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

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ZðK

ÞZðK



¼ð4i=kÞ

DAe

ðik=4ÞS



jAi

¼ð4i=kÞZðT



where K



denotes the result of replacing the

crossing by a self-touching crossing. We distinguish

this from adding a graphical node at this crossing by

using the double-star notation.

A key point is to notice that the Lie algebra

insertion for this difference is exactly what is done

(in chord diagrams) to make the weight systems for

Vassiliev invariants (without the framing compensa-

tion). Thus, the formalism of the Witten functional

integral takes one directly to these weight systems in

the case of the classical Lie algebras. In this way, the

functional integral is central to the structure of the

Vassiliev invariants.

The Loop Transform and Quantum

Gravity

Suppose that (A) is a (complex-valued) function

defined on gauge fields. Then we define formally the

loop transform

(K), a function on embedd ed loops

in three-dimensional space, by the formula

ðKÞ¼

DA ðAÞW

ðAÞ

If  is a differential operator defined on (A), then

we can use this integral transform to shift the effect

of  to an operator on loops via integration by

parts:

 ðKÞ¼

DA ðAÞW

ðAÞ

¼

DA ðAÞW

ðAÞ

When  is applied to the Wilson loop, the result can

be an understandable geometric or topological

operation. One can illustrate this situation with

operators G and H:

G ¼F

=A

ðxÞ

H ¼

ars

=A

with summation over the repeated indices. Each of

these operators has the property that its action on

the Wilson loop has a geometric or topological

interpretation. One has

G ðKÞ¼

ðKÞ

where this variation refers to the effect of varying K.

As we saw in the previous section, this means that if

(K) is a topological invariant of knots and link s,

then

G (K) = 0 for all embedded loops K. This

condition is a transform analog of the equation

G (A) = 0. This equation is the differential analog

of an invariant of knots and links. It may happen

that 

(K) is not strictly zero, as in the case of our

framed knot invariants. For example, with

ðAÞ¼exp ðik=4Þ

trðA ^dA þð2=3ÞA ^A ^AÞ



we conclude that

G (K) is zero for flat deformations

(in the sense of the previous section) of the loop K,

but can be nonzero in the presence of a twist or curl.

In this sense, the loop transform provides a subtle

variation on the strict condition G (A) = 0.

In Ashtekar

et al. (1992) and other publications by

Ashtekar, Rovelli, Smolin, and their colleagues, the

loop transform is used to study a reformulation and

quantization of Einstein gravity. The differential-

geometric gravity theory of Einstein is reformulated

in terms of a background gauge connection and in the

quantization, the Hilbert space consists in functions

(A) that are required to satisfy the constraints

G = 0andH = 0. Thus, we see that

G(K)canbe

partially zero in the sense of producing a framed knot

invariant, and that

H(K)iszerofornon-self-

intersecting loops. This means that the loop trans-

forms of G and H can be used to investigate a subtle

variation of the original scheme for the quantization

of gravity. This program is being actively pursued by

a number of researchers. The Vassiliev invariants

arising from a topologically invariant loop transform

are of significance to this theory.

Braiding, Topological Quantum Field

Theory, and Quantum Computing

The purpose of this section is to discuss in a very

general way how braiding is related to topological

quantum field theory and to the enterprise

(Freedman et al. 2002) of using this sort of theory

as a model for anyonic quantum computation. The

ideas in the subject of topological quantum field

theory are well expressed by Michael Atiyah (1990)

z = z

= (c/k)z

+ O

(1/k

)

– z

Figure 12 The difference formula.

Knot Theory and Physics 227

and Edward Witten (1989). The simplest case of this

idea is C N Yang’s original interpretation of the

Yang–Baxter equation. Yang articulated a quantum

field theory in one dimension of space and one

dimension of time, in whi ch the R-matrix giving the

scattering amplitudes for an interaction of two

particles whose (let us say) spins corresponded to

the matrix indices so that R

is the amplitude for

particles of spin a and spin b to interact and produce

particles of spin c and d. Since these interactions are

between particles in a line, one takes the convention

that the particle with spin a is to the left of the

particle with spin b, and the particle with spin c is to

the left of the particle with spin d. If one follow s the

concatenation of such interactions, then there is an

underlying permutation that is obtained by follow-

ing strands from the bottom to the top of the

diagram (thinking of time as moving up the page).

