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functions in the loop representation are functions of

a clos ed curve (more precisely of families of closed

curves, or spin networks, as we will discuss below).

We still have to deal with the diffeomorphism

and Hamiltonian constraints. The diffeomorphism

constraint when written in the loop representation

implies that the wave functions are not functions of

loops but rather of topologically invariant properties of

the loops under general diffeomorphisms of the spatial

manifold containing the loops. Such functions are

technically known in the mathematical literature as

‘‘knot invariants.’’ This is the first point of connection

between knot invariants and quantum gravity; they

constitute the kinematical arena of the theory. One still

has to deal with the Hamiltonian constraint, which has

to be imposed as an operator equation. We shall see that

knot theory also seems to have a lot to say about

solutions of the Hamiltonian constraint. This is quite

remarkable, since the Hamiltonian constraint embodies

in detail the specific dynamics of Einstein’s theory of

gravitation, and to our knowledge this is an input that

has never gone into the ideas of knot theory.

In terms of the Ashtekar variables, the Hamiltonian

constraint takes the form

H ¼ E

 E

 B

þ E

ðÞ

abc

½1

where we have used a conventional vector notation

for the frame indices and kept explicit the spatial

indices. 

abc

is the Levi-Civita totally antisymmetric

tensor. We have included a possible cosmological

constant . The Ashtekar formulation can be

constructed in different ways. In the original

formulation, the connection A

was a complex

variable and the Hamiltonian took the form we

listed above. However, the resulting theory was only

equivalent to real general relativity if the variables

satisfied certain reality conditions. One can choose

to use a real connection instead, but then the

Hamiltonian constraint has additional terms. At

the moment, we will concentrate on the constraint

as listed above. The constraint has to be implemen-

ted as a quantum operator acting on wave functions.

Since it involves the product of operators, it needs to

be regularized. Most regularization methods are

problematic in this context, since they use a metric,

and here the metric is a quantum operator, not an

external fixed quantity. If we ignore these difficul-

ties, one observes that, if one were to choose a

quantum state, for instance in the connection

representation, for which ,



½A¼

½A½2

the state would be annihilated by the Hamiltonian

constraint, and this would be true no matter what

regularization was chosen. Classically, the condition

 B

is satisfied for the de Sitter geometry, so one

could envision the state as a quantum state

associated with such geometry. The exact solution

of the above equation is given by a state that is the

exponential of the integral on the spatial slice of the

Chern–Simons form built from the connection



½A¼exp k

x tr A ^ dAð



A ^ A ^ A



½3

and the constant k needs to be chosen as k = 6= for

the state to be a solution.

One can ask, ‘‘what is the expression of this state

in the loop representation?’’ To answer this, one

needs to compute the coefficients of its expansion in

the basis of Wilson loops W



[A], where as we stated

earlier,  should be a collection of (intersecting)

loops (later we will discuss the genera lization to spin

networks). The expression for the coefficients will

be a function only of the loops  and is given by



½¼

DAW



½A

½A½4

This expression is invariant under diffeomorph-

isms of the manifold or, equivalently, under smooth

deformations of the curve . That is, it is what in the

mathematical literature is called ‘‘knot invariant.’’ In

fact, this integral has been studied by Witten in the

context of Chern–Simons theory and has been

shown to be related to the Kauffman bracket knot

polynomial, which in turn is related to the cele-

brated Jones polynomial. Therefore, the implication

of these results is that the Kauffman bracket knot

polynomial appears to be the representation in the

loop representation of a state of quantum gravity

that solves the quantum Einstein equations (with a

cosmological constant). The reader may be intrigued

by the word ‘‘polynomial’’ in this context. It should

be noted that the Cher n–Simons state 

[A]

depended on a param eter k, which had to take a

certain value for it to solve the quantum Einstein

equations. The resulting knot invariant is a poly-

nomial in exp (k). If one expands out the result, an

infinite power series in k results. There will be

infinite coefficients in the series, but they are just

combination of the finite number of coefficients of

the polynomial. Knot polynomials are a powerful

tool for analyzing and distinguishing knots. The

coefficients of the polynomials are all knot invari-

ants. Typically, for ‘‘simple’’ knots, the first few

coefficients of the knot polyn omial are nonzero. As

one considers more complicated knottings, higher

Knot Invariants and Quantum Gravity 217

coefficients become nonvanishing. The ultimate goal

of knot theory is to be able to consider two arbitrary

knots and to unambiguously determine if the two

knots are related by a smooth transformati on. The

knot polynomials appear as promising tools for

achieving this task that has remaine d elusive up

to now.

