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

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.

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Printed in Singapore.

& #L ® •![ u l£ 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 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 £ 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 < l>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'^ 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, #( 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+