Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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The puncture and dilaton equations for (p,1)

theories read

h

0;0





i¼1;n

6¼0

h



1;k



½27

h

1;0





¼ð2g  2 þ sÞh



½28

The special terms at g = 0 and g = 1 are given by

h

0;0



0;i



0;pi2

¼ 1; h

1;0

p  1

½29

Integrable Hierarchy

We now summarize some basic facts about the

integrable hierarchy (see for instance eqn [5]). We

introduce a pth order differential operator:

L ¼ D

p2

i¼0

ðxÞD

; D 

½30

where the coefficient functions u

are arbitrary

functions of x. This Lax operator describes the pth

generalized KdV hierarch y. We consider the time

evolution of the operator L by an infi nite set of

commuting H amiltonians:

¼½H

; L; n ¼ 1; 2; ... ½31

where H

is given by

¼ L

n=p



½32

Here ‘‘þ’’ denotes the non-negative part of a

pseudodifferential operator and is defined as

A ¼

i¼1

ðxÞD

; A

i¼0

ðxÞD

½33

We also use the notation

res A ¼ f

1

ðxÞ; A



1

i¼1

ðxÞD

½34

Note that x is identified as the first time variable t

that is, x = t

It is a basic result of the calculus of pseudodiffer-

ential operators that the a bove Hamiltonians satisfy

the zero-curvature condition



þ½H

; H

¼0 ½35

Note that when m is a multiple of p, H

becomes a

power of L and trivially commutes with L. Thus, the

time variables t

are absent for n  0 mod p. In the

simple case of p = 2, one has

L ¼ D

þ uðxÞ½36

and H

= D

þ (3=2)uD þ (3=4)u

. One finds

¼½H

; L¼

½37

which is the standard KdV equation.

In the case of KP hierarchy, one starts with a

pseudodifferential operator

Q ¼ D þ

i¼1

i

½38

and considers the time evolution equations

¼½H

; Q; H

¼ Q

ðÞ

½39

p-reduced KP hierarchy is obtained if one has



¼ 0 ½40

By introducing a pseudodifferential operator K, one

may bring Q to the sim ple derivative operator D as

Q ¼ KDK

1

½41

K has an expansion of the form

K ¼ 1 þ

i¼1

i

½42

After time evolution, the coefficient functions u

(x)

of the Lax operator dep end also on the variables

, t

, ... and become functions of t  {t

, t

, ...}.

These functions are expressed by the -function (t)

of the hierarchy in the following manner :

res K ¼

log ðtÞ½43

res L

i=p

@x@t

log ðtÞ½44

These residues are expressed in terms of {u

} and

their derivatives in x, and one can determine them in

terms of the -function.

268 Topological Gravity, Two-Dimensional

In the case p = 2, one has

½H

; L¼2D resðL

k=2

Þ¼DR

; k ¼ odd ½45

Here {R

} are the Gelfand–Dikii potentials

¼ u; R

ð3u

þ u

ð10u

þ 5u

þ 10uu

þ u

0000

½46

and obey the recursion relation

kþ2

þ 2ðDu þ uDÞ



½47

If one uses the relation [44], Gelfand–Dikii potentials

are identified as

¼ 2hO

i½48

By setting k = 1, we note u = 2hO

i and find that

the evolution equations [31] are all satisfied as

¼ 2

i¼2DhO

¼ DR

¼½H

; L½49

Now it is possible to identify the initial condition

for the Lax operator in the case of topological (p,1)

gravity. By using the definition

log ðtÞ¼

g¼0

exp

½50

one has

res L

i=p

ð0Þ¼hO

i; i ¼ 1; ...; p  1 ½51

From [29] one finds

resL

i=p

ð0Þ¼ix  

i;p1

½52

This gives the initial value of the Lax operat or:

Lð0Þ¼D

þ px ½53

Thus, only the lowest term u

(x) = px is nonzero

and higher coefficients all vanish at t = 0. This is the

special simplification which takes place in the

topological gravity theory.

We note a relation

D; Lð0Þ



¼ 1 ½54

This is the so-called ‘‘string equation’’ (at t = 0). At

nonzero values of t, the string equation takes the form

½P; L¼1

P ¼

ðL

1=p



k¼pþ1

ðL

ðkpÞ=p

½55

From [55], we see that (p, 1) theory corresponds to the

background value of the coupling t

pþ1

= 1=(p þ 1).

In the case of (p, q) theory, background value is given

by t

pqþ1

= 1=(pq þ 1).

Virasoro Conditions

A powerful algebraic machinery controlling

the structure of 2D gravity is the so-called ‘‘Virasoro

conditions.’’ One introduces differential operators

1

¼

k¼pþ1

kp

iþj¼p

ijt

½56

¼

pþ1

k¼1

 1

½57

By using the fact that deri vative in t

brings down

the operator O

when acting on the -function, it is

easy to show that

1

  ¼ 0 ½58

  ¼ 0 ½59

reproduce the puncture [27] and dilaton equation

[28], respectively. It is possible to show that the

1

-condition, L

1

  = 0, is equivalent to the

string equation [55].

