RMP Colloquia

This section, begun in January 1992, contains short articles intended to describe recent research of
interest to a broad audience of physicists. It will concentrate on research at the frontiers of physics,
especially on concepts able to link many different subfields of physics. Responsibility for its contents
and readability rests with the Advisory Committee on Colloquia, U. Fano, chair, Robert Cahn, S.
Freedman, P. Parker, C. J. Pethick, and D. L. Stein.

Prospective authors are encouraged to com-

municate with Professor Fano or one of the members of this committee.

Quantum physics and the topology of knots

*

Michael Atiyah

The Master’s Lodge, Trinity College, Cambridge CB2 1TQ, England

CONTENTS

I. Introduction

977

II. The Classical Story

977

III. Quantum Physics

977

IV. Topology

978

V. Quantization of Electric Charge

978

VI. Knots and Links

978

VII. Kelvin’s Vortex Atoms

978

VIII. The Alexander Polynomial

979

IX. The Jones Polynomial

979

X. Braids

980

XI. Conclusions

981

Acknowledgment

981

References

981

Further Reading Material

981

I. INTRODUCTION

There is a long history of interaction between geometry

and physics with each subject stimulating the other. In the
days of Euclid, geometry was seen as the description of
physical space, and in due course this provided the frame-
work for Newtonian mechanics. Subsequently, at a more so-
phisticated level, the theories of Maxwell and of Einstein
required a deeper understanding of geometry, and this stimu-
lated many of the developments in geometry in this century.

This, however, describes the situation of some 50 years

ago. In recent times, in the past 20 years or so, there have
been new and unexpected developments linking physics and
geometry in a quite different way. These developments are
extensive and cover a wide range. In picking on the subject
of knots, I have isolated one aspect of this new interaction. In
some ways it is typical and it is also the easiest to describe,
because knots are familiar objects whose intricacy is intu-

itively easy to understand. However, I should emphasize that
this represents just the tip of the iceberg. There is a great deal
more behind the scenes, though much of it is too technical to
explain without extensive formulas. In what follows I shall
avoid all such technicalities.

II. THE CLASSICAL STORY

In classical physics the fundamental connection with ge-

ometry lies in the interpretation of f orce as cur

vature.

This takes many forms, beginning with the Newton view
that, in the absence of external forces, a particle will move in
a straight line with uniform velocity. Any deviation from
uniform motion is therefore due to a force, and the amount of
deviation

~or curvature! measures the strength of the force.

Several centuries later, when Einstein developed his gen-

eral theory of relativity, gravitational force became explained

~or interpreted! as a distortion ~or curvature! of space-time
itself. Moreover, Maxwell’s theory of electromagnetism can
also be seen in geometrical form. The electromagnetic field
can be viewed as an appropriate form of curvature.

Just as Newton’s theory stimulated both Euclidean geom-

etry and the development of the differential calculus, so the
theories of Einstein and Maxwell stimulated the more gen-
eral development of differential geometry. This has certainly
been one of the main features of the 20th century.

III. QUANTUM PHYSICS

Ironically, just as the age-old relationship between geom-

etry and physics was being reinforced by Einstein’s theory, a
cloud appeared on the horizon in the shape of quantum
theory. Since the 1920’s we know that classical physics is
only an approximation to a more subtle and complex quan-
tum physics. In this new picture of physics the fundamental
notions are quite different. Localized particles now get re-
placed by waves, so that the very geometrical basis of phys-
ics gets undermined. In addition, various quantities can only
take on discrete values: quantities such as angular momen-
tum, spin, or electric charge.

*

This article is based on a lecture delivered at the 11th International

Congress of Mathematical Physics, held in Paris in July 1994, and
appears here courtesy of the Congress. The Proceedings, edited by
Daniel Iagolnitzer, are published under the title 11th International
Congress of Mathematical Physics

~International Press, Cambridge,

MA, USA

!.

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Reviews of Modern Physics, Vol. 67, No. 4, October 1995

0034-6861/95/67(4)/977(5)/$07.75

© 1995 The American Physical Society

The mathematics that relates most naturally to quantum

physics is very ungeometric and centers around the theory of
linear operators.

