Borromean Torus

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Asymptotic Massey products, induced currents and Borromean torus links

Peter Laurence and Edward Stredulinsky

Citation: Journal of Mathematical Physics 41, 3170 (2000); doi: 10.1063/1.533299

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Asymptotic Massey products, induced currentsand Borromean torus links

Peter Laurencea)

Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-

1185 and Dipartimento di Matematica, University of Roma ‘‘La Sapienza,’’Piazzale Aldo Moro 2, 00199, Roma, Italy

Edward Stredulinskyb)

Department of Mathematics, University of Wisconsin– Richland, Richland Center,Wisconsin 53581

Received 9 July 1999; accepted for publication 5 January 2000

We introduce a class of currents which allows a new and very explicit form for the

Massey product of a third order link as a line integral. The explicit form permits the

introduction of an asymptotic Massey product analogous to that introduced previ-

ously for Gauss’s integral by V. Arnold. The average third order asymptotic Mas-

sey product is shown to be equal to Berger’s third order helicity for divergence-free

vector fields in linked tori. © 2000 American Institute of Physics.

S0022-24880002105-8

I. INTRODUCTION AND MAIN RESULTS

An integral formula for linking numbers was discovered by Gauss. His formula for the linking

number of two closed curves C 1 and C 2 is

LK C 1 ,C 2 1

4

C 1

C 2

1 l 1• 2 l2 X l1 X l2

X l1 X l 23 dl1 d l2 , 1

where i is a unit vector along C i , i1,2. The analog for curves that are linked in a higher order

fashion, such as the Borromean rings see Fig. 1 came much later. The Borromean rings are

perhaps the simplest example of a three component link in which the linking numbers of any twocurves is zero but such that the link is nonetheless nontrivial. In 19561 Massey introduced an

analog of the linking number which equals 1 for three curves linked as in the Borromean rings,

and 0 for three curves that are unlinked. Massey’s formula in its original form involves an integral

of a divergence-free vector field over the boundary of a tubular neighborhood of one of the curves,

or in more geometric language, of a certain representative of the cohomology class of the comple-

ment in R3 or S 3 of the tubular neighborhood of the link. This invariant was little known outside

the algebraic topology community until Monastyrsky and Retakh Ref. 2 in 1985, and Berger

Ref. 3 in 1990, presented and interpreted it in a manner accessible to nontopologists. For a nice

exposition we also recommend the recent book by Arnold and Khesin.4

A. Curves

An integral expression on a tubular neighborhood of a curve is different than a line integral

over the curve itself and, to our knowledge, the first to suggest the latter in this context were Evans

and Berger in Ref. 5. We will rigorously derive their expression from Massey’s formula in Sec. II

by using Stokes’s theorem. Denoting by C i( X ),i1,2,3, the solid angle subtended by the curve

C i from the viewpoint X , the result is

aElectronic mail: [email protected] mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 5 MAY 2000

31700022-2488/2000/41(5)/3170/22/$17.00 © 2000 American Institute of Physics


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C 1

“1 AC 2AC 3 14 C 3AC 2 • d S , 2where the fundamental forms AC i

are defined below in 3 With a line integral expression for the

Massey product in hand, we are well on our way to having a generalization of the Gauss linking

integral applicable to higher order links. But there is a key missing element. The expressions in2

still involve quantities that, compared to 1, are not expressed in an explicit way in terms of the

curves themselves. Indeed in 1 the expression involves only the distance between points on the

curves and the tangent vector . On the other hand, the expressions “1( AC 2AC 3) appearing

in 2 depend a priori in a complicated way on the curves, since, given that the field AC 2AC 3, which is divergence-free in the complement of the tubular neighborhoods, has a nonzero

normal component on the boundary, the inverse curl is not given by the classical Biot–Savart

potential, but rather, a priori, involves solving boundary value problems on this domain for the

Laplacian, which provides a much less geometrically explicit description. Another approach,

which is the one taken by Berger in Ref. 3, is to find a globally divergence free extension of the

field AC 2AC 3which is defined only outside a tubular neighborhood of the link to which the

Biot–Savart formula can be applied. Actually the inverse is not, and should not, be expected to be

global in the case of curves. In Berger’s case the inverse is global because his ‘‘ Ai’s’’ are vector

potentials for nonsingular magnetic fields. In the case of curves we will see that the inverse is

strictly speaking, an inverse in the classical sense, only in R2 i13

C i . Nonetheless in Sec. III

see formula 18 we give an explicit expression for the singular part of the curl of our inverse

which is as a measure supported on the curves. Recall also that an inverse such as that given by

specializing the constructive part of Poincarè’ s formula to two-forms is not possible, since our

domain is not star shaped.

The approach we will take to finding an inverse that is as explicit as possible is motivated by

the idea due to Berger mentioned above. Berger found a continuation of the vector field AC 2AC 3 into T 2 and T 3 where T i is a tubular neighborhood of C i as a globally defined divergence-

free vector field. Our expressions for the inverse curl are related to Berger’s in the following way:

FIG. 1. The standard Borromean rings.

3171J. Math. Phys., Vol. 41, No. 5, May 2000 Asymptotic Massey products, induced currents . . .


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As he does, we continue the vector field AC 2AC 3 into the interior of the tubes. But we use a

different divergence-free extension closely connected to the Frenet triad of the curves C 2 and C 3 .