Together with the operators (n  1)

¼

1þðnþ1Þp

k¼1

kþnp

iþj¼np

they generate Virasoro algebra (L

 (1=p)L

)

½L

; L

¼ðm  nÞL

mþn

; n; m 1 ½60

It is possible to show that the (p, 1) model obeys the

Virasoro conditions [6]

  ¼ 0; n 1 ½61

It is known that (p, 1) models with p > 2 also obey

constraints of W-algebra.

The relationship of the Virasoro conditions to

KdV hierarchy is summarized as

string equation þ KdV hierarchy

() Virasoro and W-algebra constraints

Topological -Model

It is known that when the target space of a

supersymmetric nonlinear -model is a Kahler

manifold K, the theory acquires an enhanced N = 2

Topological Gravity, Two-Dimensional 269

supersymmetry. Then we can twist the theory and

converted into a topological field theory. This is the

topological -model [7]. The partition function of

the theory consists of a sum over world-sheet

instantons, that is, holomorphic maps from the

Riemann surface to the target space K. Due to

supersymmetry, functional determinants around

instantons cancel and the theory simply counts the

number of holomorphic curves inside the Kahler

manifold K. Thus, the topological  -model has a

close relationship with enumerative problems in

algebraic geometry, that is, Gromov–Witten invar-

iants and quantum cohomology theory .

When the topological  -model is coupled to topolo-

gical gravity, the BRST-invariant observables are given

by 

(

)  

 

,where

are cohomology classes

of K. Correlation functions are defined as

i¼1



ð

g;d



MðK;dÞ

g;s

i¼1

ðL

^ e



ð

Þ½62

Here



g, s

(K; d) denotes the (stable compactification

of) moduli space of degree d holomorphic maps

to K from genus g Riemann surfaces . e



is the

pullback of the evaluatio n map e

:(f ; x

, ..., x

) 2



g, s

(K; d) ! f (x

) 2 K by f where f is a holo-

morphic map. Correlation functions [62] give

topological (symplectic) invariants of the manifold

K. In the cases n

= 0(i = 1, ..., s), they are known as

Gromov–Witten invariants.

Equation [62] is nonvanishing if the selection rule

i¼1

ðn

þq

Þ¼dim M

g;s

ðK; dÞ

¼ c

ðKÞd þð3 dim KÞðg 1Þþs ½63

is obeyed, where q

is the degree of cohomology

class 

and c

(K) is the first Chern class of the

tangent bundle of K.

We see that there is a close parallel between the

topological -model and (p, 1) topological gravity.

If we formally set q

= ‘

=p, c

(K) = 0, and dim K =

(p  2)=p, eqn [63] agrees with eqn [25]. Based on this

analogy, Eguchi, Hori, and Xiong proposed the

Virasoro conjecture [8], that is, generating functions

of the number of holomorphic maps to arbitrary

Kahler manifolds are annihilated by the Virasoro

operators which are constructed by taking an analogy

with those of (p, 1) gravity. The Virasoro conjecture is

a natural generalization of Witten’s conjecture, and

has recently been rigorously proved in the case of

curves and projective spaces.

Excellent reviews on the theory of 2D topological

gravity are given in Witten (1991) and Dijkgraaf (1991).

See also: Axiomatic Approach to Topological Quantum

Field Theory; Large-N and Topological Strings; Mirror

Symmetry: A Geometric Survey; Moduli Spaces: An

Introduction; Riemann Surfaces; Topological Sigma

Models; WDVV Equations and Frobenius Manifolds.

Further Reading

Bre´zin E and Kazakov V (1990) Exactly soluble field theory of

closed strings. Physics Letters B 236: 144.

Dijkgraaf R (1991) Lectures Presented a t Carge´ se Summer

School on New Symmetry Principles in Quantum Field

Theory, hep-th/9201003.

Dijkgraaf R, Verlinde E, and Verlinde H (1991) Loop equations

and Virasoro constraints in non-perturbative 2d quantum

gravity. Nulcear Physics B 348: 435.

Douglas M and Shenker S (1990) Strings in less than one dimension.

Nuclear Physics B 335: 635.

Eguchi T and Yang S-K (1990) N = 2 superconformal models

as topological field theories. Modern Physics Letters A 5: 1693.

Eguchi T, Hori K, and Xiong C-S (1997) Quantum cohomology

and Virasoro algebra. Physics Letters B 402: 71.

Fukuma M, Kawai H, and Nakayama R (1991) Continuum

Schwinger–Dyson equations and universal structures in two-

dimensional quantum gravity. International Journal of Mod-

ern Physics A 6: 1385.

Gelfand IM and Dikii LA (1975) Asymptotic behavior of the

resolvent of Sturm–Liouville equations and the algebra of the

Korteweg–de Vries equations. Russian Mathematical Surveys

30(5): 77–113.