Returning to quantum gravity, to have a well-known

knot polynomial as a solution of the quantum Einstein

equations is a remarkable fact. The first connection we

outlined between knot theory and quantum gravity was

less unexpected: if one describes a theory that is

diffeomorphism invariant in terms of loops, the

appearance of knots is inevitable. But we are now

finding that knot invariants from the mathematical

literature, which were constructed without any knowl-

edge of the details of the dynamics of the Einstein

equations, seem to manage to solve such equations. This

is either a big coincidence or a pointer to some

unexplained deep connection yet to be understood.

Notice, for instance, that other theories of gravity would

not have the Kauffman bracket as a quantum state.

There is a certain technicality about the Kauffman

bracket that makes it diffic ult to argue with precision

that it is a state of quantum gravity. To underst and

this technicality better, it is perhaps best to concen-

trate on the form of the quantum state written above

if the connection is an abelian connection. In that

case, the integral in question,



CS abelian

½¼



exp iA

ðÞ

 exp

x

abc



½5

by turning it into a Gaussian integral. The result is



CS abelian

½¼





abc

ðx  yÞ

jx  yj

½6

This integral has problems, since the integrand is

ill-defined when x = y. Notice that the integral

would be well defined if the two contour integrals

were evaluated on different, nonintersecting curves.

The result would be the well-known formula for

the Gauss linking number of the curves, yielding

zero if they are not linked and and integer multiple

of 4 if they were. So the integral we were trying to

compute was actually the Gauss linking number of

the curve with itself. Such a quantity is not well

defined for ordinary curves. To deal with this

problem, mathematicians introduced the con cept of

framed knots. A framed knot is a curve with a

prescription to determine a second curve from it.

One way to see it is to construct another curve that

is ‘‘infinitesimally close’’ in space to the original

one. It is clear that there is no canonical way to

compute such a second curve. Then, when one

considers quantities like the self-linking number,

one makes them well defined by evaluating the two

integrals on the two curves, the original one and

the one yielded by the prescription. In reality, the

notion of framing is a bit more elaborate than what

we hint at here, since one could consider invariants

constructed with more than two integrals and could

still be ill-defined if one only considers two curves.

The notion has to be extended as well to handle

intersections in the curves. We will ignore these

subtleties in this discussion.

The Kauffman bracket knot invariant is an

invariant of framed knots, just like the self-linking

number. It is not well defined for a single curve. It

requires a framing of the knot. In quantum gravity,

there is no compelling reason to consider framed

curves. It is true that framed curves arise naturally in

q-deformed field theories and perhaps a q-deformed

version of quantum gravity is what needs to be

considered to accommodate the Chern–Simons state,

but at the moment there are no proposals along

these lines that have widespread consensus.

So, it appears the Kauffman bracket does not have

a natural role to play as a state of quantum gravity.

However, it is known that the frame dependence of

the Kauffman bracket knot polynomial can be

captured in an overall factor that depends on the

self-linking number. If one strips the polynomial of

this factor, one gets the Jones polynomial, which is a

knot invariant of single curves. Could it be that this

polynomial has a chance of being a solution of the

quantum Einstein equations?

To determine this, the analogy with Chern–

Simons theory is no longer useful, sinc e there is no

straightforward way to transform the relation

between the Kauffman and Jones polynomials into

relations between states in the connection represen-

tation. To analyze if the Jones polynomial could be

a solution of the quantum Einstein equations, one

needs to write the quantum Einstein equations

directly in terms of loops.

There have been several attempts to rewrite the

quantum Einstein equations directly in the loop

representation. In one of these attempts, the curva-

ture that appears in the Hamiltonian constraint was

represented by the ‘‘loop derivative.’’ This is a

differential operator that can be introduced in the

space of loops by considering that two loops that

differ by a small element of area are ‘‘close.’’ One

can build an attractive differential calculus in loop

space that actually encodes many of the kinematical

properties that are useful to formulate Yang–Mills

theory.

218 Knot Invariants and Quantum Gravity

The Hamiltonian constraint in terms of the loop

derivative is an operator that has an explicit form.

The coefficients of the Jones polynomial can also be

given an explicit form by computing perturbatively

the integral in the Chern–Simons theory. The results

are generalizations of the types of integrals that arise

in the self-linking number, but involving a larger

number of integrals. One can therefore envisage

carrying out an explicit computation in which one

checks if the coefficients of the Jones polynom ial are

annihilated or not by the Hamiltonian co nstraint of

quantum gravity in the loop representa tion. Such a

calculation has been carried out for the first few

coefficients. It turns out that the second coefficient

(the first coefficient is normalized to unity, so it

trivially satisfies the constraint) is indeed annihilated

by the Hamiltonian constraint of vacuum quantum

gravity (with zero cosmological constant). It has

been shown that the third coefficient is not, and

there are good arguments to indicate that other

coefficients will not be states of quan tum gravity.