We then renormalize in an appropriate way and consider the limit as the radius of the tubes tends

to zero. Our expression for the inverse curls can then be obtained as pointwise almost everywhere,

but highly nonuniform close to the curves themselves, limits of a family of these specialdivergence-free extensions. However, pointwise convergence does not suffice to guarantee the

distributional identity we are after, and, in fact, we were unable to push through the proof using

the intuitive extension-blowup argument. The interested reader can find this argument in Ref. 6. A

rigorous proof that the currents obtained really are inverse curls of the field AC 2AC 3 takes a

different, more direct but less intuitive approach, and is given in Sec. III. Our expression for the

inverse is

A2,3 X

AC 2AC 3K X ,Y dY C 3

C 2 Y K X ,Y dlC 2

C 3 Y K X ,Y dl,

where K is defined in 5. If we now plug this expression for the inverse into 2, we obtain a new

and very geometrical form for the Massey product of three curves. The expression obtained

involves solid angles C i,i1,2,3, subtended by curve C i , i1,2,3. In this sense, too, it consti-

tutes a natural generalization to third order links of the connection that exists between Gauss

linking number and solid angle. Indeed, recall that the fundamental one-forms

AC i X

1

4

C i

X Y

X Y 3 dl 3

have the property that

C i4 AC i

for all X C i . 4

On the other hand, in our formula 3, solid angle plays the role of a weight on the same kernel

( X Y )/ X Y 3 that appears in the fundamental one-forms. Also, for any family of curves, weshow that these solid angles in turn have an expression as line integrals. Note that in the case of

a Borromean link of tori, as mentioned in the next paragraph, potentials needed to give a volume

integral representation for the third order helicity admit a representation as a very geometric

integral, subject only to certain restrictions that the three tori do not get too ‘‘wild.’’ This will be

studied in Sec. IV.

B. Links of tori and asymptotic third order linking number

In 1974 Arnold7 introduced the notion of asymptotic linking number. This notion provides a

natural extension of Gauss linking numbers to the context where the trajectories of a divergence-

free vector field are not closed. Using Birkhoff’s ergodic theorem, Arnold showed that the space

average of the asymptotic linking number equals the helicity. An application of Arnold’s formulawas given in Ref. 8 to show that the mutual helicity of linked tori is equal to the linking number

of the core curves of the tori times the product of the fluxes of the magnetic fields. Analogously,

denoting the third order helicity of three magnetic fields Bi that leaves tori T i invariant by

H(B1 ,B2 ,B3), we have

H B1 ,B2 ,B3 i1

3

F iM C 1 ,C 2 ,C 3,

where M is the third order Massey product of the curves, which when applied to the a Borromean

link of tori becomes H(B1 ,B2 ,B3)F 1F 2F 3 .

3172 J. Math. Phys., Vol. 41, No. 5, May 2000 P. Laurence and E. Stredulinsky


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In Sec. IV, using the explicit geometric form of the Massey product for curves, we introduce

an asymptotic version of the Massey product and then demonstrate a relation between a volume

averaged form see 14 of this asymptotic invariant and the third order helicity of the link. This

relation is the analog of the one that exists in the Gauss linking number context, between volume

averaged asymptotic linking number and ordinary helicity. We present such a generalization fora certain subclass of Borromean torus links, those that satisfy a translation property that is best

expressed in terms of an admissible cone of translations see Sec. III B. This subclass is quite

broad but excludes, roughly speaking, Borromean links of tori for which the tori wrap around each

other in certain pathological ways and/or have tentacles that protrude in all possible directions.

The technical reason for such a specialization is simple. The line integral definition of Massey

product involves the multi-valued potentials i . The standard definition of these as line integrals

choosing some base point of the fundamental one-forms Ai does not interact well with ergodic

theorems since it introduces an additional line integral into these expressions, which is not a line

integral along the trajectories of a divergence-free vector field. We are not aware of ways to get

around this difficulty in full generality. Thus we look for a class of Borromean links of tori for

which the potentials solid angles can be expressed as volume integrals of explicit kernels, i.e., an

analog of the Biot–Savart potential for scalar potentials. We show that such a representation is

possible provided that the tori satisfy the translation invariance property mentioned above.A starting point for our investigation was to ask whether the availability of an explicit formula

such as Massey’s could, in the case of Brunian links, provide complementary and possibly sharper

inequalities to those obtained by Freedman and He in Ref. 9. In a companion paper,10 we use the

new form of the third order helicity see 13 to derive the following lower bound on the magnetic

energy of magnetic fields normalized by their flux:

1

16i1

3

E 3/22/3

Bi 16 2/3

j1,2,3

E 3/22/3

B j i1,2,3

E 3/22

Bi ,where (E 3/2)

2/3 denotes the L3/2 norm.

Finally, an obvious next step is to consider higher order Massey products. The methods and

results carry over in a straightforward way for fourth order links. In the case of n th order Massey

products, we only conjecture the same to be true. We gratefully acknowledge the permission of Rob Scharein at UBC to adapt his beautiful images in the figures. Any defects in these figures are

due to our editing. The originals can be found on his KnotPlot site at the University of British

Columbia, VC, computer science department.

II. PRELIMINARIES

A. The fundamental one-forms

Throughout this paper we will denote by K( X ,Y ) the kernel

K X ,Y 1

4

X Y

X Y 3

1

4 X

1

r ,

where r X Y . Given a C 1 curve C we can associate to it the following one-form

AC X C Y K X ,Y dl Y ,

where is a unit tangent vector. AC ( X ) is a harmonic form in the complement of the curve, i.e.,

•AC “AC 0 in R3 C . Its distributional curl has a singular part that is supported on the

curve and for which one can give an explicit expression. This is a good warm-up for the more

complicated expressions we derive in connection with the Massey product and so we give it here.

We have



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AC C

• d l, 5

where is a vector test function and we use the notation to denote the action of a distri-

bution D

(R3

) on the test function . In a more geometric form, denoting the one-dimensional Hausdorff measure supported on the curve C by H1C , we have “AC ( X )

( X )d H1C , where is a unit vector on the curve C . The proof is a good warm up for the more

complicated derivations in Sec. III. The interested reader may find it in Ref. 6.

B. The Massey product for curves

Given three C 1 closed curves C i , i1,2,3, whose Gauss linking numbers LK(C k , C l), k

l, are zero, one can define the Massey product associated to these curves as follows.1– 3 Let

U i( i) be small tubes of radius i around the curves C i , and for i1,2,3 using the convention

i11, for i3, define Ai ,i1 by

Ai ,i1“1

AC i

AC i1. 6

Let

1 , 2 , 3R3 i1,2,3U i i.