Gross D and Migdal A (1990) Non-perturbative two dimensional

quantum gravity. Physical Review Letters 64: 17.

Kontsevich M (1991) Intersection theory on the moduli space of

curves and the matrix Airy function. Communications in

Mathematical Physics.

Witten E (1988) Topological sigma models. Communications in

Mathematical Physics 118: 411.

Witten E (1990) On the topological phase of two dimensional

gravity. Nuclear Physics B 340: 281.

Witten E (1991) Two-dimensional gravity and intersection theory of

moduli space. Surveys in Differential Geometry 1: 243–310.

270 Topological Gravity, Two-Dimensional

Topological Knot Theory and Macroscopic Physics

L Boi, EHESS and LUTH, Paris, France

Introduction to the Physical and

Mathematical Contexts and Issues

One of the most exciting developments of mathema-

tical physics in the last three decades has been the

discovery of numerous intimate relationships between

the topology and the geometry of knot theory and the

dynamics of many domains of ‘‘classical’’ and ‘‘new’’

macroscopic physics. Indeed, complex systems of

knotted and entangled filamentary structures are

ubiquitous in nature and arise in such disparate

contexts as electrodynamics, magnetohydrodynamics,

fluid dynamics (vortex structures), superfluidity,

dynamical systems, plasma physics, cosmic string

theory, chaos of magnetic flows and nonlinear

phenomena, turbulence, polymer physics, and mole-

cular biology. In the recent years, mathematical tools

have been developed to identify and analyze the

geometrical and topological complex structures and

behaviors of such systems and relate this information

to energy levels and stable states.

The influence of geometry and topology on

macroscopic physics has been especially fruitful in

the study and comprehension of the following topics.

1. Knots and braids in dynamical systems.Itis

now clear that the chaotic behavior of the He´non–

Heiles system and other nonlinear systems is driven

and controlled by topological properties. For example,

it has been found that trajectories in the phase space

form hyperbolic knots. The finding of knots in the

Lorenz equations is another important theme closely

related to the previous. By varying the Rayleigh

number r, a parameter in the Lorenz equations, both

chaotic and periodic behavior is observed. In the recent

years, the knots (notably several torus knots) corre-

sponding to the different periodic solutions of the

system have been found and classified. By finding

hyperbolic knots and in particular hyperbolic figure-8

knot as a solution to the Lorenz equations the

suspicion that there exists a new route to chaos

would be strengthened.

2. Topological structures of electromagnetic fields.

Progress in the field of space physics, astronomy, and

astrophysics over the last decade, increasingly reveals

the significance of topological magnetic fields in these

areas. In particular, the interaction of plasma and

magnetic field can create an astonishing variety of

structures, which often exhibit linked and knotted

forms of magnetic flux. In these complex structures of

the fields, huge amounts of magnetic energy can be

stored. It is, however, a typical property of astro-

physical plasmas, that the dynamics of magnetic fields

is alternating between an ideal motion, where all forms

of knottedness and linkage of the field are conserved

(topology conservation), and a kind of disruption of

the magnetic structure, the so-called magnetic recon-

nection. In the latter, the magnetic structure breaks up

and reconnects, a process often accompanied by

explosive eruptions, where enormous amounts of

energy are set free. Magnetic reconnection is in close

analogy to splitting of knots, which makes us

confident that the global dynamics of magnetic and

electromagnetic fields can be characterized with the

help of such topological quantities as well.

3. Knotting and unknotting of phase singularities.

It has long been known that dislocation lines can be

closed, and recently it was shown that they can be

knotted and linked. Moreover, Berry and Dennis

(2001) constructed exact solutions of the Helmhotz

equation representing torus knots and links; in fact,

a straightforward application of this idea led to

knotted and linked dislocation lines in stationary

states of electrons in hydrogen. As a parameter,

called , is varied, the topo logy of dislocation lines

can change, leading to the creation of knots and

links from initially simple dislocation loops, and the

reverse process of unknotting and unlinking. The

main purpose here is to elucidate the mechanism of

these changes of topology. All waves are solutions of

monochromatic wave equations, that is, stationary

waves, and  is an external parameter that could be

manipulated experimentally. However,  could

represent time, and then the analogous solutions of

time-dependent wave equations would describe

knotting and linking events in the history of waves.

The methods of Berry and Dennis are based on exact

stationary solutions of wave equations, and lead to

knots and links threaded by multistranded helices.

The Origins of Topological Vortex

Dynamics Ideas

The intimate relationship between three-dimensional

vortex dynamics and topology was recognized as early

as 1869 by W Thomson (Lord Kelvin) who tried to

elaborate a theory of matter in which atoms were

thought to be tiny vortex filaments embedded in an

elastic-like fluid medium, called ether. Accordingly,

the infinite variety of possible chemical compounds

was given by the endless family of topological

Topological Knot Theory and Macroscopic Physics 271

combinations of linked and knotted vortices. Kelvin

was inspired by the work of Gauss, who in an attempt

to describe topologically the behavior of two insepar-

ably closed linked circuits carrying electric current,

found a relationship between the magnetic action

induced by the currents and a pure number that

depends only on the type of link, and not on the

geometry: this number is the first topological invariant

now known as the linking number.