However, there is a branch, or offshoot, of geometry called

topology which shares some of the features exhibited by
quantum physics, notably the emphasis on nonlocal aspects
and the appearance of discrete quantities. Topology is a re-
cent subject, having emerged in the 19th century and blos-
somed in the 20th century, and it has turned out that its
superficial resemblance to quantum physics is not mislead-
ing. There are deep links which are only now being under-
stood and developed, as I shall indicate via the example of
knots. First, however, we must digress to explain what topol-
ogy is.

IV. TOPOLOGY

As the word indicates, geometry is the study of the mea-

surement

~of distance, angles, curvatures, . . . ). It is an es-

sentially quantitative subject. Topology, on the other hand, is
qualitative: it is, so to speak, what is left of geometry when
measurement is removed. The reason why this is not trivial is
that topology concerns itself with nonlocal or global features.

The best example of what this can involve is the globe

itself. The distinction between a ‘‘flat earth’’ and a round or
spherical earth is a topological one and is qualitative. The
exact size of the earth is irrelevant to the question. One
might therefore say that the early navigators were the first
‘‘experimental topologists.’’

Another example of topology concerns the number of

times a closed piece of string on a table winds round a given
central point. This number is essentially independent of the
length or precise shape of the string. It is qualitative and
discrete: it can take any integer value, including negative
ones to allow for reversal of direction.

I will not attempt a formal definition of topology, but it

should already be clear that it concerns global and discrete
questions: precisely what turns up in quantum physics. Later,
when we come to knots, our topology will become more
complex and we will have more interesting examples; but
meanwhile we return to physics and establish the first sub-
stantial linkage.

V. QUANTIZATION OF ELECTRIC CHARGE

The first significant use of topological ideas in quantum

physics was made by Dirac more than 50 years ago. He put
forward a theoretical explanation, based on topology, for the
experimental fact that all particles in nature appear to have
an electric charge which is an integer multiple of the charge
of the electron. In other words, he explained the ‘‘quantiza-
tion’’ or discreteness of electric charge.

Dirac’s argument went roughly as follows. In Maxwell’s

theory of electromagnetism, electricity and magnetism ap-
pear formally on a similar footing

~a ‘‘duality’’ which is now

being probed at a deeper level: cf. Seiberg and Witten, 1994

!.

Isolated point-sources

~e.g., electrons! exist, and theoreti-

cally they could have magnetic counterparts, which are
termed ‘‘magnetic monopoles.’’ Dirac considered the quan-
tum theory of a particle moving in the background field of

such a mythical magnetic monopole. He realized that the
topology around the monopole

~a three-dimensional version

of the winding number in a plane

! would affect the wave

function of the particle, and this in turn would lead to the
quantization of its electric charge. Thus the discreteness of
charge is directly related to the discreteness of topological
‘‘winding numbers.’’

One might have expected more to be made of this insight

that topology was relevant to quantum theory, but in fact it
took another 40 years for this to be realized. Knot theory was
one of the first examples of the new interaction; so let us
now introduce knots.

VI. KNOTS AND LINKS

To a topologist a knot is a closed piece of string without

loose ends. The length, thickness, and precise shape are of no
interest. What is significant is the extent to which the string
is really knotted and whether it can be to some extent un-
tangled. For example, given two superficially different knots,
can I twist one around so that it eventually looks like the
other? In particular, can I make it look like a circle

~which is

called the unknot

!?

In practical terms, if we are presented with a tangle of

string

~without loose ends!, it may not be clear how many

pieces there are. The word ‘‘link’’ is used as a generalization
of ‘‘knot’’ in this case. The number of pieces and the number
of times each piece links or winds round another are numeri-
cal invariants of the link. That is to say, they are discrete
quantities which can be calculated and are independent of the
particular presentation of the link

~i.e., the way it is presented

to us or the way we choose to look at or manipulate it

!.