Yang designed the Yang–Baxter equation for R so

that the amplitudes for a composite process depend

only on the underlying permutation corresponding

to the process and not on the individual sequences of

interactions.

In taking over the Yang–Baxter equation for

topological purposes, we can use the same inter-

pretation, but think of the diagrams with their

under- and over-crossings as modeling events in a

spacetime with two dimensions of space and one

dimension of time. The extra spatial dimension is

taken in displacing the woven strands perpendicular

to the page, and allows the use of braiding operators

R and R

1

as scattering matrices. Taking this picture

to heart, one can add other particle properties to the

idealized theory. In particular, one can add fusion

and creation vertices where, in fusion, two particles

interact to become a single particle and, in creation,

one particle changes (deca ys) into two particles.

Matrix elements corresponding to trivalent vertices

can represent these interactions (see Figure 13).

Once one introduces trivalent vertices for fusion

and creation, there is the question how these

interactions will behave in respe ct to the braiding

operators. There will be a matrix expression for the

compositions of braiding and fusion or creation as

indicated in Figure 14. Here we will restrict

ourselves to showing the diagrammatics with the

intent of giving the reader a flavor of these

structures. It is natural to assume that braiding

intertwines with creation as shown in Figure 15

(similarly with fusion). This intertwining identity is

clearly the sort of thing that a topologist will love,

since it indicates that the diagrams can be inter-

preted as embeddings of graphs in three-dimensional

space. Figure 16 illustrates the Yang–Baxter equa-

tion. The intertwining identity is an assumption like

the Yang–Baxter equation itself, which simplifies the

mathematical structure of the model.

It is to be expected that there will be an operator

that expresses the recoupling of vertex interactions

as shown in Figure 17 and labeled by Q. The actual

formalism of such an operator will parallel the

mathematics of recoupling for angular momentum

(see, e.g., Kauffman (1994)). If one just considers

the abstract structure of recoupling then one sees

that for trees with four branches (each with a single

root) there is a cycle of length 5, as shown in

Figure 13 Creation and fusion.

= R

Figure 14 Braiding.

Figure 17 Recoupling.

RIRI

Figure 16 Yang–Baxter equation.

Figure 15 Intertwining.

228 Knot Theory and Physics

Figure 17. One can start with any pattern of three

vertex interactions and go through a sequence of five

recouplings that bring one back to the same tree

from which one started. It is a natural simplifying

axiom to assume that this composition is the identity

mapping. This axiom is called the pentagon identity

(Figure 18).

Finally, there is a hexagonal cycle of interactions

between braiding, recoupling and the intertwining

identity as shown in Figure 19. One says that the

interactions satisfy the hexagon identity if this

composition is the identity.

A three-dimensional topological quantum field

theory is an algebra of interactions that satisfies the

Yang–Baxter equation, the intertwining identity, the

pentagon identity and the hexagon identity. There is

no room in this summary to detail the way that

these properties fit into the topology of knots and

three-dimensional manifolds, but a sketch is in

order. For the case of topological quantum field

theory related to the group SU(2) there is a

construction based entirely on the combinatorial

topology of the bracket polynomial (see the section

‘‘Kno ts, braids , and bracket polynom ial’’). For more

information on this approach, the reader is referred

to Kauffman (1994, 2002).

It turns out that the algebraic properties of a

topological quantum field theory give it enough

power to rigourously model three manifold invar-

iants described by the Witten integral. This is done

by regarding the 3-manifold as a union of two

handlebodies with boundary an orientable surface

of genus g. The surface is divided up into

trinions as illustrated in Figure 20.Atrinionisa

surface with boundary that is topologically equiva-

lent to a sphere with three punctures. In Figure 20

we illustrate two trinions, the second shown as a

neighborhood of a trivalent vertex, and a surface

of genus 3 that is decomposed into three t rinions.