In modern mathematical terms, Gauss introduced

an invariant of a link consisting of two simple closed

curves 

, 

in R

, namely the signed number of turns

of one of the curves around the other, the linking

coefficient {

, 

} of the link. His formula for this is

N ¼f

;

4



ð½d

ðtÞ; d

ðtÞ;

 

Þ=

j

ðtÞ

ðtÞj

½1

where [ , ] denotes the vector (or cross) product of

vectors in R

and ( , ) the Euclidean scalar product.

Thus, this integral always has an integer value N.If

we take one of the curves to be the z-axis in R

and

the other to lie in the (x, y)-plane, then the formula

[1] gives the net number of turns of the plane curve

around the z-axis. It is interesting to note that the

linking coefficient [1] may be zero even though the

curves are nontrivially linked. Thus, its having

nonzero value represents only a sufficient condition

for nontrivial linkag e of the loops. This last

consideration leads naturally to the mathematical

concepts of knots and links whose most striking

properties have been investigated in our introduc-

tory article (see Mathematical Knot Theory).

The other source of inspiration of Kelvin’s theory

of matter was the Helmholtz’s laws of vortex

motion, which state that in an ideal fluid (where

there is no viscosity) vortex lives forever: two closed

vortex rings, once linked, will always be linked. The

classical results obtained by Helmholtz are basic to

understanding the dynamics of Euler motion s. The

vorticity of a velocity field is its curl and is denoted

(z):= curl(X(z, t)). In two dimensions, the vorticity

is a real-valued function and !

= , where  is

the stream function of X(z, t). Recall that the push-

forward of a scalar field (0-form) s under a

diffeomorphism f is f



s = s  f

1

. These results, in

modern terms, can be stated as follow s:

Theorem (Helmholtz–Kelvin). An incompres sible

fluid motion (M

, 

) with velocity field X and

vorticity !

is Euler if and only if its vorticity is

passively transported,





¼!

and circulation around all smoot h simple closed

curves C are preserved under the flow,



ðCÞ

X  dr ¼0

One knows that in three dimensions, the Helmholtz–

Kelvin theorem says that the vorticity (now a vector

field) is transported. Thus, with generic initial

vorticity a 3D time-periodic Euler fluid motion

preserves a nontrivial vector field. One very interest-

ing question that remains to be elucidated is the

following: are there any chaotic, time-periodic Euler

flows with stationary boundaries?

The Connection between Topological and

Numerical Invariants of Knots and the

Physical Helicity of Vector Fields

The writhing number of a curve in Euclidean three-

dimensional space is the standard measure of the

extent to which the curve wraps and coils around

itself; it has proved its importance for electrody-

namics and fluid mechanics in the study of the

knotted structures of magnetic vortices and

dynamics flows, and for molecular biologists in the

study of knotted duplex DNA and the enzymes

which affect it. The helicity of a divergenceless

vector field defined on a domain in Euclidean

3-space, introduced by Woltjer in 1958 in an

astrophysical context and coined by Moffat in

1969 in the study of its topological meaning, is the

standard measure of the extent to which the field

lines wrap and coil around one another; it plays

important roles in fluid mechanics, magnetohydro-

dynamics, and plasma physics. The ‘‘Biot–Savart

operator’’ associates with each current distribution

on a given domain the restriction of its magnetic

field to the domain. When the domain is simply

connected, the divergence-free fields which are

tangent to the boundary and which minimize energy

for given helicity provide model s for stable force-

free magnetic fields in space and laboratory plasmas;

these fields appear mathematically as the extreme

eigenfields for an appropri ate modification of the

Biot–Savart operator. Information about these fields

can be converted into bounds on the writhing

number of a given piece of DNA.

Recent researches (Cantarella et al. 2001)

obtained rough uppe r bounds for the writhi ng

number of a knot or link in terms of its length and

thickness, and rough upper bounds for the helicity

of a vector field in terms of its energy and the

geometry of its domain. It was also showed that in

the case of classical electrodynamics in vacuum, the

272 Topological Knot Theory and Macroscopic Physics

natural helicity invariant, called the electromagnetic

helicity, has an important particle meaning: the

difference between the numbers of right- and left-

handed photons. Recently, a topological model of

classical electrodynamics has been proposed in

which the helicity is topologically quantized, in a

relation that connects the wave and particle aspects

of the fields (Trueba and Ran˜ ada 2000).