So, a remarkable result has been found in that one

of the coefficients of the Jones polynomial (related

to the Arf and Casson invariants) is annihilated by a

version of the quantum Hamiltonian constraint of

general relativity. The result is quite nontrivial; it

requires a fair amount of calculation to actually

show that the coefficient is annihilated. The mean-

ing of this quantum state and the deep reason why it

is annihilated remain at present a mystery.

The quantum Hamiltonian constraint based on the

loop derivative makes certain assumptions about the

space of functions one is using to quantize the theory.

In quantum field theory, not all classical operators

have a well -defined quantum counterpart. The choice

being made is to assume that the curvature F

is a

well-defined quantum operator defined by the loop

derivative. Differentiability of knot polynomials is

not a new idea. It is the core idea of the Vassiliev knot

invariants, which are defined by a set of identities,

one of them acting as a ‘‘derivative in knot space.’’ It

can be shown that the loop derivative is a concrete

implementation of the Vassiliev derivative and, there-

fore, Vassiliev invariants are the ‘‘arena’’ in which this

version of quantum gravity takes place.

The Hamiltonian based on the loop derivative has

problems, in the sense that it is obtained by a

regularization procedure that requires extra external

geometric structures. This is common practice in

Yang–Mills theory, where one has at hand a fixed

external background metric. However, in gravity the

geometry is a dynamical object and, if one con-

structs expressions that resort to some fixed external

geometry, one gets inconsistencies. In particular, it is

expected that the Hamiltonian based on the loop

derivative will not reproduce the correct Poisson

algebra of canonical general relativity. This sort of

problem plagued early attempts to construct a

quantum version of the Hamiltonian constraint in

the early 1990s.

A point that we mentioned earlier but did not

elaborate upon, is that the Wilson loops constitute an

overcomplete basis of states. Therefore, if one takes a

quantum state and expands it on such a basis, one gets

that the coefficients of the expansion satisfy certain

identities, called the Mandelstam identities. These are

nonlinear identities that states in the loop representa-

tion have to satisfy. These identities are very incon-

venient at the time of constructing quantum states. The

identities stem from the fact that if one chooses a

matrix representation of the group of interest, the fact

that one is in a given representation is indicated by

certain identities the matrices satisfy. To break free

from these constraints, one possibility is to consider

multiple representations when constructing Wilson

loops. To do this, one considers piecewise-continuous

graphs with intersections (the nonintersecting case is

a trivial subcase). Along the lines connecting the

intersections one considers holonomies in a given

representation for a given line. In the case of the group

SU(2), which is the one of interest in quantum gravity,

such representations are labeled by a (half-) integer.

One then considers invariant tensors in the group to

‘‘tie the holonomies together’’ at intersections. The

resulting object is a gauge-invariant object for a given

connection based on a ‘‘spin network.’’ The latter

is an embedded piecewise-continuous graph with an

assignment of integers to each of its lines and an

assignment of ‘‘intertwiners’’ at each intersection (if

the intersections are trivalent or lower, one can choose

canonical intertwiners and forget about them).

One can then consider the ‘‘spin network represen-

tation’’ in which one expands gauge-invariant states

in terms of the basis of Wilson nets. Knot polynomials

for these types of graphs have been considered in the

mathematical literature (‘‘polynomials of colored

graphs’’). The construction with the Chern–Simons

state can be repeated, and there exist suitable general-

izations of the Kauffman bracket and Jones polyno-

mials. The Hamiltonian based on the loop derivative

can also be introduced in this context; again, its action

is well defined on suitable generalizations of Vassiliev

invariants for these kinds of graphs. This opens the

possibility of encoding the quantum dynamics of

general relativity as a combinatorial action in the

space of Vassiliev invariants.

An alternative Hamiltonian based on assuming that

the holonomies and the volume operators are well

defined quantum mechanically (but not the curvature)

has been introduced that has the advantage of not

Knot Invariants and Quantum Gravity 219

requiring external structures for its regularization. In

fact, it can be explicitly checked that it satisfies the

correct Poisson algebra without anomalies at the

quantum level. The exploration of the action of this

Hamiltonian constraint on knot polynomials has not

been carried out as systematically as for the one based

on the loop derivative, but it has been explicitly shown

that the first coefficient in the expansion of the Jones

polynomial is annihilated by this Hamiltonian con-

straint. The first coefficient, written in terms of loops,

was simply the numeral 1 and was automatically

annihilated. In terms of spin network states, the first

coefficient is the ‘‘chromatic evaluation’’ of the net-

work (the result of computing the Wilson loop on a

connection that is pure gauge). It is somewhat

nontrivial to show that this quantity is actually

annihilated by the Hamiltonian constraint in question.