It turns out that there is a way to associate a

vector space V(S

)toasurfacewithatrinion

decomposition, defined in terms of the associated

topological quantum field t heory, such that the

isomorphism class of the vector space V(S

) does

not depend upon the choice of decomposition.

This independence is guaranteed by the braiding,

hexagon, and pentagon identities in such a way

that one can associate a well-de fined vector jM



i in

V(S

)wheneverM is a 3-manifold whose boundary is

. Furthermore, if a closed 3-manifold M

is decom-

posed along a surface S

into the union of M



and M

where these parts are otherwise disjoint 3-manifolds

with boundary S

, then the inner product I(M) =



i is, up to normalization, an invariant of the

3-manifold M

. With the definition of topological

quantum field theory given above, knots and links can

be incorporated as well, so that one obtains a source of

invariants I (M

, K) of knots and links in orientable

3-manifolds.

The invariant I(M

, K) can be formally compared

with the Witten integral

ZðM

; KÞ¼

DAe

ðik=4ÞSðM;AÞ

ðAÞ

Figure 18 Pentagon identity.

Figure 19 Hexagon identity.

Figure 20 Decomposition of a surface into trinions.

Knot Theory and Physics 229

It can be shown that up to limits of the heuristics,

Z(M, K) and I(M

, K) are essentially equivalent for

appropriate choice of gauge grou ps.

This point of view leads to more abstract

formulations of topological quantum field theories

as ways to associate vector spaces and linear

transformations to manifolds and cobordisms of

manifolds. (A cobordism of surfaces is a 3-manifold

whose boundary consists of these surfaces.)

As the reader can see, a three-dimensional TQFT is,

at base, a highly simplified theory of point-particle

interactions in (2 þ 1)-dimensional spacetime. It can be

used to articulate invariants of knots and links and

invariants of 3-manifolds. The reader interested in the

SU(2) case of this structure and its implications for

invariants of knots and 3-manifolds can consult

Kauffman (1994, 2002)andCrane (1991). One expects

that physical situations involving 2 þ 1 spacetime will

be approximated by such an idealized theory. It is

thought, for example, that aspects of the quantum Hall

effect will be related to topological quantum field

theory (Wilczek 1990). One can imagine a physics

where the geometrical space is two dimensional and the

braiding of particles corresponds to their interactions

through circulating around one another in the plane.

Anyons are particles that do not just change their wave

functions by a sign under interchange, but rather by a

complex phase or even a linear combination of states. It

is hoped that TQFT models will describe applicable

physics. One can think about the possible applications

of anyons to quantum computing. The TQFTs then

provide a class of anyonic models where the braiding is

essential to the physics and to the quantum

computation.

The key point in the application and relationship

of TQFT and quantum information theory is, in our

opinion, contained in the structure illustrated in

Figure 21. There we show a more complex braiding

operator, based on the composition of recoupling

with the elementary braiding at a vertex. (This

structure is implicit in the hexagon identity of

Figure 19.) The new braiding operator is a source of

unitary representations of braid group in situations

(which exist mathematically) where the recoupling

transformations are themselves unitary. This kind of

pattern is utilized in the work of Freedman et al.

(2002) and in the case of classical angular momentum

formalism has been dubbed a ‘‘spin-network quantum

simulator’’ by Rasetti and collaborators (see, e.g.,

Marzuoli and Rasetti (2002). Kauffman and Lomo-

naco (2006) show how certain natural deformations

(Kauffman 1994)ofPenrose (1969) spin networks can

be used to produce such the Freedman–Kitaev model

for anyonic topological quantum computation. It is

legitimate to speculate that networks of this kind are

present in physical reality.

Quantum computing can be regarded as a study of

the structure of the preparation, evolution, and

measurement of quantum systems. In the quantum

computation model, an evolution is a composition of

unitary transformations (usually finite-dimensional

over the complex numbers). The unitary transforma-

tions are applied to an initial state vector that has been

prepared prior to this process. Measurements are

projections to elements of an orthonormal basis of

the space upon which the evolution is applied. The

result of measuring a state j i, written in the given

basis, is probabilistic. The probability of obtaining a

given basis element from the measurement is equal to

the absolute square of the coefficient of that basis

element in the state being measured.