Consider two disjoint closed space curves, C and

, and the Gauss’ integral formula for their linking

number

LkðC; C

Þ¼

4

C C

ðdx=ds dy=dtÞ

 x  y= jx  yjds dt ½2

The curves C and C

are assumed to be smooth and

to be parametrized by arclength. Now the question

is to know what happens to this integral when the

two space curves C and C

come together and

coalesce as one curve C. At first glance, the

integrand looks like it might blow up along the

diagonal of C C

, but a careful calculation shows

that in fact the integrand approaches zero on the

diagonal, and so the integral converges. Its value is

the writhing number Wr(C)ofC defined above:

WrðCÞ¼

4

C C

ðdx=ds dy=dtÞ

 x  y= jx  yj ds dt ½3

Here is the very useful result, due to Fuller (1978).

The writhing number of a knot K is the average

linking number of K with its slight perturbations in

every possible direction:

WrðKÞ¼

4

W2S

LkðK; K þ "WÞdðareaÞ½4

This is helpful for getting a quick approximation to

the writhing number of a knot which almost lies in a

plane; in the example of a trefoil knot, Wr(K)  3.

Here, a very important result must be recalled, a

‘‘bridge theorem,’’ proved by Berger and Field

(1984), see also Ricca and Moffatt (1992), which

connects helicity of vector fields to writhing of knots

and links, and which can be used to convert upper

bounds on helicity into upper bounds on writhing.

Proposition (Berger and Field). Let K be a smooth

knot or link in 3-space and =N(K, R) a tubular

neighborhood of radius R about K. Let V be a

vector field defined in , orthogonal to the cross-

sectional disks, with length depending only on

distance from K. This makes V divergence-free and

tangent to the boundary of . Then the writhing

number Wr(K) of K and the helicity H(V) of the

vector field V are related by the formula

HðVÞ¼FluxðVÞ

WrðKÞ

In the formula, Flux(V) denotes the flux of V

through any of the cross-sectional disks D,

FluxðVÞ¼

V  n dðareaÞ

where n is a unit normal vector field to D.

A key feature of this formula is that the helicity of V

depends on the writhing number of K,butnotany

further on its geometry; in particular, such quantities

as the curvature and torsion of K do not enter into the

formula. Berger and Field actually showed that the

helicity H(V) is a sum of two terms: a ‘‘kink helicity,’’

which is given by the right-hand side of the above

formula, and a ‘‘twist helicity,’’ which is easily shown

in our case to be zero. Their proof assumes K is a knot,

but it is straightforward to extend it to cover links.

Let  be a compact domain in 3-space with

smooth bounda ry @; we allow both  and @ to be

disconnected. Let V be a smooth vector field (where

‘‘smooth’’ means of class C

), defined on the

domain . The helicity H(V) of the vector field V

is defined by the formula

HðVÞ¼

4

 

VðxÞVðyÞx  y=jx  yj

 dðvol Þ

dðvolÞ

½5

Clearly, helicity for vector fields is the analog of

writhing number for knots. Both formulas are

variants of Gauss’ integral formula for the linking

number of two disjoint closed space curves.

In order to understand this formula for helicity,

think of V as a distribution of electric current, and

use the Biot–Savart law of electrodynamics to

compute its magnetic field:

BSðVÞðyÞ¼

4



VðxÞy  x=jy  xj

dðvolÞ

½6

Then the helicity of V can be expressed as an integrated

dot product of V with its magnetic field BS(V):

HðVÞ¼

4



VðxÞVðyÞx y=jy xj

dðvolÞ

dðvolÞ



VðgÞ

4



VðxÞy x=jx yj

dðvolÞ



dðvolÞ



VðyÞBSðVÞðyÞdðvol Þ



V BSðVÞdðvolÞ

Topological Knot Theory and Macroscopic Physics 273

Cantarella et al. (2001) found two very interesting

results.

Theorem 1 Let K be a smooth knot or link in

3-space, with length L and with an embedded

tubular neighborhood of radius R. Then the wri-

thing number Wr(K) of K is bounded by

jWrðKÞj < 1=4ðL=RÞ

4=3

For the proof, see Cantarella et al. (2001).

Theorem 2 The helicity of a unit vector field V

defined on the compact domain  is bounded by

jHðVÞj < 1=2volðÞ

4=3

Let us now give a brief overview of the methods

used to find sharp upper bounds for the helicity of

vector fields defined on a given domain  in

3-space. As usual,  will denote a compact domain

with smooth boundary in 3-space. Let K() denotes

the set of all smooth divergence-free vect or fields

defined on  and tangent to its boundary. These

vector fields, sometimes called ‘‘fluid knots,’’ are

prominent for several reasons: (1) They are natural

vector fields to study in a ‘‘fluid dynamics

approach’’ to geometric knot theory. (2) They

correspond to incompressible fluid flows inside a

fixed container. (3) They are vector fields most often

studied in plasma physics. (4) For given energy

(equivalently minimize energy for given helicity),

they provide models for stable force-free magnetic

fields in gaseous nebulaes and laboratory plasmas.