At the moment, the issue of what the correct

Hamiltonian constraint is that describes a realistic

and physically correct theory of quantum gravity is

still open to debate. There are certain concerns that

the action of the operators considered up to now is

too simple to encompass the true dynamics of

general relativity. Constructing a semiclassical the-

ory that could confirm or deny the viability of the

proposals is a complicated task, since one has to

make contact with physics that is not diffeomor-

phism invariant in the context of a theory that is.

Moreover, in canonical quantum gravity, there

exists the ‘‘problem of time.’’ Since the Hamiltonian

vanishes, the dy namics implied by it is trivial, and

one has to disentangle the true dynamics by

relational constructions among the variables of the

theory. One then needs to compare the resulting

predictions with classical general relativity.

Whether the current proposals are viable and

whether knot theory will play a role at a ‘‘kinematical

level’’ or it will actually play a key role in the detailed

dynamics of quantum general relativity is yet to be

seen. It is reassuring that in partial constructions,

celebrated knot polynomials have appeared to have

some knowledge of the dynamics of the Einstein

equations.

Quantum gravity being an unfinished symphony,

we cannot entirely conclude how great an impact

knot theory will have on it in the end. One can only

note that beautiful mathematical results seem to tie

in naturally with the partial constructions that have

been carried out thus far.

See also: BF Theories; Braided and Modular Tensor

Categories; Finite-Type Invariants; Finite-type invariants

of 3-Manifolds; The Jones Polynomial; Knot Theory and

Physics; Loop Quantum Gravity; Mathematical Knot

Theory; Quantum Dynamics in Loop Quantum Gravity;

Quantum Geometry and its Applications; Yang–Baxter

Equations.

Further Reading

Ashtekar A (2005) Gravity and the quantum. New Journal of

Physics 7: 200.

Gambini R and Pullin J (1996) Loops, Knots, Gauge Theories

and Quantum Gravity. C ambridge: Cambridge University

Press.

Rovelli C (2001) Notes for a brief history of quantum gravity. In:

Ruffini R (ed.) Rome 2000, Recent Developments in Theo-

retical and Experimental General Relativity, Gravitation, and

Relativistic Field Theories, p. 742. Singapore: World Scientific.

Rovelli C (2004) Quantum Gravity. Cambridge: Cambridge

University Press.

Smolin L (2001) Three Roads to Quantum Gravity. New York:

Basic Books.

Thiemann T, Introduction to Modern Canonical General Rela-

tivity. Cambridge University Press (arXiv.org:gr-qc/0110034)

(to appear).

Knot Theory and Physics

L H Kauffman, University of Illinois at Chicago,

Chicago, IL, USA

Introduction

This article is an introduction to some of the relation-

ships between knot theory and theoretical physics.

Knots themselves are macroscopic physical phenomena

in three-dimensional space, occurring in rope, vines,

telephone cords, polymer chains, DNA, certain species

of eel, and many other places in the natural and man-

made world. The study of topological invariants of

knots leads to relationships with statistical mechanics

and quantum physics. This is a remarkable and deep

situation where the study of a certain (topological)

aspects of the macroscopic world is entwined with

theories developed for the subtleties of the microscopic

world. The present article is an introduction to the

mathematical side of these connections, with some

hints and references to the related physics.

We begin with a short introduction to knots,

links, braids, and the bracket polynomial invariant

of knots and links. The article then discusses

Vassiliev invariants of knots and links, and how

these invariants are naturally related to Lie algebras

and to Witten’s gauge-theoretic approach. This part

220 Knot Theory and Physics

of the article is an introduction to how Vassiliev

invariants in knot theory arise naturally in the

context of Witten’s functional integral.

The article is divided into several sections beyond

the introduction . Section two is a quick introduction

to the topology of knots and links. The third one

discusses Vassiliev invariants and invariants of rigid

vertex graphs. The fourth section introduces the

basic formalism and shows how Witten’s funct ional

integral is related directly to Vassiliev invariants.

The fifth section discusses the loop transform and

loop quantum gravity in this context. The final

section is an introduction to topological quantum

field theory, and to the use of these techniques in

producing unitary representations of the braid

group, a topic of intense interest in quantum

information theory.

Knots, Braids, and Bracket Polynomial

The purpose of this section is to give a quick

introduction to the diagrammatic theory of knots,

links, and braids. A knot is an embedding of a circle in

three-dimensional space, taken up to ambient isotopy.

That is, two knots are regarded as equivalent if one

embedding can be obtained from the other through a

continuous family of embeddings of circles in 3-space.

A link is an embedding of a disjoint collection of

circles, taken up to ambient isotopy. Figure 1 illus-

trates a diagram for a knot. The diagram is regarded

both as a schematic picture of the knot, and as a plane

graph with extra structure at the nodes (indicating

how the curve of the knot passes over or under itself

by standard pictorial conventions).