It is remarkable that the above lines constitute an

essential summary of quantum theory. All applications

of quantum theory involve filling in details of unitary

evolutions and specifics of preparations and measure-

ments. Such unitary evolutions can be seen as approxi-

mated arbitrarily closely by representations of the Artin

braid group. The key to the anyonic models of quantum

computation via topological quantum field theory, or

via deformed spin networks, is that all unitary evolu-

tions can be approximated by a single coherent method

for producing representations of the braid group. This

beautiful mathematical fact points to a deep role for

topology in the structure of quantum physics.

The future of knots, links, and braids in relation

to physics will be very exciting. There is no question

that unitary representations of the braid group and

quantum invariants of knots and links play a

fundamental role in the mathematical structure of

quantum mechanics, and we hope that time will

show us the full meaning of this relationship.

Acknowledgments

It gives the author pleasure to thank the National

Science Foundation for support of this research

–1

= Q

–1

Figure 21 A more complex braiding operator.

230 Knot Theory and Physics

under NSF Grant DMS-0245588. Much of this

effort was sponsored by the Defense Advanced

Research Projects Agency (DARPA) and Air Force

Research Laboratory, Air Force Materiel Command,

USAF, under agreement F30602-01-2-05022. The

views and conclusions contained herein are those of

the authors and should not be interpreted as

necessarily representing the official policies or

endorsements, either expressed or implied, of the

Defense Advanced Research Projects Agency, the Air

Force Research Laboratory, or the US Government.

See also: Chern–Simons Models: Rigorous Results;

Functional Integration in Quantum Physics; Knot

Homologies; Knot Invariants and Quantum Gravity;

Large-N and Topological Strings; Loop Quantum Gravity;

Mathematical Knot Theory; Schwarz-Type Topological

Quantum Field Theory; The Jones Polynomial;

Topological Knot Theory and Macroscopic Physics;

Topological Quantum Field Theory: Overview;

Two-Dimensional Conformal Field Theory and Vertex

Operator Algebras; von Neumann Algebras: Introduction,

Modular Theory, and Classification Theory; Yang–Baxter

Equations.

Further Reading

Alexander JW (1923) Topological invariants of knots and links.

Transactions of the American Mathematical Society 20: 275–306.

Atiyah MF (1990) The Geometry and Physics of Knots.

Cambridge: Cambridge University Press.

Ashtekar A, Rovelli C, and Smolin L (1992) Weaving a classical

geometry with quantum threads. Physical Review Letters 69: 237.

Bar-Natan D (1995) On the Vassiliev knot invariants. Topology

34: 423–472.

Crane L (1991) 2-d physics and 3-d topology. Communications

Mathematical Physics 135(3): 615–640.

Freedman MH, Kitaev A, and Wang Z (2002) Simulation of

topological field theories by quantum computers. Communica-

tions in Mathematical Physics 227: 587–603 (quant-ph/0001071).

Jones VFR (1985) A polynomial invariant for links via von

Neumann algebras. Bulletin of the American Mathematical

Society 129: 103–112.

Kauffman LH (1987) State models and the Jones polynomial.

Topology 26: 395–407.

Kauffman LH (1994) Temperley–Lieb Recoupling Theory and

Invariants of Three-Manifolds, Annals Studies, vol. 114.

Princeton: Princeton University Press.

Kauffman LH (1991) Knots and Physics (2nd edn., 1993, 3rd

edn., 2002). Singapore: World Scientific Publishers.

Kauffman LH (1995) Functional integration and the theory of

knots. Journal of Mathematical Physics 36(5): 2402–2429.

Kauffman LH (2002) Quantum computation and the Jones

polynomial. In: Lomonaco S, Jr. (ed.) Quantum Computation

and Information, AMS CONM/305. Providence, RI: American

Mathematical Society pp. 101–137.

Kauffman LH and Lo monaco SJ, Jr. (2002) Q uantum entangle-

ment and topological entanglement. New Journal of Physics

4: 73.1–73.18 (http://www.njp.org).

Kauffman LH and Lomonaco SJ, Jr. (2006) Spin networks and

anyonic topological quantum computing (in preparation).