(5) The search for these helicity-maximizing fields

can be converted to the task of solving a system of

partial differential equations. (6) The fluid knots can

reveal some fundamental and still unknown

mechanisms, which characterize the phenomenon

of phase transition, and in particular the transition

from chaotic (unstable) phases and behaviors of

matter to ordered (stable) ones.

Knots and Fluid Mechanics (Vortex Lines,

Magnetic Helicity, and Turbulence)

The Kelvin’s theory of explaining atoms as knotted

vortices in fluid ether was seminal in the develop-

ment of topological fluid mechanics. The recent

revival (starting in the 1970s) is mainly due to the

work of Moffat, on topological interpretation of

helicity, and Arnol’d, on asymptotic linking number

of space-filling curves. Modern developments have

been influenced by recent progress in the theory of

knots and links.

Influence of Geometry and Topology

on Fluid Flows

Ideal topological fluid mechanics deals essentially

with the study of fluid structures that are

continuously deformed from one configuration to

another by ambient isotopies. Since the fluid flow

map ’ is both continuous and invertible, then

’

(K)and’

(K) generate isotopies of a fluid

structure K (e.g., a vortex filament) for any

, t

} 2 I. Isotopic flows generate equivalence

classes of (linked and knotted) fluid structures. In

the case of (vortex or magnetic) fluid flux tubes,

fluid actions induce continuous deformations in D.

One of the simplest deformations is local stretch-

ing of the tube. From a mathematical viewpoint,

this deformation corresponds to a time-dependent,

continuous reparametrization of t he tube center-

line. This reparametrization (via homotopy classes)

generates ambient isotopies of the flux tube, with

a continuous deformation of the integral curves.

Moreover, in the context of the Euler equations,

the Reidemeister moves (or isotopic plane deforma-

tions), whose changes conserves the knot topology,

are performed quite naturally by the action of local

flows on flux tube strands. If the fluid in (D  K)is

irrotational, then these fluid flows (with velocity u)

must satisfy the Dirichlet problem for the Laplacian

of the stream function ’, that is,

u ¼r’ in ðD  KÞ

’ ¼0

½7

with normal component of the velocity to the tube

boundary u

given. Equations [7] admit a unique

solution in terms of local flows, and these flows are

interpretable in terms of Reidemeister’s moves

performed on the tube strands. Note that boundary

conditions prescribe only u

, whereas no condition

is imposed on the tangential component of the

velocity. This is consistent with the fact that

tangential effects do not alter the topology of the

physical knot (or link). The three type of Reidemeister’s

moves are therefore performed by local fluid flows,

which are solutions to [7], up to arbitrary tangential

actions.

Knotted and Linked Tubes of Magnetic Flux

Let T be the standard solid torus in R

given by

ðð2 þ " cos Þcos ’; ð 2 þ " cos Þsin ’; " sin ÞÞ ½8

where 0  , ’<2, and 0  "<1. For relatively

prime integers p and q, let F

p, q

denote the foliation

274 Topological Knot Theory and Macroscopic Physics

of T by the curves 

", 

(where 0  "  1 and

0  <2) given by



";

ðsÞ¼ð2 þ " cosð þ qsÞÞcosðpsÞ;

ð2 þ " cosð þ qsÞÞsinðpsÞ;"sinð þ qsÞ½9

where 0  s < 2.

Definition A magnetic tubular link (or magnetic

link) is a smooth immersion into R

of finitely many

disjoint standard solid tori [

i = 1

L : [

i ¼1

! R

and a smooth magnetic field B on R

such that

(i) L is an imbedding when restricted to the

interior of [

i = 1

(ii) the bounding surface of [

L(T

), that is,

[

L(@T

) is a magnetic surface, and

(iii) for each component LT

, there exist relatively

prime nonzero integers p

and q

such that L

maps the foliation F

, q

of T

onto the integral

curves of B in LT

Remark Thus, for every fixed i and j, the linking

number between an arbitrary field line in LT

and

an arbitrary field line in LT

is the same regardless

of which integral curves are chosen from LT

and

, respectively. This is true even when i = j.

It follows that a magnetic link [

remains a

magnetic link under the action of the fluid flow, that

is, [

is a magnetic link for t  0.

Keeping that the magnetic field B is frozen in the

fluid, we can now find and study those properties of

magnetic links that are invariant under the action of

fluid flow. One obvious invariant is the volume V

of each flux tube g

, that is,

¼VolðLT

Þ¼Volðg

Þ¼

ZZZ

dðvolÞ½10

which remains unchanged because of incompressibility.

Another invariant of fluid flow is defined as

follows:

Definition Let L be a magnetic link. For each solid

torus T

, choose a meridional disk D

. The magnetic

flux 

=(LT

) in the ith component is the surface

integral defined as



¼ðLT

Þ :

B  U dðareaÞ

where U deno tes the normal to the surface LD

pointing in the positive direction induced by the B

field.