Ambient isotopy is mathematic ally the same as

the equivalence relation generated on diagrams by

the Reidemeister moves. These moves are illustrated

in Figure 2. Each move is performed on a local part

of the diagram that is topologically identical to the

part of the diagram illustrated in this figure (these

figures are representative examples of the types of

Reidemeister moves) without changing the rest of

the diagram. The Reidemeister moves are useful in

doing combin atorial topology with knots and links,

notably in working out the behavior of knot

invariants. A knot invariant is a function defined

from knots and links to some other mathematical

object (such as groups or polynomials or numbers)

such that equivalent diagrams are mapped to

equivalent objects (isomorphic groups, identical

polynomials, identical number s).

Another significant structure related to knots and

links is the Artin braid group. A braid is an

embedding of a collection of strands that have

their ends in two rows of points that are set one

above the other with respect to a choice of vertical.

The strands are not individually knotted and they

are disjoint from one another . S ee Figures 3–5 for

illustrations of braids and moves on braids. Braids

can be multiplied by attaching the bottom row of

one braid to the top row of the other braid. Taken

up to ambient isotopy, fixing the endpoints, the

braids form a group under this notion of multi-

plication. In Figure 3 we illustrate the form of the

basic generators of the braid group, and the form of

the relations amon g these generators. Figure 4

illustrates how to close a braid by attaching the

top strands to the bottom strands by a collection of

parallel arcs. A key theorem of Alexander states that

every knot or link can be represented as a closed

braid. Thus, the theory of braids is critical to the

Figure 1 A knot diagram.

III

Figure 2 The Reidemeister moves.

Braid generators

= s

–1

= 1

Figure 3 Braid generators.

Knot Theory and Physics 221

theory of knots and links. Figure 5 illustrates the

famous Borrowmean rings (a link of three unknotted

loops such that any two of the loops are unlinked)

as the closure of a braid.

We now discuss a significant example of an

invariant of knots and links, the bracket polynomial.

The bracket polynomial can be normalized to

produce an invariant of all the Reidemeister moves.

This normalized invariant is known as the Jones

(1985) polynomial. The Jones polynomial was

originally discovered by a different method than

the one given here.

The bracket polynomial, hKi= hKi (A), assigns to

each unoriented link diagram K a Laurent poly-

nomial in the variable A, such that

1. If K and K

are regularly isotopic diagrams, then

hKi= hK

2. If K q O denotes the disjoint union of K with an

extra unknotted and unlinked component O (also

called ‘‘loop’’ or ‘‘simple closed curve’’ or

‘‘Jordan curve’’), then

hK q Oi¼hKi

where

 ¼A

 A

2

3. hKi satisfies the following formulas:

hi¼AhiþA

1

hÞði

i¼A

1

hiþAhÞði

where the small diagrams represent parts of

larger diagrams that are identical except at the

site indicated in the bracket. We take the

convention that the letter chi, , denotes a

crossing where the curved line is crossing over

the straight segment. The barred letter denotes

the switch of this crossing, where the curved line

is undercrossing the straight segment.

In computing the bracket, one finds the following

behavior under Reidemeister move I:

hi¼A

h^i

and

i¼A

3

h^i

where  denotes a curl of positive type as indicated

in Figure 6,and

 indicates a curl of negative type,

as also seen in this figure. The type of a curl is the

sign of the crossing when we orient it locally. Our

convention of signs is also given in Figure 6. Note

that the type of a curl does not depend on the

orientation we choose. The small arcs on the right-

hand side of these formulas indicate the removal of

the curl from the corresponding diagram.

The bracket is invariant under regular isotopy and

can be normal ized to an invariant of ambient

isotopy by the definition

ðAÞ¼ðA

wðKÞ

hKiðAÞ

where we chose an orientation for K, and where

w(K) is the sum of the crossing signs of the oriented

link K. w(K) is called the writhe of K. The

convention for crossing signs is shown in Figure 6.

The State Summation

In order to obtain a closed formula for the bracket,

we now describe it as a state summation. Let K be

any unoriented link diagram. Define a state, S,ofK

to be a choice of smoothing for each crossing of K.

There are two choices for smoothing a given

crossing, and thus there are 2

states of a diagram

with N crossings. In a state we label each smoothing

with A or A

1

as in the expansion formula for the

bracket. The label is called a vertex weight of the

: or

–

Figure 6 Crossing signs and curls.

Hopf link

Figure-8 knot

Trefoil knot

Figure 4 Closing braids to form knots and links.

b CL(b)

Figure 5 Borromean rings as a braid closure.

222 Knot Theory and Physics

state. There are two evaluations related to a state.

The first one is the product of the vertex weights,

denoted hKjSi. The second evaluation is the number

of loops in the state S, denoted kSk.