Note that AC i, i1,2,3, represent cohomology classes in Rot( 1 , 2 , 3) , i.e., in

H 1( 1 , 2 , 3:R) , and AC iAC j represents a cohomology class in Sol( 1 , 2 , 3), i.e., in

H 2( 1 , 2 , 3:R). Put simply, A i,i1 is not uniquely defined, being defined up to a single-valued

gradient, and A i,i1 is not uniquely defined, being defined up to the curl of a single valued vector

field. The Massey product associated to the cohomology classes in the complement of 1 , 2 , 3 is

defined as follows. Let

V i(1 )AiAi1,i2

and

V i(2 )Ai2Ai ,i1 ,

and define the divergence-free vector field V i by

V iV i(1 )V i

(2 ) 7

V i is divergence-free and represents an element in the cohomology class in Sol( 1 , 2 , 3). The

Massey product is given by any one of the following surface integrals:

U i( i)V i•ndS , i1,2,3. 8

C. The third order helicity associated to divergence-free vector fields

Suppose that instead of the three curves C i , i1,2,3, we are given three tori which cannot

necessarily be expressed as ‘‘tubes’’ around the curves C i , but are linked in the same fashion. We

call this a Borromean link of tori. More precisely, we have the following.

Definition 1: Let T S (T 1¯ ,T 2¯ ,T 3¯ ) denote the standard Borromean link of tori. A Borromean

link of tori T (T 1 ,T 2 ,T 3) is a link of three tori such that there exists a diffeomorphism fromR3

to R3 carrying T S to T .



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Equivalently, the tori T i , i1,2,3, are smooth and unknotted , any two pairs of tori ( T j ,T k ) are

unlinked, and the axes of the tori form a standard Borromean link.

Definition 2: Given a Borromean link of tori T S and a divergence-free vector field B

i13 Bi , where Bi is zero in the complement of T i and has zero normal component on T i let Ai

be a globally defined vector potential for Bi . Then we define the third order helicity of B,

H(B1 ,B2 ,B3), by

H B1 ,B2 ,B3 T i

V i•n dS ,

where the V i were defined in Eq. (7).

Thus the third order helicity is the result of applying the expression for the Massey product to

the vector potentials of the magnetic fields Bi instead of to the fundamental one-forms AC iassociated to the curve C i .

Remark: The third order helicity is well defined for L1 vector fields. Moreover, when these

vector fields Bi are divergence-free in the sense of distributions, and have zero normal component

in the sense of traces

on the boundary of T i , the fundamental formula

14

holds true. This canbe easily shown using a standard approximation argument.

An example of a nonstandard Borromean link, still with nonzero Massey product, but with

two of its components linked as in a Whitehead link, is illustrated in Fig. 2.

In the case of tori one may, following Berger,3 consider as the fundamental object a diver-

gence free-vector field B(B1 ,B2 ,B3) with the property that B i is supported in T i and is tangen-

tial on the boundary of T i . Then we may consider the cohomology in the complement of any one

of the tori T i . As particular representatives of the cohomology class, we may consider the Biot–

Savart potentials A iBiot of the magnetic field B i and, in the definition of the Massey product given

earlier see 7 and 8, we use ABiot instead of AC i. Berger’s contribution was to show how the

resulting expression can be transformed into a volume integral in T i , i1,2,3. To achieve this

FIG. 2. A third order Brunnian link with two components forming a Whitehad link.



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goal he found divergence-free extensions of vector fields of the form A iA j into the tori T i and

T j , in such a way that the extended globally defined vector field were globally divergence-free in

the distributional sense.

1. The Biot –Savart potential

Given a divergence-free field B, the Biot–Savart vector potential is defined by

ABiot B X 1

4

R3

B Y X Y

X Y 3 dY

R3

B Y K X ,Y dY ,

It satisfies

“AbiotB.

In order to find divergence-free extensions of vector fields of the form AiA j to simplify

notation we drop the superscript ‘‘Biot’’ when there is no ambiguity into tori T i and T j , in such

a way that the extended globally defined vector field is globally divergence-free in the distribu-

tional sense, it suffices to ensure that the normal component of the globally defined vector fieldvaries continuously as we cross the boundary of the tori T i , T j .

D. Extension „Berger’s extension…

We use the convention that i11 when i3, i.e., that sums are calculated mod 3. We

define the globally divergence free vector field F i,i1 by

F i,i1 AiAi1 iBi1 for X T i1 ,

AiAi1 i1Bi for X T i ,

AiAi1 elsewhere,

where i is any single valued in T i solution of iAi . Note that the free constant in the

definition of i does not affect the value of the integral above since the mutual helicity T 3A1•B3 is zero. It is possible to find other extensions into T i . See Ref. 6.

E. Transformations of the expression for the Massey product into a volume form

Consider one of the three equivalent up to permutation of the indices terms T iV i , say with

i1. Given the expression

T 1

A3“1 A1A2•n dS

corresponding to T 1V 1(2 )•n dS , use the divergence theorem, the vector identity •(VW)

W•“VV•“W, and the extensions associated with the fields to transform the latter into

the following volume integral in T 1 :

T 1

A1•F 1,2 dV T 1

A1A2•A3 d V T 1

A1•B1 3 d V . 9

Use the divergence theorem on T 1V 1(1 )•n dS to transform it into a volume integral inside T 1 .