It can be shown that 

is independent of the

chosen meridional disk. It also can be shown that

each 

is a fluid flow invariant, that is,



ðg

Þ¼

B  U dðareaÞ½11

is independent of t.

One more fluid invariant that will play a central

role in the energy minimization of magnetic links is

given by the following definition.

Definition The helicity of a magnetic link L is

defined as

HðLÞ¼

ZZZ

[

A  B dðvolÞ

The term helicity was first introduced in a fluid

context by Moffat, and it was previously used in

particle physics for the scalar product of the momen-

tum and spin of a particle. In another connection, note

that the helicity H(L) is the same as the Chern–Simons

action:

HðLÞ¼

A ^ dA

trðA ^ dA þ

A ^ A ^ AÞ½12

where A now denotes the magnetic vector potential

as a 1-form.

It can be shown that H(L) is gauge invariant, and

hence well defined.

Theorem (Moffat). The helicity is invariant under

fluid flow, that is,

Hðg

LÞ¼0

Arnol’d (1998) defines the helicity in a more abstract

setting and shows that it is invariant under the group

S(Diff) of volume-preserving diffeomorphisms.

The following theorem summarizes the many

results due to Moffat, Ricca, Berger, Lomonaco,

Hornig, Kauffman, and others, relating the helicity

of magnetic links to linking and to magnetic flux.

Theorem Let L be a magnetic link. Then

HðLÞ¼

1in



þ 2

1i<jn



where SL

denotes the self-link ing number of

the axis curve of the tube LT

with respect to the

framing F

induced by the integral curves of the

magnetic field B within LT

, and LK

denotes

the linking number between any integral curve of

Topological Knot Theory and Macroscopic Physics 275

the magnetic field B in LT

with any integral curve

of the magnetic field B in LT

Remark In fact, SL

is the same as the linking

number between any two integral curves of the

magnetic field B within the tube LT

Thus, as many authors have showed, the helicity

does reflect the topology and the geometry of the

magnetic lines of force within a magnetic link. If, for

example, L has only one component, that is, L is a

magnetic knot, then

HðLÞ¼

ðCÞ½13

where SL

curve C of the knotted tube with respect to the

framing F induced by the integral curves of the

magnetic field B within the magnetic knot. If, for

example, the tube is knotted in the form of a trefoil

and if the magnetic lines of force appear to be

parallel to the axis curve when the trefoil is placed

on a plane flat surface, then SL = 3 and

H ¼3

½14

On the other hand, if for example, the magnetic

lines of force induce the trivial framing in each

component, then

HðLÞ¼2

1i<jn



½15

Thus, if L is a magnetic two-component Hopf link

with no twisting of the integral curves of the

magnetic field within the components of L, then

HðLÞ¼2



½16

because the self-linking number based on the B-field

framing is zero for each component, and the linking

number between the two components is 1.

Energy of Magnetic Knots and Links

Let us conclude this section with the definition of

the energy of a magnetic link.

Definition The magnetic energy E

(L) of a mag-

netic link L is defined by the classical formula

ðLÞ¼

8

ZZZ

[

jBj

dðvolÞðGaussian unitsÞ

Although the energy E

is not flow invariant, it will

play a central role in magnetic relaxation of knots

and minimum energy magnetic links.

Consider a magnetic link L in a perfectly

conducting, incompressible, viscous fluid. As a result

of dissipative frictional fluid forces, the magnetic

energy E

L)ofg

L will decrease with time t.In

losing energy, the magnetic lines of force will

contract. On the other hand, since this is a volume-

preserving process, the cross sections of the flux

tubes of g

L will at the same time expand. These

changes of topology occur while the flux , volume

V, and helicity of g

L will remain the same. In other

words, knotted magnetic flux tubes left free to

evolve in such a fluid will do so by conserving their

magnetic flux  and volume V, but converting their

magnetic energy into kinetic energy, which in turn

dissipates by internal friction. Magnetic links and

knots evolve from high to low magnetic energy

levels, conserving topology; and because of the

induced shortening of field lines under conservation

of volume, they become fatter and fatter, with an

increase of the average tube cross section.

This process cannot continue indefinitely. Even-

tually, the magnetic flux tubes of g

L must make

contact with each other. In other words, the topology

of the magnetic link g

L, as expressed in knotting and

linking, creates a barrier to the full dissipation of the

magnetic link’s energy, that is, E

L) has a positive

lower bound that results from the topology of g

That means, in other words, that relaxation is

obstructed by the knottedness and entanglement of

the field lines, and a minimum magnetic energy is

reached. Thus, the magnetic link will reach a

nontrivial stable and invariant energy state, much as

Kelvin conjectured his atomic vortices would.

Various estimates of magnetomechanical energy in

terms of topological quantities have been put forward

in recent years (see Freedmann and He (1994)). These

relations give lower bounds for the energy levels

attainable by knot or link types by taking into account

the effects that linking numbers and number of

crossings have on the energy of the relaxed state.