Define the state summation, hK i, by the formula

hKi¼

hKjSi

kSk1

It follows from this definition that hKi satisfies the

equations

hi¼AhiþA

1

hÞði

hK q Oi¼hKi

hOi¼1

The first equation expresses the fact that the entire

set of states of a given diagram is the union, with

respect to a given crossing, of those states with an

A-type smoothing and those with an A

1

-type

smoothing at that crossing. The second and the

third equation are clear from the formula defining

the state summation. Hence, this state summation

produces the bracket polynom ial as we have

described it at the beginning of the section.

Remark By a change of variables one obtains the

original Jones polynomial, V

(t), for oriented knots

and links from the normalized bracket:

ðtÞ¼f

ðt

1=4

Remark The bracket polynomial provides a con-

nection between knot theory and physics, in that the

state summation expression for it exhibits it as a

generalized partition function defined on the knot

diagram. Partition functions are ubiquitous in

statistical mechanics, where they express the sum-

mation over all states of the physical system of

probability weighting functions for the individual

states. Such physical partition functions contain

large amounts of information about the correspond-

ing physical system. Some of this information is

directly present in the properties of the function,

such as the location of critical points and phase

transition. Some of the information can be obtained

by differentiating the partition funct ion, or perform-

ing other mathematical operations on it.

In fact, by defining a genera lization of the bracket

polynomial, defined on knot diagrams but not

invariant under the Reidemeister moves, we can

capture significant partition functions that are

physically meaningful. There is no room in this

survey to detail how this generalization can be used

to express the Potts model for planar graphical

configurations, and how it expresses the relationship

between the Potts model and the Temperley–Lieb

algebra in diagrammatic form. There is much more

in this connection with statistical mechanics in that

the local weights in a partition function are often

expressed in terms of solutions to a matrix equation

called the Yang–Baxter equation, that turns out to

fit perfectly invariance under the third Reidemeister

move. As a result, there are many ways to define

partition functions of knot diagrams that give rise to

invariants of knots and links. The subject is

intertwined with the algebraic structure of Hopf

algebras and qua ntum groups, useful for producing

systematic solutions to the Yang–Baxter equation.

In fact, Hopf algebras are deeply connected with

the problem of constructing invariants of three-

dimensional manifolds in relation to invariants of

knots. We have chosen, in this survey article, not to

discuss the details of these approaches, but rather to

proceed to Vassiliev invariants and the relationships

with Witten’s functional integral. The reader is

referred to Kauffman (1987, 1994, 2002), Jones

(1985),andReshetikhin and Turaev (1991) for

more information about relationships of knot theory

with stat istical mechanics, Hopf algebras, and

quantum groups. For topology, the key point is

that Lie algebras can be used to construct invariants

of knots and links. This is shown nowhere more

clearly than in the theory of Vassiliev invariants that

we take up in the next section.

Vassiliev Invariants and Invariants

of Rigid Vertex Graphs

In this section we study the combinatorial topology

of Vassiliev invariants. As we shall see, by the end of

this section, Vassiliev invariants are directly con-

nected with Lie algebras, and representations of Lie

algebras can be used to construct them. This aspect

of link invariants is one of the most fundamental for

connections with physics. Just as symmetry con-

siderations in physics lead to a fundamental rela-

tionship with Lie algebras, topological invariance

leads to a fundamental relationship of the theory of

knots and links with Lie algebras.

If V(K) is a (Laurent polynomial valued or, more

generally, commutative ring valued) invariant of

knots, then it can be naturally extended to an

invariant of rigid vert ex graphs by defining the

invariant of graphs in terms of the knot invariant via

an ‘‘unfolding of the vertex.’’ That is, we can regard

the vertex as a ‘‘black box’’ and replace it by any

tangle of our choice. Rigid vertex motions of the

graph prese rve the contents of the black box, and

hence implicate ambient isotopies of the link

obtained by replacing the black box by its contents.

Knot Theory and Physics 223

Invariants of knots and links that are evaluated on

these replacements are then automatically rigid vertex

invariants of the corresponding graphs. If we set up a

collection of multiple replacements at the vertices

with standard conventions for the insertions of the

tangles, then a summation over all possible replace-

ments can lead to a graph invariant with new

coefficients corresponding to the different replace-

ments. In this way, each invariant of knots and links

implicates a large collection of graph invariants.

The simplest tangle replacements for a 4-valent

vertex are the two crossings, positive and negative, and

the oriented smoothing. Let V(K) be any invariant of

knots and links. Extend V to the category of rigid

vertex embeddings of 4-valent graphs by the formula

VðK



Þ¼aVðK

ÞþbVðK



ÞþcVðK

where K

denotes a knot diagram K with a specific

choice of positive crossing, K



denotes a diagram

identical to the first with the positive crossing

replaced by a negative crossing and K



denotes a

diagram identical to the first with the positive

crossing replaced by a graphical node.