Marzuoli A and Rasetti M (2002) Spin network quantum

simulator. Physics Letters A 306: 79–87.

Penrose R (1969) Angular momentum: an approach to combina-

torial spacetime. In: Bastin T (ed.) Quantum Theory and

Beyond. Cambridge: Cambridge University Press.

Reshetikhin NY and Turaev V (1991) Invariants of three

manifolds via link polynomials and quantum groups. Inven-

tiones Mathematicae 103: 547–597.

Wilczek F (1990) Fractional Statistics and Anyon Superconduc-

tivity. Berlin–Heidelberg–New York: Springer-Verlag.

Witten E (1989) Quantum field Theory and the Jones Polynomial.

Communications in Mathematical Physics 121: 351–399.

Kontsevich Integral

S Chmutov and S Duzhin, Petersburg Department

of Steklov Institute of Mathematics, St. Petersburg,

Russia

Introduction

The Kontsevich integral was invented by Kontsevich

(1993) as a tool to prove the fundamental theorem of

the theory of finite-type (Vassiliev) invariants (see Bar-

Natan (1995a)). It provides an invariant exactly as

strong as the totality of all Vassiliev knot invariants.

The Kontsevich integral is defined for oriented

tangles (either framed or unframed) in R

; therefore,

it is also defined in the particular cases of knots,

links, and braids (see Figure 1).

As a starter, we give two examples where simple

versions of the Kontsevich integral have a

straightforward geometrical meaning. In these

examples, as well as in the general construction of

the Kontsevich integral, we represent 3-space R

the product of a real line R with coordinate t and a

complex plane C with complex coo rdinate z.

Example 1 The number of twists in a braid with

two strings z

(t) and z

(t) placed in the slice 0  t  1

(see Figure 2) is equal to

2i

 dz

 z

Figure 1 A tangle, a braid, a link, and a knot.

Kontsevich Integral 231

Example 2 The linking number of two spatial

curves K and K

(see Figure 3) can be computed as

lkðK; K

Þ¼

2i

m<t<M

dðz

ðtÞz

ðtÞÞ

ðtÞz

ðtÞ

where m and M are the minimum and the maximum

values of t on the link K [ K

, j is the index that

enumerates all possible choices of a pair of strands

of the link as functions z

(t), z

(t) corresponding to K

and K

, respectively, and "

= 1 according to the

parity of the number of chosen strands that are

oriented downwards.

The Kontsevich integral can be regarded as a far-

going generalization of these formulas. It aims at

encoding all information about how the horizontal

chords on the knot (or tangle) rotate when moved in

the vertical direction. From a more general view-

point, the Kontsevich integral represents the mono-

dromy of the Knizhnik–Zamolodchikov connection

in the complement to the union of diagonals in C

(see Bar-Natan (1995a) and Ohtsuki (2002)).

Chord Diagrams and Weight Systems

Algebras A(p)

The Kontsevich integral of a tangle T takes values in

the space of chord diagrams supported on T.

Let X be an oriented one-dimensional manifold,

that is, a collection of p numbered oriented lines and

q numbered oriented circles. A chord diagram of

order n supported on X is a collection of n pairs

of unordered points in X, considered up to an

orientation- and component-preserving diffeo-

morphism. In the vector space formally generated

by all chord diagrams of order n, we distinguish the

subspace spanned by all four-term relations

–

+–

where thin lines designate chords, while thick lines are

pieces of the manifold X. Apart from the fragments

shown, all the four diagrams are identical. The

quotient space over all such combinations is denoted

by A

(X) = A

(p, q). Let A(p, q) = 

n = 0

(p, q)

and let

A(p, q) be the graded completion of A(p, q)

(i.e., the space of formal infinite series

i = 0

with

(p, q)). If, moreover, we divide A(p, q)byall

‘ ‘framing independence’ ’ relations (any diagram with

an isolated chord, i.e., a chord joining two adjacent

points of the same connected component of X,issetto

0), then the resulting space is denoted by A

(p, q), and

its graded completion by

(p, q).