That is we have

T 1

A1•A2A3 d X 1T 1

A1•ABiot F 2,3 dX 1 , 10



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where the second term in 10 may, by the definition of F 2,3 , be written

T 1A1•ABiot F 2,3 dV T 1B1•

R3

A2A3 Y K Y X 1 dY

T 2 3 X 2 B2 X 2K X 2 X 1 dX 2

T 3

2 X 3 B3 X 3K X 3 X 1 dX 3 dX 1 . 11

F. Volume integral form 1 of third order helicity

Now subtracting the result of substituting 11 into 10 from 9 we get the first of two

equivalent expressions for the Massey product as a volume integral. We will refer to the following

volume form as the third order helicity, and denote it by H(B1 ,B2 ,B3):

T 1

B1•¦

R3

A2A3 Y K X 1Y d Y d X 1

T 2

3 X 2 B2 X 2K X 1 X 2 dX 2

T 3

2 X 3 B3 X 3K X 1 X 3 dX 3

T 2

3 X 1B2 X 2K X 1 X 2 § dX 1. 12G. Volume form 2 of third order helicity

Alternatively, we may use Fubini’s theorem i.e., change the order of integration in the

multiple integral

and the definition of the Biot–Savart potential to obtain the second volume form

R

3 A2A3 Y K Y X 1 d Y d X 1

T 1

A3•B1 2 d V T 2

A1•B2 3 d X 2T 3

A1•B3 2 d X 3 . 13This form will turn out to be particularly useful in obtaining lower estimates on the magnetic

energy in Borromean links. We also record the following result proved in Appendix D in Ref. 6.

Theorem 1: Let B(B1 ,B2 ,B3) be an integrable divergence-free vector field supported in a

Borromean torus link T (T 1 ,T 2 ,T 3) and tangential to the boundary of the link. Then we have

H B1 ,B2 ,B3 F 1F 2F 3 , 14

where F i is the flux through torus T i .

H. Line integral form of the Massey product

Starting from the form 15 with i1 for the Massey product, using Stokes’s theorem in the

two-dimensional surface T 1 , after a cut has been introduced to make the potential 1 single

valued recall 1A1, one obtains

C 1

“1 A2A3•dlC 1

3A2•dl. 15



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The details are contained in Ref. 6, Appendices A and B.

III. MAIN RESULTS

A. Definition of an inverse curl and an explicit form for the Massey product

In this section we begin to present our main results. Our aim is to exhibit in explicit form avector potential A2,3

global with the property that for any X in the open set ̂ R3 C 1C 2C 3 onehas

A2,3global

X 2 3 X AC 2AC 3 X in ̂ .

We will show that A 2,3 is given by

A2,3 X

AC 2AC 3 K X ,Y dY C 3

2 Y K X ,Y dlC 2

3 Y K X ,Y dl

:ab 2b3 , 16

where i( X ) i( X )/4 , the solid angle subtended by curve C i viewed from point X and

normalized to have total variation 1 if C i is linked once. In the sequel we will write A iAC i, for

brevity. In analogy with 5 we will establish the following elegant expression for the curl in the

distributional sense:

“A2,3A2A3 d H1C 2 d H1C 3 , 17

or, equivalently, if is a vector test function,

R

3A2,3• dV 18

R3 A2A3• d V C 2 •

dlC 3 • dl. 19

Remark 1: Two remarks are in order concerning (16). The first is that there is a singular contri-

bution given by the last two terms, even when the curl is taken at a point not on the two curves C 2and C 3 . Moreover the distributional curl has a singular part supported on the curves C 2 and C 3 .

The second is that in the proof given below, we assume the curves are C 2. It is not clear what

minimal regularity is needed for the formula to hold .

We now give an exact statement of our theorem.

Theorem 2: Given two closed C 2 curves C 2 and C 3 , and given the two-form formed by the

exterior product of the two one-forms AC 2 and AC 3 associated with the curves we define a current

through the formula (16). Then the distributional curl of (16) is (19). In particular, in

R3 (C 2C 3) we have that “A2,3A2,3 in the classical sense.

An immediate consequence of Theorem 2 is the following form for the Massey productCorollary 1 (Semi-explicit form Massey product): Given three C 2 closed curves with zero

pairwise linking numbers, their Massey product is given by the expression

C 1 R3

A2A3 Y K X ,Y dY C 3

2 X 3K X 1 , X 3 dl3

C 2

3 X 2K X 1 , X 2dl2 •dl1

C 1

3 X 1 C 2

K X 1 , X 2 dl2 •dl1 . 20



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Proof of corollary: The corollary follows immediately by using the line integral form for the

Massey product given at the end of Sec. II, in conjunction with the expressions for the inverse

curls given by Theorem 2.

Proof of Theorem 2: We must show for , a three vector test function with compact support

in R

3

, that

̂

ab 2b3 • dX ̂

A2 X A3 X • d X i1,2, ji

C i

1 i j X •dl.

First note that

R

3 •

R3 A2A3 K X ,Y dY dX lim

→0

R

3 •

( ) A1A2 K X ,Y dY dX

: lim →0

t ,

where for simplicity we let

1 , 2 , 3.

This can be seen as follows. Let B R be a ball that contains the curves C 2 and C 3 , and hence

the singularities of A2A3 . Split the integral over R3R

3 into four integrals over regions: B R B R , B R B R

c , B Rc B R , B R

c B R

c . Since K L 3/2 ( B R B R) for any 0, and A2A3 is in

L loc2 ( B R), the contribution from B R B R is finite by Young’s inequality. The other contributions

are easily seen to be finite due to the decay of A2A3 and K at infinity.

We begin by pulling the operator out from under the integral, i.e.,

( )

A2A3 Y K X ,Y dX X ( )

A2A3 Y

X Y dY ,

where here BA2A3 , and where we have used the identity ( f C) f C f C.

We now use the divergence theorem and the identity

• CDD•CC•D,

which shows that “ is a symmetric operator with respect to the inner product in L 2 when acting

on functions which are zero on the boundary. Note that a sharp characterization of not only the

symmetry property but also the ‘‘maximal’’ closed subspace of solenoidal L 2 fields for which “

can be extended to a self-adjoint operator has been given by Yoshida and Giga in Ref. 11. Thus

we reexpress t ( ) as follows:

R

3 ( ) A2A3 Y

X Y dY • X X dX .