There is a rich class of graph invariants that can

be studied in this manner. The Vassiliev invariants

(Bar-Natan 1995) constitute the important special

case of these graph invariants where a = þ1, b =  1

and c = 0. Thus, V(G) is a Vassiliev invariant if

VðK



Þ¼VðK

ÞVðK



Call this formula the exchange identity for the

Vassiliev invariant V. See Figure 7.

V is said to be of finite type k if V(G) = 0

whenever jGj > k, where jGj denotes the number of

(4-valent) nodes in the graph G. The notion of finite

type is of extraordinary significance in studying

these invariants. One reason for this is the following

basic lemma.

Lemma If a graph G has exactly k nodes, then the

value of a Vassiliev invariant v

of type k on G,

(G), is independent of the embedding of G.

Proof Omitted.

The upshot of this lemma is that Vassiliev

invariants of type k are intimately involved with

certain abstract evaluations of graphs with k nodes.

In fact, there are restrictions (the four-term relations)

on these evaluations demanded by the topology and

it follows from results of Kontsevich (see Bar-Natan

(1995) that such abstract evaluations actually deter-

mine the invariants. The knot invariants derived from

classical Lie algebras are all built from Vassiliev

invariants of finite type. All of this is directly related

to Witten’s functional integral (Witten 1989).

In the next few figures we illustrate some of these

main points. In Figure 8 we show how one

associates a so-called chord diagram to represent

the abstract graph associated with an embedded

graph. The chord diagram is a circle with arcs

connecting those points on the circle that are welded

to form the corresponding graph. In Figure 9 we

illustrate how the four-term relati on is a conse-

quence of topological invarianc e.

In Figure 10 we show how the four-term relation is a

consequence of the abstract pattern of the commutator

)

V(K

) = V(K

) – V(K

–

)

⏐

⏐⏐⏐

)

⏐

–

)

⏐

Figure 7 Exchange identity for Vassiliev invariants.

Figure 8 Chord diagrams.

Figure 9 The four-term relation from topology.

224 Knot Theory and Physics

identity for a matrix Lie algebra. That is, we show how

a diagrammatic version of the formula

 T

¼ f

fits directly with the four-term relation. The formula

we have quoted here states that the commutator of

the matrices T

and T

is equal to a sum of the

matrices T

with coefficients (the structure coeffi-

cients of the Lie algebra) f

. Such a relation is the

most concrete way to define a matrix Lie algebra.

There are other levels of abstraction that can be

employed here. The same diagrammatic can be

interpreted directly in terms of the Jacobi identity

that defines a Lie algebra. We shall con tent

ourselves with this matrix point of view here, and

add that it is assumed here that the structure

coefficients are invariant under cyclic permutation,

an assumption that is not needed in the general case.

The four-term relation is directly related to a

categorical generalization of Lie algebras.

Figure 11 illustrates how the weights are assigned

to the chord diagrams in the Lie algebra case – by

inserting Lie algebra matrices into the circle and

taking a trace of a sum of matrix products. The

relationship between Vassiliev invariants and Lie

algebras has been known since Bar-Natan’s thesis

(see also Kauffman (1995).InBar-Natan (1995) the

reader will find a good account of Kontsevich’s

theorem, showing how Lie algebra weight systems,

and in fact any weight system satisf ying the four-

term relation, can be used to construct knot

invariants. Conceptually, the ideas behind the

Kontsevich theorem are directly related to Witten’s

approach to knot invariants via quantum field

theory. We give an exposition of this approach in

the next section of this article.

Example Let P

(t) = f

)(A = e

)wheref

(A)is

the normalized bracket polynomial invariant dis-

cussed in the last section. Then P

(t) is e xpressed as

a power series in t with coefficients v

(K),

n = 0, 1, 2, ..., that are invariants of the knot or

link K. It is not hard to show that these coefficient

invariants (extended to graphs so that the Vassiliev

exchange identity is satisfied) are Vassiliev invar-

iants of finite type. In fact, most of the so-called

polynomial invariants of knots and links (relatives

of the bracket and Jones polynomials) give rise to

Vassiliev invariants in just this way. T hus, Vassiliev

invariants of finite type are ubiquitous in this area

of knot theory. One can think of Vassiliev

invariants as building blocks for the other invar-

iants, or that these invariants are sources of

Vassiliev invariants.

