , 148 of a knot, 178

(c) (/,
absolute circular self-linking number a la Gauss, 254 admissible polygonal knot, 75 admissible solid torus, 75 annulus cross-separating annulus, 158 Lorentzian modulus of — , 156 maximal cross-separating annulus, 158 maximal non-trivial —, 232 maximal non-trivial — for links, 241 modulus of — , 156 non-trivial —, 232 non-trivial — for links, 241 area of the twice tangent spheres, 223 argument of the infinitesimal cross ratio, 204 asymptotic crossing number, 261 average crossing number, 249

K n + 1 - 1 , 136 R1(K), 118 Sl(f), 246 Thk(K), 118 Wr(f), 246 Atwt(K), 223 LkiK^K?), 244 ess.dcsd(/i), 131 ess.sd(/i), 131 -
39

A(K), 124 Cc (2 ' 2) (ii:), 180 Ccw(K), 184 e», see energy E*, see energy GI(f,g), 244 Itv, 255 r_c*-modified e-orT diagonal electrostatic energy, 15 r~™-modified e-self avoiding voltage, 15 r _ a -modified electrostatic energy, 22 V»*, see voltage

base circls, 177, 180, 183, 184 sphere (circle, or points), 152 basic, 7 285

286

beads radius, 131 beta function, 79 Brylinski's beta function, see beta function canonical symplectic form of a cotangent bundle, 191 captured by a polygonal knot, 75 center of a pencil, 152 charge, 5 circular Gauss map, 253 combined angle, 60 complete family of N, r-polygonal knots, 76 complete system of solid tori, 75 concircular set of (2,2)-— points, 180 set of — points in S 3 , 147 set of — points of a knot, 177 set of generalized — points, 186 set of osculating — points, 184 configuration space, 147 conformal ( l - c o s 0 ) energy, 40, 208 (1 — cos9)k energy, 61 sin X-energy, 59 sin 9 energy, 213 1 — cos X-energy, 59 angle, 41, 204 angle 9K, 39 diffeomorphism, 138 group, 136 modulus of a solid torus, 260 modulus of a solid torus, 260 transformation, 35, 136 transportation, 41 unit — transportation, 40, 173 conformally transported unit vector, 40 vector, 41 conformally transported unit vector, 173 cosine formula, 39

Index counter term, see knot energy functional cross-separating annulus, 158 sphere, 158 cross-strand, 232 cross ratio of four ordered points, 149 curvature, see geodesic curvature de Sitter space, 137 distortion, 28 doubly critical self distance essential — ,131 elastica, 263 energy (a,p)-energy functional, 67 a-energy, 20, 28 a-energy polynomial, 79 e-off diagonal electrostatic energy, 11 Ea'f ',64 £ < * > • ,46 E(a) , 2 0 EW , 12 £<2> , 2 8 Ei2) , 29, ea-p, 64 £

( i - cos 6)

Eia) , 15

Esing, 213 Emnts(K), 225 Eapm, 101 r _ a -modified e-off diagonal electrostatic energy, 15 r~ a -modified electrostatic energy, 22 beta function, 79 conformal (1 — cos 9) energy, 208 conformal sin 9 energy, 213 imaginary cross ratio energy, 213 Mobius energy, 29 mutual conformal (1 — cos#) energy, 209

Index mutual conformal sin 9 energy, 222 renormalization of electrostatic energy, 12 scale invariant (a,p) energy, 114 unit density (a,p)-energy, 101 unit quantity (a,p) energy, 114 X-energy, 59 energy minimizer, 6 essential doubly critical self distance, 131 essential self distance, 131 finiteness of knot types, 7 Gauss absolute circular self-linking number a la Gauss, 254 circular Gauss map, 253 linking number formula, 243 geodesic curvature, 154 global radius of curvature at a point, 121 minimum — , 124 Gromov's distortion, see distortion ideal knot, 118 imaginary cross ratio energy, 213 index, 20, 64, 101 infinitesimal cross ratio argument of — , 204 of a knot, 203 of the complex plane, 189 inflation radius, 118 intersecting (two spheres), 154 inversion in a sphere, 34 inverted open knot, 37 knot energy functional counter term, 13, 19, 20 principal term, 20 with respect to the C 2 -topology, 5 L-orthogonal, 136 light-cone, 136

287

light-like, 136 linking number GI(f,g), 244 absolute circular self-linking number a la Gauss, 254 Gauss formula, 243 self-linking number, 246 Lorentz group, 111, 136 Lorentzian exterior product, 143 modulus of an annulus, 156 modulus of four points, 158 pseudo inner product, 110, 135 quadratic form, 135 Mobius energy, 29 group, 138 transformation, 34, 138 maximal cross-separating annulus, 158 non-trivial annulus, 232 non-trivial annulus for links, 241 measure of non-trivial spheres, 225 minimum distance, 128 Minkowski space, 136 modulus conformal — of a solid torus, see conformal modulus Lorentzian — , see Lorentzian of a knot, 232 of a link, 241 of an annulus, 156 of four ordered points, 158 mutual conformal (1 — cos 9) energy, 209 conformal sm9 energy, 222 nested (two spheres), 154 non-trivial annulus, 232 annulus for links, 241 sphere, 175 sphere in the strict sense, 175