The spaces A(p,0)= A(p) have the structure of an

algebra (the product of chord diagrams is defined by

concatenation of underlying manifolds in agreement

with the orientation). Closing a line component into a

circle, we get a linear map A(p , q) !A(p  1, q þ 1)

whichisanisomorphismwhenp = 1. In particular,

A(S

) ﬃA(R

) has the structure of an algebra; this

algebra is denoted simply by A; the Kontsevich integral

of knots takes its values in its graded completion

A. Another algebra of special importance is

A(3) =

A(3, 0), because it is where the Drinfeld

associators live.

Hopf Algebra Structure

The algebra A(p) has a natural structure of a Hopf

algebra with the coproduct  defined by all ways to

split the set of chords into two disjoint parts. To give

a convenient description of its primitive space, one

can use generalized chord diagrams. We now allow

trivalent vertices not belonging to the supporting

manifold and use STU relations (Bar-Natan 1995a)

=–

to express the generalized diagrams as linear combi-

nations of conventional chord diagrams, for example,

=–

Then the primitive space coincides with the sub-

space of A(p) spanned by all connected generalized

chord diagrams (‘‘connected’’ means that they remain

connected when the supporting manifold X is

disregarded).

(t )

Figure 2 Counting the number of twists.

(t )

′

Figure 3 Counting the linking number.

232 Kontsevich Integral

Weight Systems

A ‘‘weight system’’ of degree n is a linear function

on the space A

. Every Vassiliev invariant v of

degree n defines a weight system symb(v)ofthe

same degree called its ‘‘symbol.’’

Algebras B(p)

Apart from the spaces of chord diagrams modulo four-

term relations, there are closely related spaces of Jacobi

diagrams. A Jacobi diagram is defined as a unitrivalent

graph, possibly disconnected, having at least one

vertex of valency 1 in each connected component and

supplied with two additional structures: a cyclic order

of edges in each trivalent vertex and a labeling of

univalent vertices taking values in the set {1, 2, ..., p}.

The space B(p) is defined as the quotient of the vector

space formally generated by all p-colored Jacobi

diagrams modulo the two types of relations:

Antisymmetry: IHX:

–

The disjoint union of Jacobi diagrams makes the

space B(p)intoanalgebra.

The symmetrization map 

: B(p) !A(p), defined

as the average over all ways to attach the legs of color i

to ith connected component of the underlying manifold

is an isomorphism of vector s paces (the formal

PBW isomorphi sm ( Bar-Natan 1995a, Le and

Murakami 1995) which is not compatible with

the multiplication. The relation between A(p)and

B(p) very much resembles the relation between

the universal enveloping algebra and the sym-

metric algebra of a Lie algebra. The algebra

B= B(1) is used to write out the explicit formula

for the Kontsevich integral of the unknot (see

Bar-Natan et al. (2003) and below).

The Construction

Kontsevich’s Formula

We will explain the construction of the Kontsevich

integral in the classical case of (closed) oriented

knots; for an arbitrary tangle T, the formula is the

same; only the result is interpreted as an element of

A(T). As above, represent three-dimensional space

as a direct product of a complex line C with

coordinate z and a real line R with coordinate t.

The integral is defined for Morse knots, that is,

knots K embedded in R

= C

 R

in such a way

that the coordinate t restricted to K has only

nondegenerate (quadratic) critical points. (In fact,

this condition can be weakened, but the class of

Morse knots is bro ad enough and convenient to

work with.)

The Kontsevich integral Z(K) of the knot K is the

following element of the complet ed algebra

ZðKÞ¼

m¼0

ð2iÞ



min

< t

< < t

max

are noncritical

P¼fðz

Þg

ð1Þ

 D

j¼1

 dz

 z

Explanation of the Constituents

The real numbers t

min

and t

max

are the minimum and

the maximum of the function t on K.

The integration domain is the m-dimensional

simplex t

min

< t

< < t

< t

max

divided by the

critical values into a certain number of ‘‘connected

components.’’ For example, Figure 4 shows an

embedding of the unknot where, for m = 2, the

integration domain has six connected components.