Vassiliev Invariants and Witten’s

Functional Integral

Edward Witten (1989) proposed a formulation

of a clas s of 3-manifold invariants as generalized

Feynman integrals taking the form Z(M), where

ZðMÞ¼

DAe

ðik=4ÞSðM;AÞ

Here M denotes a 3-manifold without boundary and

A is a gauge field (also called a gauge potential or

gauge connection) defined on M. The gauge field is a

1-form on a trivial G-bundle over M with values in a

representation of the Lie algebra of G. The group G

corresponding to this Lie algebra is said to be the

gauge group. In this integral, the action S(M, A)is

taken to be the integral over M of the trace of the

Chern–Simons 3-form A ^ dA þ (2=3)A ^ A ^ A.

(The product is the wedge product of differential

forms.) Z(M) integrates over all gauge fields modulo

gauge equivalence.

The formalism and internal logic of Witten’s

integral supports the existence of a large class of

topological invariants of 3-manifolds and associated

invariants of knots and links in these manifolds.

–

–=

–==

Figure 10 The four-term relation from categorical Lie algebra.

tr( Σ T

)

Figure 11 Calculating Lie algebra weights.

Knot Theory and Physics 225

The invariants associated with this integral have

been given rigorous combinatorial descriptions but

questions and conjectures arising from the integral

formulation are still outstanding. Specific conjec-

tures about this integral take the form of just how it

implicates invariants of links and 3-manifolds, and

how these invariants behave in certain limits of the

coupling constant k in the integral. Many conjec-

tures of this sort can be verified through the

combinatorial model s. On the other hand, the really

outstanding conjecture about the integral is that it

exists! At the present time there is no measure

theory or generalization of measure theory that

supports it in full generality. Here is a formal

structure of great beauty. It is also a structure

whose consequences can be verified by a remarkable

variety of alternative means.

The formalism of the Witten integral implicates

invariants of knots and links corresponding to each

classical Lie algebra. In order to see this, we need to

introduce the Wilson loop. The Wilson loop is an

exponentiated version of integrating the gauge field

along a loop K in three space that we take to be an

embedding (knot) or a curve with transversal self-

intersections. For this discussion, the Wilson loop

will be denoted by the notation

ðAÞ

to denote the dependence on the loop K and the

field A. It is usually indicated by the symbolism

tr(Pe

). Thus,

ðAÞ¼tr





Here the P denotes path ordered integration – we

are integrating and exponentiating matrix valued

functions, and so must keep track of the order of the

operations. The symbol tr denotes the trace of the

resulting matrix. This Wilson loop integration exists

by normal means and does not require functional

integration.

With the help of the Wilson loop functional on

knots and links, Witten writes down a functional

integral for link invariants in a 3-manifold M:

ZðM; KÞ¼

DAe

ðik=4ÞSðM;AÞ

tr Pe



DAe

ðik=4ÞS

ðAÞ

Here S(M, A) is the Chern–Simons Lagrangian, as in

the previous discussion. We abbreviate S(M, A)asS

and write W

(A) for the Wilson loop. Unless

otherwise mentioned, the manifold M will be the

three-dimensional sphere S

An analysis of the formalism of this functional

integral reveals quite a bit abou t its role in knot

theory. One can determine how the Witten integral

behaves under a small deformation of the loop K.

Theorem

(i) Let Z(K) = Z(S

, K) and let Z(K) denote the

change of Z(K) under an infinitesimal change in

the loop K. Then

ZðKÞ¼ð4i=kÞ

dAe

ðik=4ÞS

½VolT

ðAÞ

where Vol = 

rst

The sum is taken over repeated indices, and

the insertion is taken of the matrices T

at the

chosen point x on the loop K that is regarded

as the center of the deformation. The volume

element Vol = 

rst

is taken with regard

to the infinitesimal directions of the loop

deformation from this point on the original

loop.

(ii) The same formula applies, with a different

interpretation, to the case where x is a double

point of transversal self-intersection of a loop K,

and the deformation consists in shifting one of

the crossing segments perpendicularly to the

plane of intersection so that the self-intersection

point disappears. In this case, one T

is inserted

into each of the transversal crossing segments so

that T

(A) denotes a Wilson loop with

a self-intersection at x and insertions of T

x þ 

and x þ 

, where 

and 

denote small

displacements along the two arcs of K that

intersect at x. In this case, the vo lume form is

nonzero, with two directions coming from the

plane of movement of one arc, and the perpen-

dicular direction is the direction of the other arc.

Remark One shows that the result of a topological

variation has an analytic expression that is zero if

the topological variation does not create a local

volume. Thus, we have shown that the integral of

(ik=4)S(A)

(A) is topologically invariant as long as

the curve K is moved by the local equivalent of

regular isotopy.

In the case of switching a crossing, the key point is

to write the crossing switch as a composition of first

moving a segment to obtain a transversal intersec-

tion of the diagram with itself, and then to continue

the motion to complete the switch. Up to the choice

of our conventions for constants, the switching

formula is, as shown below (see Figure 12).

226 Knot Theory and Physics