288 normal projection energy symmetric — , 60 optimal embedding, 7 osculating — sphere map, 186 circle, 119 sphere, 119 pencil base sphere (circle, or points) of — , 152 center of — , 152 Poncelet — , 152 Poncelet pencil, 152 points, 153 principal part, see knot energy functional pull-tight, 7 renormalization electrostatic energy, 12 of r~a-modified electrostatic energy, 20 of r _ a -modified voltage, 19 residue formula, 57 ropelength, 118 scale invariant (a,p) energy, 114 self C 2 self-repulsive, 5 -linking number, 246 essential self-distance, 131 set of concircular points, see concircular space-like, 136 sphere area of the twice tangent spheres, 223 cross-separating sphere, 158 inversion in a sphere, 34 measure of non-trivial spheres, 225 non-trivial sphere, 175

Index non-trivial sphere in the strict sense, 175 osculating — ,119 twice tangent — ,173 twice tangent — of a knot, 179 twice tangent sphere, 119 strand cross-strand, 232 minimal cross-strand, 232 symmetric normal projection energy, 60 thickness, 117 tight, 7 time-like, 136 total variation of the conformally transported unit tangent vectors, 255 transportation, see (unit) conformal transportation twice tangent sphere, 119 of a knot, 179 twice tangent sphere map, 181 unit density (a,p)-energy, 101 quantity (a,p) energy, 114 voltage e-self avoiding voltage, 10 r~ Q -modified e-self avoiding voltage, 15 V{a), 19 V£(a), 15 wasted length of an inverted open knot, 39 writhe —, 246 direction —, 246

SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman (Univ. of Illinois, Chicago)

The Series on Knots and Everything: is a book series polarized around the theory of knots. Volume 1 in the series is Louis H Kauffman's Knots and Physics. One purpose of this series is to continue the exploration of many of the themes indicated in Volume 1. These themes reach out beyond knot theory into physics, mathematics, logic, linguistics, philosophy, biology and practical experience. All of these outreaches have relations with knot theory when knot theory is regarded as a pivot or meeting place for apparently separate ideas. Knots act as such a pivotal place. We do not fully understand why this is so. The series represents stages in the exploration of this nexus. Details of the titles in this series to date give a picture of the enterprise.

Published: Vol. 1:

Knots and Physics (3rd Edition) L. H. Kauffman

Vol. 2:

How Surfaces Intersect in Space — An Introduction to Topology (2nd Edition) J. S. Carter

Vol.3:

Quantum Topology edited by L. H. Kauffman & R. A. Baadhio

Vol. 4:

Gauge Fields, Knots and Gravity 7. 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. Millett & 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 Modern View edited by J. D. Edmonds, Jr. Vol.13: Entropic Spacetime Theory J. Armel Vol. 14: Diamond — A Paradox Logic N. S. Hellerstein Vol. 15: Lectures at KNOTS ' 96 S. Suzuki Vol. 16: Delta — A Paradox Logic N. S. Hellerstein Vol. 19: Ideal Knots A. Stasiak, V. Katritch & L. H. Kauffman Vol. 20: The Mystery of Knots — Computer Programming for Knot Tabulation C. N. Aneziris Vol. 24: Knots in HELLAS '98 — Proceedings of the International Conference on Knot Theory and Its Ramifications edited by C. McA Gordon, V. F. R. Jones, L. Kauffman, S. Lambropoulou & J. H. Przytycki Vol. 26: Functorial Knot Theory — Categories of Tangles, Coherence, Categorical Deformations, and Topological Invariants by David N. Yetter Vol. 27: Bit-String Physics: A Finite and Discrete Approach to Natural Philosophy by H. Pierre Noyes; edited by J. C. van den Berg Vol. 28: Beyond Measure: A Guided Tour Through Nature, Myth, and Number by J. Kappraff Vol. 29: Quantum Invariants — A Study of Knots, 3-Manifolds, and Their Sets by Tomotoda Ohtsuki Vol. 30: Symmetry, Ornament and Modularity by Slavik Vlado Jablan Vol. 31: Mindsteps to the Cosmos by Gerald S Hawkins Vol. 32: Algebraic Invariants of Links by J. A. Hillman Vol. 33: Energy of Knots and Conformal Geometry by Jun O 'Hara

ENERGY OF KNOTS AND CONFORMAL GEOMETRY Energy of knots is a theory that was introduced to create a "canonical configuration" of a knot — a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a "canonical configuration" of a knot in each knot type. It also considers this problem in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments.

ISBN 981-238-316-6

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Conformal Groups in Geometry and Spin Structures

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Conformal Geometry of Generalized Metric Spaces

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