Then use the identity

to obtain from the standard distributional identity

1

X Y 4 X Y .

Then



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t R

3 A2 X A3 X • X dX

R3

( ) X •

A2A3

X Y dY • dX :t 1t 2 .

21

Note that we have legitimately brought the operator underneath the integral sign, since thereis no singularity of A2A3 in the domain ( ). Now set

s ( )

X •A1A2

X Y dY

using the fact that

X •1

X Y Y

1

X Y .

We may integrate by parts using the divergence theorem in the inner integral in s( ) and

exploit the fact that for Y ( ) but not in all of R3

, •(A2A3)0. It is easily seen that theboundary term vanishes at infinity because of the sufficiently rapid decay of A2 and A3 , and thus

we are left with

t 2 R

3 ( )

A2A3•n Y

X Y dS y • X dX .

We now use the fact that (( )) U 2( ) U 3( ) and write for brevity

R

3• X

( )

A2A3•n Y

X Y dS y

i2,3R

3• X

U i

A2A3•n Y

X Y dS y

:R3• X s 2 X , s 3 X , dX .Now calculate the limit as tends to zero of R3• s3( X , ). The limit of R3

• s2( X , ) is treated in the same way.

Applying Stokes’ formula in the two-dimensional surface U 3 we obtain

U 3( )

A2A3•n

X Y dS y 22

U 3( )

3•n Y A2 X Y d S y 23

U 3cut

( ) •

2 Y l

X Y l dl

U 3cut

( ) 3 S • n Y 2 Y X Y d S y 24

:s31

X , s32

X , . 25

For X in R3 C 3 we clearly have that lim →0s 31( X , )0. But, due to the singularity of the kernel

K at C 3 , the limit in the distributional sense is different from zero, as we will see below.

We next treat s 32. To deal with the fact that 1/ X Y is singular for X on C 3 , we use Fubini,

and S •(n 2)0, and we rearrange the triple product to get



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R

3• X s 3

2 X , 26

R3

U 3

cut( )

3

S •

n Y 2 Y

X Y • X dX recall that

S n

20

U 3

cut( ) 3 Y n Y 2 Y dY •

R3

• X K X ,Y dY dX . 27The inner integral is clearly bounded and the outer integral is of order , and so tends to zero as

→0. Thus s32( )→0 as →0.

To prove our claim that “A2,3A2A3 d H1C 2 d H1C 3 , we must show that

lim →0

R

3 U 3

cut( )

• 2 Y l

X Y l dl y• dX 28

R3C 3 2 K X ,Y dl y•“ dX C 3 2

•dl, 29

and analogously for the analogously defined term s22( ) just interchange the indices 2 and 3.

The line integral in the inner integral in the first term on the right hand side may be written

X C 3

2

X Y Y dl y ,

and so, integrating by parts, we may rewrite the first term in the right hand side of 28 and 29

as

R3

C 3

2

X Y ““ dX . 30

To treat the left hand side in 25, note that we may reexpress the line integral appearing there as

U 3

cut( )

• 2 Y X Y dl y U 3cut( ) 2 Y • Y 1

X Y dl y . 31Note that the first term in 31 vanishes, since it is the integral of a perfect differential on a closed

curve. In the second term we replace Y by X and pull the • operator out of the line integral

so that using X • f (Y )V( X )V• f , it may be written as the divergence of a line integral . Then

after an integration by parts we are left with a left hand side equal to

R3 U 3

cut( )

2

X Y d l y • dX . 32

The curve U 3cut( ) is a longitude on U 3 and can be obtained when C 2 is C

2 by following an

appropriate vector p, in the normal–binormal plane to the curve, out a distance . So easy esti-

mates show that the contribution due to the difference between the line integral on U 3cut and the

line integral on C 3 are vanishingly small as →0. Thus combining the contributions 30 and 32

we obtain

C 3

2 Y

X Y dl

R3

X dX . 33



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Reversing the order of integration and using the fact that

R

3

X

X Y dX Y ,

we see that we may write 33 as

C 3

2 l l dl ,

and this establishes the equality of the two sides appearing in 16.

B. Third order torus links with a translation property

In this section, in order to recover results connecting the threefold averaged third order

Massey product over all trajectories in the three tori, with the third order helicity, we restrict

ourselves to a family of third order torus links which satisfy an additional property which we refer

to as a translation property or, better yet, an admissible ‘‘cone of translations.’’

Definition 3: We will say that a standard Borromean link of tori is ‘‘tame’’ if we can choosetwo pairs of indices, which, without loss of generality, we denote by (1,3) and (2,3) from the three

combinations of two indices, such that for the correponding ordered pair of tori (T 1 ,T 3), resp.

(T 2 ,T 3), there exist directions a1,3 and a2,3 (represented by unit vectors on the unit sphere) for

which the translation of T 1 in the direction a1,3 does not intersect the torus T 3 and analogously for

T 2 and T 3 .

Note that a2,3 is admissible for the pair (T 2 ,T 3) if and only if a2,3 is admissible for the pair

(T 3 ,T 2). In order for such a choice not to be possible the tori need to be quite ‘‘wild’’ as

illustrated by Figs. 1 and 3 of a ‘‘not wild’’ and a ‘‘wild’’ Borromean link of tori. Also, it is clear

that when there is a pair of unit vectors ( a1,3 ,a2,3) with the above properties, there is actually a

cone Co1,3 , Co 2,3 . See Fig. 4.

FIG. 3. An illustration of a Brunnian link that does not satisfy the translation property.