The study of knots and links is an excellent illustration of

what topology is about. It also indicates the difficulty of the
problem. The numerical invariants described above are only
a small step towards the understanding or classification of
links. A few pictures begin to illustrate the complexities

~see

Fig. 1

!.

VII. KELVIN’S VORTEX ATOMS

The history of knot theory is both interesting and relevant

to our present topic. Clearly, knots have been familiar in
various practical situations, notably to sailors. However, they
were not seriously studied until the 19th century, and that
study was directly stimulated by a potential use in physics.

In the mid-19th century the nature of atoms was a great

mystery. On the other hand, the theory of fluids was well
advanced and the significance of vortices there was appreci-
ated. In particular, the fact that vortices avoided crossing
each other was well known. With this background Lord
Kelvin

~see Thompson, 1869! put forward the theory ~or con-

jecture

! that atoms might be knotted vortex tubes of ‘‘the

ether.’’ This theory was taken seriously for a number of years
and it had two main merits:

~a! It explained the ‘‘stability’’ of

matter, since knots retain their essential knottiness while be-
ing able to move or stretch;

~b! it offered an explanation of

the variety of different atoms as coming from the different
possible knots.

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Michael Atiyah: Quantum physics and the topology of knots

Rev. Mod. Phys., Vol. 67, No. 4, October 1995

In view of

~b!, an understanding or classification of knots

became an important problem. Tait, a collaborator of Kelvin,
devoted many years of his life to a tabulation of knots. If we
draw pictures of knots

~with over and/or under crossings! as

though they were pieces of string laid on the table, the num-
ber of crossings gives an initial measure of complexity. With
up to 10 crossings, there are 165 different knots, and these
were essentially tabulated

~in minimal form! by Tait. How-

ever, the numbers rise very rapidly as we allow more cross-
ings. For example, with 13 crossings there are more than
10 000 different knots, so that even a modern computer rap-
idly runs out of enthusiasm.

In the course of his tabulations, Tait discovered a number

of empirical facts which were subsequently termed the ‘‘Tait
conjectures,’’ and these remained as a theoretical challenge
for mathematicians for more than a century. They have only
recently been established as a result of the new interaction
with quantum physics which I shall be describing.

A page from Tait’s tables is reproduced

~see Fig. 2! so that

the complexity of the problem can be appreciated. If
Kelvin’s theory had turned out to be correct, then such tables
would be on the walls of all chemistry laboratories instead of
the rather dull periodic table!

VIII. THE ALEXANDER POLYNOMIAL

Kelvin’s theory of vortex atoms was, of course, discarded,

and atomic physics took a different direction. This meant that
knots were no longer of potential interest to physicists, but
the work of Tait left a substantial legacy for the mathemati-
cians to ponder over.

As topology developed in the 20th century, new machin-

ery was developed, much of it arising from generalizations
of the modest ‘‘winding number.’’ Among other things, this
led to some progress with knot theory. In 1928, J. W. Alex-

ander showed that one could define an invariant for a knot k
which consisted of a polynomial A

k

(t) with integer coeffi-

cients. This was useful for the classification of knots, be-
cause if two knots had different Alexander polynomials, then
they had to be different knots. As a simple example for the
trefoil knot k,

A

k

~t!5t211t

21

,

whereas for the unknot u

A

u

~t!51

~note: inverse powers of t are allowed!. This proves formally
that the trefoil knot k is different from the unknot; i.e., it
cannot be ‘‘unknotted.’’

The Alexander polynomial of a knot or link can be calcu-

lated by a simple recurrence algorithm as follows. We first
orient each component of the link by putting arrows indicat-
ing the direction to be traversed. We then look at a particular
crossing and consider the three pictures

~Fig. 3!: We now

imagine three versions of our link which look locally like
L

1

, L

2

, or L

0

~not that L

0

will actually have one less cross-

ing

!. Our recurrence formula then relates the Alexander poly-

nomials of these three links. We have

A

1

2A

2

5~t

1/2

2t

21/2

!A

0

.

~Note: For knots, the square roots of t eventually cancel out,
but they will remain for some links.