These bounds are expressed by relationship of the kind

min

 ðC

min

; ; V; NÞ½17

where E

min

is the equilibrium energy and  gives the

relationship between physical quantities – such as

total flux ,numberoftubesN,magneticvolumeV –

and topology, given here by the minimal possible

number of crossing C

min

. These relations offer

numerous advantages due to the explicit dependence

on qualitative properties of the flow field. A simple

example is provided by the analysis of three braids,

which shows that magnetic energy grows quadrati-

cally in time due to random braiding. This means

that the least possible amount of magnetic energy

that can be attained by the physical knot or link is

determined purely by its topology. If topological

information sets the levels of minimum energy

accessible to the knot or link, geometric properties

276 Topological Knot Theory and Macroscopic Physics

may also influence the relaxation process. Consid-

erations of helicity and linking numbers, for

example, demonstrate that internal rearrangement

of magnetic field geometry leads to a spectrum of

different asymptotic endstates with the same topol-

ogy. Moreover, magnetic knots have a natural

tendency to get rid of excessive torsion of field

lines and S-shaped tube geometries, and this may

influence the relaxation process.

Since the helicity H(g

L) is both an invariant of

fluid flow and an expression of the magnetic link

L’s topology, the following theorem, first stated by

Moffat, is a mathematical expression of this

topological bounds.

Theorem Let L be a magnetic link. Then

ðLÞq

jHðLÞj

where q

is a nonzero constant that is independent

of the magnetic link.

Freedman and He (1991) obtain more subtle and

tighter topological bounds on the minimum energy

of magnetic links. For example, for a magnetic knot

K, they prove that

ðKÞ

4

5=4

ðKÞ

3=2

acðKÞ

3=4

VðKÞ

1=3



4

5=4

ðKÞ

3=2

ð2gðKÞ1Þ

VðKÞ

1=3

ðGaussian unitsÞ½18

where V(K) denotes the volume of the magnetic knot

K, (K)denotesthefluxinK,ac(K) is the asymptotic

crossing number, and g(K) is the genus of the knot K.

Freedman and He conjecture that ac(K) = c(K ), where

c(K) is the crossing number, that is, the minimum

number of crossings among all plane diagrams repre-

senting the knot K. Besides, Moffat (1990) suggests

that the minimum energy spectrum of a magnetic knot

can be used to construct new knot invariants.

Topological Changes, Dissipation,

and Reconnection in Fluid Patterns

As we saw above, topological changes do occur

when dissipative effects become predominant over

the coherency of structures. When this happens,

there is a dramatic change of fluid patterns, often on

small timescales compared to evolution. The change

occurs through the formation and disappearance of

physical reconnections in the fluid pattern. In real

fluids, for example, vortex and magnetic tubes do

interact and reconnect freely. From a dynamical

system viewpoint, reconnection s take place when the

vector field lines (streamlines, vortex lines, or

magnetic lines) cross each other. If two field lines

meet, the point of crossing is a true nodal point, like

a bifurcation in a path. Dissipative effects allow the

reconnection to proceed through such points.

In dissipative fluids, mathematical and physical

properties are no longer conserved, and during the

process we lose part of the original information.

However, some of the invariants are rather robust

and may only degrade slowly. One of them is magnetic

helicity, the magnetic analog of the kinetic helicity. Its

dissipation during reconnection can be modest; in

particular, if the reconnection timescale is small

compared to classical dissipation times, then helicity

loss will be negligible. The robustness of magnetic

helicity plays a central role in fusion plasma physics

and in many astrophysical contexts. On the other hand,

large changes in kinetic helicity are intimately related to

qualitative changes in the topology of vortex flows.

Under Euler’s equations, the helicity of a vortex

tube of vorticity ! and velocity u is defined by

H =

u  ! dV. The integral is taken over the tube

volume V occupied by !. Now, for n knotted and

linked vortex tubes, each of (constant) strength

(total vorticity) 

(1  i  N), the helicity of the

whole system can be expressed in terms of linking

numbers Lk

H ¼



Lk

which is equal to Lk

; this is a topological invariant

whose value does not change under continuous

deformation of the fluid structure. Since helicity

and flux-tube strength are measurable conserved

quantities, the above equation provides useful

information about the topology of the flow field

and flow structures. In addition, by direct measure-

ments of helicity and application of conservation of

topology, one can estimate average geometric

quantities, such as the mean twist of field lines,

and their contribution to the total energy.

Brief Conclusion

In this article, we have made an attempt to indicate

how ‘‘classical’’ field theories, which have been

successfully used to describe physics of fundamental

structures and forces of nature, can also be used to

study geometry and topology of low-dimensional

manifolds. These developments not only provide new

insights into old problems of topology of these

manifolds but also have been responsible for pro-

foundly interesting new mathematics (fluid

mechanics, dynamical flows, and polymer biophysics

Topological Knot Theory and Macroscopic Physics 277