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To motivate the definition consider the volume integral form 1 for the Massey product, which

we repeat here for convenience:

T 1

B1•¦

R3

A2A3 Y K X 1Y dY dX 1

T 2

3 X 2B2 X 2 K X 1 X 2 dX 2

T 3

2 X 3B3 X 3 K X 1 X 3 dX 3

T 2

3 X 2B2 X 2 K X 1 X 2 § dX 1 . 34We will see below that

i X j

T i

B X i•ASo X j X i dX i 35

where, by definition the kernel ASo( X j X i) is given by

ASo X j X i X j X ia j ,i

X j X ia j ,i2 X j X i•a j,i X j X i 1

X j X i a j ,i

X j X i X j X ia j ,i

in terms of constant unit vectors a j ,i . The kernel ASo( X j X i) is only singular when the vector

X j X i and the vector a j ,i point in opposite directions. Thus we will say that the Borromean torus

link (T 1 ,T 2 ,T 3) is tame when it is possible to choose a1,2 so that for no point ( X 1 , X 2) in T 1T 2 does the vector X 1 X 2 point in the same direction as the vector a1,2 and when it is possible

FIG. 4. A Brunnian link that satisfies the translation property and an accompanying admissible pair of translations vectors

a1,3 and a2,3 .



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to choose in an analogous way a2,3 and so a3,2 for the tori T 2 and T 3 . In fact, as mentioned

previously, when the tori are tame, there is not just one unit vector but rather a cone of possible

translations.

The condition of being tame guarantees that we may define the solid angles using volume

integrals and so express the third order helicity of the three tori by plugging the expressions 36into 34.

C. The derivation of the volume integral expressions for the scalar potential for theBiot–Savart potential for the magnetic field

The main object of this section is, given a magnetic field B supported in a torus T and given

its Biot–Savart potential ABiot as defined in the preliminaries, to find an expression for a scalar

potential for A ( A) in the form of a volume integral over T . To prepare for this we will

make use of a little known inverse curl of the gradient of the fundamental solution.

Lemma 1: Given a unit vector a, if X is not parallel to a and is not zero, then we have

1 X

X 3

X a

X a2 X •a

X 1 X a

X X •a X . 36

Proof: We use the identity

“ f V f V f “V.

The lemma follows easily from the following relations:

X a2a, 37

X •a X 1 a

X

X

X 3 X •a, 38

X a22 X 2 X •aa, 39

X •a X 1 1

X a2 X a, 40

and so we get

X •a

X 1 1

X a2 X •a

X 1 X a2

X a4 X aª 1 2 .

After some simplification and using the identities X a2 X 2a2( X •a) 2 and a•(bc) (a•c)b(a•c)b we get

1 X

X 3.

The second term gives



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221

X a4 X •a

X 1 X 2a X •a X X •a2a X •a X )

2a

X a4

X •a

X 1

X a2

2a1

X a2 X •a

X 1 .

But the remaining term in the calculation of “ ( X a)/ X a2 ( X •a / X 1) , i.e., the terminvolving “( X a), yields exactly 2 , so we are done.

It follows immediately that if we define, abusing notation a bit, a vector potential ASo( X ,Y ,a)

by

ASo X ,Y ,aASo X Y ,a, 41

then we have

X ASo X ,Y ,a4 K X ,Y , 42

where

ASo X ,Y ,a X Y a

X Y X Y •a X Y .

Let C be a closed curve and S a smoothly embedded surface bounded by C . Relative to the

viewing point Y the solid angle is given by

C Y 4 S C

K•n dS .

Using Stokes’ theorem and ASo( X ,Y ,a) this may be written recalling that C 4 C

C Y a

C ASo X l ,Y ,a dl. 43

D. An integral expression for the scalar potential of the Biot–Savart vector potentialand for the fundamental one-form

Using the expressions derived above for the inverse curl of the gradient of the fundamental

solution of Laplace’s equation, we derive for each a the following integral expression for a

solution i of

C i

aA

C i.

The analogous formula in the case of scalar potentials for solid tori is

iaAi

Biot ,

44

i X a

T i

Bi Y •ASo X ,Y ,a dY .

This solution is well defined only if ( X Y )a does not vanish for Y T i and X in the domain in

consideration. Note that as discussed in the section on potentials and cuts, i is actually defined in



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the complement of T i minus a cut but will not be represented by formula 44 except in subregions

having the property that for each X in the subregion, ( X Y )a does not vanish for any Y

T i , i.e., X Y is never parallel to a.

Formula 44 is easily checked. Indeed, taking the gradient of i , bringing the gradient under

the integral sign and using the identity

U•VU•VV•UU“VV“U,

we get

i X T i

B Y •“Y ASo X ,Y ,adY T i

B Y “ASo X ,Y ,adY

T i

B Y K X Y dY ,

where we have used formula 36 to simplify the second integral and where the first integral

vanishes since

T i

B Y •“Y ASo X ,Y ,a dY T i

B Y •“ X ASo X ,Y ,a dY

T i

“•BASo dY T i

B•nASo dS ,

and the latter equals zero since “ •B0 and B•n0 on T i .

Corollary (explicit form for the Massey product): The line integral expression for the Massey

product of three curves C 1 ,C 2 ,C 3 given at the end of Sec. II, Eq. 15, for A ˆ 2,3 , may be rewritten

as

C 1

R3 A2A3 Y K X ,Y dY

C 3

C 2

2 X 2•ASo X 3 , X 2 ,adl 2K X 1 , X 3dl3

C 2

C 3

3 X 3 •ASo X 2 , X 3 ,adl3K X 1 , X 2 dl2 •dl1

C 1

C 3

3 X 3 •ASo X 1 , X 3 ,adl3 C 2

K X 1 , X 2dl2 •dl1 . 45IV. ASYMPTOTIC THIRD ORDER LINKING NUMBER

In this section we first show that there is a well defined asymptotic version of the third orderMassey product of three curves. We then show that given a ‘‘tame’’ Borromean link of tori

(T 1 ,T 2 ,T 3) and given a smooth vector field B with support in the link and tangential to the

boundary of the link, the triple volume average over the three tori of the asymptotic third order

Massey product equals the third order helicity of the three tori.

A. Definition of the asymptotic third order linking number

We begin by recalling some well known facts and introducing some notation.