!

The Alexander polynomial was useful but by no means

definitive for the classification of knots. Notably, it failed to
distinguish between knots and their mirror images, a serious
defect. Formally taking the mirror image leads to replacing
the indeterminate t in the Alexander polynomial by its in-
verse t

21

. As it happens, the Alexander polynomial is sym-

metrical in t and t

21

and so is unchanged by taking mirror

images.

IX. THE JONES POLYNOMIAL

The world of knot theory was astonished when in 1984, 60

years after the appearance of the Alexander polynomial,
Vaughan Jones

~1987! produced another polynomial which

was a new knot invariant. Moreover, the Jones polynomial
represented a crucial improvement over the Alexander poly-
nomial in that it could distinguish between mirror images.
Thus for the trefoil knot the Jones polynomial turns out to be

J

~t!52t

4

1t

3

1t.

Taking mirror images replaces t by t

21

and produces a quite

different polynomial, demonstrating that the trefoil knot and
its mirror image are indeed different knots.

Like the Alexander polynomial, there is a simple recur-

rence formula for the Jones polynomial of a link. With a
similar notation to that in the previous section,

t

21

J

1

2tJ

2

5~t

1/2

2t

21/2

!J

0

.

The Jones polynomial has been extensively studied and

we now know

~1! it is the first of a whole family of new

polynomial invariants

~related to Lie groups!; ~2! it can be

used to solve the Tait conjectures;

~3! it is not related to

FIG. 1. Some pictures of knots.

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Michael Atiyah: Quantum physics and the topology of knots

Rev. Mod. Phys., Vol. 67, No. 4, October 1995

classical topological machinery

~such as winding numbers!;

and

~4! it is related to algebraic aspects of various parts of

physics.

The relations in

~4! with parts of physics were clearly very

intriguing, but were only fully clarified in 1988 by Witten

~1989!, who gave an elegant quantum theory interpretation
of the Jones polynomial. To explain this we must first make
a digression to discuss braids.

X. BRAIDS

A braid is similar to a knot or a link, but it differs essen-

tially in that a direction of space

~the vertical! is singled out.

Moreover, a braid consists of N pieces or strands, each of
which is constantly moving upwards. The strands can twist
and intertwine with each other, but the final and initial posi-
tions are assumed to coincide

~up to order!.

A braid thus has ‘‘loose ends.’’ If these are joined together

in a standard ordered way, we end up with a link. Moreover,
all links arise in this way, so that a link invariant will auto-

matically give a braid invariant. The converse, however, is
not true; for example, the number N of strands of a braid
does not give a link invariant. However, braid invariants
which satisfy some additional conditions do give link invari-
ants, and the Jones invariants can be derived in this way.

The following diagrams

~Fig. 4! give an example of a

braid

~with three strands! and its associated link.

The best way to think of a braid is as the space-time graph

of N distinct points moving in a plane

~and returning as a set

to their original position

!. The distinguished ‘‘vertical’’ axis

FIG. 3.

FIG. 2. Extract from Tables of Knots by P. G. Tait. From Scientific Papers, Vol. 1, by P. G. Tait

~Cambridge University, Cambridge, England,

1898

!, Plate VI facing p. 334.

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Michael Atiyah: Quantum physics and the topology of knots

Rev. Mod. Phys., Vol. 67, No. 4, October 1995

now appears as the time variable.

This in fact is the physical picture that Witten adopts in his

interpretation of the Jones polynomial. Thus Witten consid-
ers three-dimensional space as

~211!-dimensional space-

time, with physical space being two dimensional. This can,
of course, relate to ‘‘real’’ physics either by considering thin
layers or by taking cross sections of three-dimensional situ-
ations which are uniform in one direction.

Witten then chooses an appropriate quantum theory of

fields and applies it to a given braid. The expectation values
of the braid then turn out to give information equivalent to
the Jones polynomial

~actually the values of the polynomial

at a suitable sequence of points

!. Without going into the de-

tails, I shall make some general comments.

Witten’s quantum theory is of a fairly standard type much

studied by physicists. In particular, it is a ‘‘chiral’’ theory that
can distinguish left-handedness from right-handedness, and
is similar in that respect to the models that explain the break-
ing of parity predicted by Lee and Yang. It is also a fully
relativistic theory

~in the three-dimensional sense!, and this

explains why we get link invariants and not just braid invari-
ants. Finally, Witten’s theory is actually a degenerate form of
physical models which contain a quadratic term in their en-
ergy. This term is dropped in Witten’s model, and this ex-
plains heuristically why we get purely topological

~as op-

posed to geometric

! information. What is perhaps surprising

and requires some delicate handling is that such a degenerate
theory can be made meaningful.

XI. CONCLUSIONS

There are now many examples of similar situations in

quantum theory related to topology. A new field of ‘‘topo-
logical quantum field theory’’ is being developed which con-
centrates on the topological

~and most fundamental! aspects

of quantum physics. This is shedding new light on traditional
problems in geometry as well as providing clues for realistic
physical models.

In retrospect, Kelvin’s ideas on vortex atoms take on a

new light. Although he was on the wrong track in identifying

atoms with knots, his insight is now being justified at a
deeper level. Fundamental physics at the subatomic level ap-
pears to have links with topology which are perhaps ulti-
mately related to the stability of matter as Kelvin was specu-
lating. For further, more recent developments in this area, see
Seiberg and Witten

~1994! or Collins ~1995!.

ACKNOWLEDGMENT

The author is grateful to Daniel Iagolnitzer for facilitating

dual publication of this paper in both Reviews of Modern
Physics and the Proceedings of the 11th International Con-
gress of Mathematical Physics, of which he is the editor.

REFERENCES

Collins, G. P., 1995, ‘‘Supersymmetric QCD sheds light on quark

confinement and the topology of 4-manifolds,’’ Phys. Today 48,
~3! 17.

Jones, V. F. R., 1987, ‘‘Hecke algebra representations of braid

groups and link polynomials,’’ Ann. Math. 126, 335.

Seiberg, N., and E. Witten, 1994, ‘‘Electric-magnetic duality, mono-

pole condensation, and confinement in N

52 supersymmetric

Yang-Mills theory,’’ Nucl. Phys. B 426, 19; 430, 485

~E!.

Tait, P. G., 1898, Scientific Papers, Vol. 1

~Cambridge University,

Cambridge, England

!.

Thompson, W. H., 1869, ‘‘On vortex atoms,’’ Trans. R. Soc. Edin-

burgh 34, 15.

Witten, E., 1989, ‘‘Some geometrical applications of quantum field

theory,’’ in Proceedings of the IXth International Congress on
Mathematical Physics, held July 1988, Swansea, Wales, edited by
B. Simon, A. Truman, and I. M. Davies

~Hilger, Bristol/New

York

!, p. 77.

FURTHER READING MATERIAL

Recent Developments in Gauge Theories

,

edited by G.

’t Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P. K. Mitter, I. M.
Singer, and R. Stora

~Plenum, New York, 1980!.

Braid Group, Knot Theory and Statistical Mechanics, edited by

C. N. Yang and M. L. Ge

~World Scientific, Singapore, 1989!.

New Developments in the Theory of Knots

,

edited by Toshitake

Kohno

~World Scientific, Singapore, 1990!.

Common Trends in Mathematics and Quantum Field Theories,

1990 Yukawa International Seminar, Kyoto, Japan, edited by T.
Eguchi, T. Inami, and T. Miwa, Prog. Theor. Phys. Suppl. No. 102
~1991!.

Topological Methods in Quantum Field Theories, edited by W.

Nahm, S. Randjbar-Daemi, E. Sezgin, and E. Witten

~World Scien-

tific, Singapore, 1991

!.

Knots and Physics, by Louis H. Kauffman

~World Scientific,

Singapore, 1991

!.

FIG. 4.

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Michael Atiyah: Quantum physics and the topology of knots

Rev. Mod. Phys., Vol. 67, No. 4, October 1995