Definition 4: Given a domain , we say that a system of curves X ,Y connecting X and Y

is a system of w-short paths if the length of a curve in X ,Y is bounded by a constant that is

independent of X and Y .



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Systems of w-short paths exist in any smooth domain. The existence of short paths that, for a

given vector field, also make vanishing contributions to integrals of Gauss type, is a much more

subtle question, being related to delicate questions about the critical set of the vector fields. See

Ref. 4, pp. 145–146. In the present paper the stronger concept of short paths is not required

because the vector fields considered have support in disjoint tori.Let (x,t ) denote the trajectory X (x,s), 0st where X (x,s) is the solution of the ODE

X (x,s)/ s B( X (x,s)) with initial condition X (x,0)x, and denote by t (x) the end point

X (x, t ). The curve ct (x) is obtained by adding a curve in x, X (t ,x) of the system of short

paths. Denote the end point of the curve obtained in this way by ct (x).

Let i , i1,2,3, be trajectories of magnetic fields Bi , where B i is tangent to the boundary

of T i and is Lipschitz continuous. In the case of multi-parameter ergodic theorems with multiple

‘‘times’’ it is necessary to put some restrictions on the way that the parameters tend to infinity.

Loosely speaking it is not permissible for some of the t i’s to tend to infinity much slower than

others. To make this precise, we use a notion due to Becker 12 which generalizes that of Wiener.13

Definition 5: We say that a set of parameters t i , i1,2,3 tends nicely to infinity if there is an

increasing family of open sets U with 0 such that the three-tuple (t 1 ,t 2 ,t 3)U and such

that for each X R3

lim →

X U U

U 0,

where denotes the symmetric difference and denotes the three-dimensional Lebesgue mea-sure.

This constrains the parameter set to increase in a fairly symmetric way. Two special cases of

families of parameters ( t i , t 2 ,t 3) that tend nicely to infinity are given by choosing

i

U T t 1 ,t 2 , t 3 R3:t 1

2t 2

2t 3

2 2

which is the case considered by Wiener, and

ii

U T t 1 ,t 2 ,t 3 R3:t 1 ,t 2 c2 , t 3 c3,

where c2 and c 3 are arbitrary constants.

The asymptotic third order linking number of the three curves is defined by

lim →(t 1 , t 2 , t 3)U

1

U M ˆ t 1, ˆ t 2, ˆ t 3,

where ˆ i is i completed to a closed curve by the addition of a short path. Becker’s generalization

of Wiener’s result is that the limit above exists almost everywhere and the convergence is domi-

nated, where M is the explicit form of the Massey product 45 and where ˆ i is i completed to

a closed curve by the addition of a short path.

In the next section we will need the following well known result. If ˜ is a multi-valued

function that increases by 1 on any degree one curve in a torus T , and if B is a divergence-free

field that is tangential to T , then

T

B• ˜ dV Flux of BF .

As pointed out by Freedman and He in Ref. 9, this has the following corollary:



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T

˜ t x ˜ 0 xdV 46

T 0s d

dt ˜

s

x ds dV 47

T

0

t

B• ˜ dsdV 0

t

dsT

B• ˜ d V 48

t F . 49

B. Multiplicativeness of third order linking number

We will need the following result:

Lemma 2: Let (T 1 ,T 2 ,T 3) be a Borromean link of tori (wild or not). If C i is a closed curve

of degree d i in T i , i1,2,3, then

M C 1 ,C 2 ,C 3 d 1d 2d 3 . 50

This lemma in the case d 11, d 21 and d 3n where n is an arbitrary integer is proved in Ref.

14 and already mentioned in Massey.1 To prove the lemma in the more general case, one may

generalize Stein’s proof, or argue directly using a generalization of the argument in Appendix D

in Ref. 6, where the Massey product is connected with the signed intersections of one of the

components of the link with Seifert surfaces bounded by one of the other components. When this

component has degree n this contribution is easily seen to be n times its value when the compo-

nent has degree 1, since the trajectories will traverse all n sheets of the corresponding Seifert

surface.

C. Average of third order asymptotic linking number is equal to third order helicity

We are now ready to establish the following result:

Theorem 3: Let T (T 1 ,T 2 ,T 3) be a tame Borromean link and let B be a smooth divergence-

free vector field, with support in T and tangential to the boundary of T . Then, if the parameters

( t 1 , t 2 ,t 3) tend nicely to infinity in particular, if t 1t 2t 3, then the volume average of the

time average of the asymptotic Massey product in the link is equal to the third order helicity of the

link, i.e.,

T 1

T 2

T 3

lim nicely→

1

t 1

1

t 2

1

t 3M

c

t 1 a1 , ct 2 a2, c

t 3 a3 d a3d a2d a1 ,

We will first establish a lemma that allows us to deal with the set of null points of the vector

field B. Indeed, away from these null points, the trajectories field lines of B are smooth, and so

we can use the line integral form of the Massey product mentioned in Sec. II and derived inAppendix B of Ref. 6. But this may not be so at a null. The lemma allows us, roughly speaking,

to show that the set of null points has a negligible influence.

D. The set of critical points is of zero measure in Lagrangian space

The idea of the proof is analogous to that which uses the coarea formula, in the context of

Lipschitz scalar functions, to show that the Hausdorff measure of the set of critical points is zero

on almost all levels.

Let



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W B aS B:If t a is a trajectory of B beginning at a,

then for t →, t a→null point of B ,

where S (B) denotes the support of B. By the ergodic theorem, we have

W ( B)

B a d 3aW (B)

d a limt →

1

t

0

t

B t a d t .

Now denote the first null point encountered on the trajectory t (a) by N (a). Then since

limt → t (a) N (a), and since B is smooth, given , there exists a T so large that for t T we

have

B t a

2 .

Now choose T 2T maxW B / , so we have

1

T 0T

B X a,t d t 1

T 0T

B X a, t d t T T

B X a, t d t

T

T maxB

T T

T

2

.

Thus, if we set

M B,a, t 1

t

0

t

B t a d t ,

we have

M→0 pointwise as t →aW B.

Thus we have

W (B)

B a d 3a0,

as claimed, and so B0 almost everywhere on the set W (B). We summarize this as a theorem.

Theorem 4: Let B be a smooth divergence-free vector field in a torus T , tangential at T .

Then B is zero almost everywhere on the set of points which tend asymptotically to a null point. In particular, the trajectory issuing from almost any point not in the null set of B does not tend

asymptotically to a null point of B.

We now can begin the proof of Theorem 2.

Using the mean value theorem, we have

˜ it x ˜ i x ˜ ic

t x ˜ i x ˜

t x ct x 51

˜ ict x ˜ i x ˜ i

t x ˜ ict x 52

˜ ict x ˜ i x sup

x

˜ i x x, X x, t 53



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ct “ ˜ i .•dlC , 54

where C is a constant. Similarly estimating the difference in 51 from below we obtain

˜ i X x, t ˜ i x ˆ t

˜ i .•dlC . 55

To deduce 54 form 53 we used the fact that ˜ i is uniformly bounded and the fact thatmembers of the system of short curves are uniformly bounded. Thus, combining 54 and 55

˜ it x ˜ i xdeg c

t x C . 56

The two inequalities together imply

deg ct xC ˜ i X x,t ˜ i xdeg

t xC . 57

Integrate 57 over T i and use 49 for i

1 to get

T i

deg ct x(i) C T itF 1

T i

deg ct x(i)C .

Dividing by t and letting t → we have

limt →

1

t

T i

deg ct x(i) dX F i . 58

Now if ( ct ) (i) are closed curves in T i , i1,2,3, and if we apply formula 50, we obtain

M ct x (1 ), c

t (2 ), ct (3 ) deg

c

t 1(1 ) x1 deg ct 2(2 ) x2deg c

t 3 (3 ) x3.

Integrate both sides in turn over T i , i1,2,3, to obtain

i1

3

T i

deg c

t 1 ( i) xi T 3

T 2

T 1

M c

t 1(1 ) x1, ct 2(2 ), x2 , c

t 3(3 ) x3 d x1d x2d x3 .

At this point we use the lemma as follows: If X ( X 1 , X 2 , X 3) is a point in T 1T 2T 3 which is

such that the trajectory emanating from any of the X i encounters a null point, the corresponding

degree on the left hand side will be bounded. For such a point modify the definition of

M(( ct ) (1 )(x1), ( c

t ) (2 )(x2), ( ct ) (3 )(x3)) to be zero.

Now divide by t 1t 2t 3 and let t i→ , i1,2,3. Using 58 the left hand side clearly tends to

F 1F 2F 2 . 59

As noted before and demonstrated in detail in Appendix D of Ref. 6, this quantity is equal to the

third order helicity. The limit of the right hand side corresponds to the average of the asymptotic

linking number. Note in the argument above we do not need to use Becker’s multi-parameter

generalization of Birkhoff’s theorem to the effect that the volume average of the time average

equals the volume average, but the proof may also be concluded in that way.

V. CONCLUSIONS

Our main result is a new and explicit form for the Massey product given by 45. It expresses

the Massey product in terms of purely geometric quantities such as a suitably normalized vector



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joining pairs of points one on each curve. The expression involves four terms. One is the line

integral of a line integral, and the other three are triple line integrals. This explicit form of the

Massey product was derived in two stages. The first step was to obtain a semi-explicit form for the

line integral form of the Massey product 20 by expressing the inverse curls appearing there as a

volume integral against an explicit kernel and two line integrals involving solid angles. Thesecond step was then to express these solid angles in turn as line integrals by using the expressions

45.

This explicit form of the Massey product was then used to define an asymptotic third order

Massey product in Sec. IV. For a certain, rather general, subclass of Borromean torus links

possessing a translation property, it is possible to establish an analog, in the present context, of a

result of Arnold for Hopf links: The volume averaged asymptotic third order Massey product is

equal to Berger’s third order helicity.

ACKNOWLEDGMENTS

E.S. was supported under National Science Foundation Grant No. DMS-9622923 and with the

assistance of the Italian CNR. We thank an anonymous referee for constructive suggestions that

allowed us to improve this paper.

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Circle, Chicago, Il, 1968 pp. 174–205.2 M. I. Monastyrsky and V. Retakh, ‘‘Topology of linked defects in condensed matter,’’ Commun. Math. Phys. 103,

445–459 1986.3 M. A. Berger, ‘‘Third order link integrals,’’ J. Math. Gen. 23, 2787–2793 1990.4 V. I. Arnold and B. Khesin, ‘‘Topological methods in fluid dynamics,’’ in Applied Mathematical Sciences, Vol. 125

Springer Verlag, New York, 1998.5 N. W. Evans and M. A. Berger, ‘‘A hierarchy of linking integrals,’’ Topological aspects of the dynamics of fluids and

plasmas Santa Barbara, CA, 1991, pp. 237–248; Adv. Sci. Inst. Ser. E. Appl. Sci, 218 Kluwer Academic, Dordrecht.6 P. Laurence and E. Stredulinsky, ‘‘Asymptotic Massey products, induced currents and Borromean links of tori,’’ Uni-

versity of Rome report 30/99, http://mercurio.mat.uniroma1.it// laurence//massey.ps 1999.7 V. I. Arnold, ‘‘The Asymptotic Hopf invariant and its applications’’ in Russian, Proc. Summer School in Differential

Equations, Eravan 1974 English translation: Sel. Math. Sov. 5, 3271986.8 P. Laurence and M. Avellaneda, ‘‘A Moffatt-Arnold formula for the mutual helicity of linked flux tubes,’’ Geophys.

Astrophys. Fluid Dyn. 69, 243–256

1993.9 M. H. Freedman and Z. X. He, ‘‘Divergence free fields: energy and asymptotic crossing number,’’ Ann of Math2 134,

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Borromean Torus