The number of summands in the integrand is

constant in each connected component of the

integration domain, but can be different for different

components. In each plane {t = t

}  R

choose an

unordered pair of distinct points (z

, t

) and (z

, t

)on

K, so that z

) and z

) are continuous branches of

the knot. We denote by P = {(z

, z

)} the collection of

such pairs for j = 1, ..., m. The integrand is the sum

over all choices of the pairing P. In the example

above for the component {t

min

< t

< c

, c

< t

max

}, we have only one possible pair of poin ts on

the levels {t = t

} and {t = t

}. Therefore, the sum

over P for this component consists of only one

summand. Unlike this, in the component {t

min

< c

, c

< t

< c

}, we still have only one

max

min

max

min

max

min

Figure 4 Connected components.

Kontsevich Integral 233

possibility for the level {t = t

}, but the plane {t = t

}

intersects our knot K in four points. So we have



= 6 possible pairs (z

, z

), and the total number

of summands is six (see Figure 5).

For a pairing P, the symbol ‘‘#

’’ denotes the

number of points (z

, t

)or(z

, t

)inP, where the

coordinate t decreases along the orientation of K.

Fix a pairing P.ConsidertheknotK as an oriented

circle and connect the points (z

, t

)and(z

, t

)bya

chord. Up to a diffeomorphism, this chord does not

depend on the value of t

within a connected

component. We obtain a chord diagram with m

chords. The corresponding element of the algebra A

is denoted by D

. Figure 5, for each connected

component in our example, shows one of the possible

pairings, the corresponding chord diagram with

the sign (1)

and the number of summands of the

integrand (some of which are equal to zero in A

due

to the framing independence relation).

Over each connected component, z

and z

are

smooth functions of t

j¼1

 dz

 z

we mean the pullback of this form to the integration

domain of variables t

, ..., t

. The integration

domain is considered with the orientation of the

space R

defined by the natural order of the

coordinates t

, ..., t

By convention, the term in the Kontsevich integral

corresponding to m = 0 is the (only) chord diagram

of order 0 with coefficient 1. It represents the unit of

the algebra A

Framed Version of the Kontsevich Integral

Let K be a framed oriented Morse knot with writhe

number w(K). Denote the corresponding knot

without fram ing by



K. The framed version of the

Kontsevich integral can be defined by the formula

ðKÞ¼e

ðwðKÞ=2Þ

 Zð



KÞ2

where  is the chord diagram with one chord and the

integral Z(



K) 2

is understood as an element of the

completed algebra

A (without one-term relations) by

virtue of a natural inclusion A

!Adefined as identity

on the primitive subspace of A

(see Goryunov

(1999) and Le and Murakami (1996)).

Basic Properties

Constructing the Universal Vassiliev Invariant

The Kontsevich integral Z(K)

1. converges for any Morse knot K,

2. is invar iant under deformations of the knot in the

class of Morse knots, and

3. behaves in a predictable way under the deforma-

tion that adds a pair of new critical points to a

Morse knot:

= Z(H

)

Here the first and the third pictures depict two

embeddings of an arbitrary knot, differing only in

the shown fragment, H =

is the ‘‘hump’’ (unknot

embedded in R

in the specified way), and the

product is the product in the completed algebra

6 summands

(–1)

1 summand

(–1)

36 summands

(–1)

1 summand

(–1)

6 summands

(–1)

1 summand

(–1)

Figure 5 Pairings and chord diagrams.

234 Kontsevich Integral

of chord diagrams. The last equality allows one to

define a genuine knot invariant by the formula

IðKÞ¼ZðKÞ=ZðHÞ

c=2

where c denotes the number of critical points of K and

the ratio means the division in the algebra

according

to the rule (1 þ a)

1

= 1  a þ a

 a

þ .

The expression I(K) is sometimes referred to as

the ‘‘final’’ Kontsevich integral as opposed to the

‘‘preliminary’’ Kontsevich integral Z(K). It repre-

sents a universal Vassiliev invariant in the following

sense: Let w be a weight system, that is, a linear

functional on the algebra

. Then the composition

w(I(K)) is a numerical Vassiliev invariant, and any

Vassiliev invariant can be obtained in this way.

The final Kontsevich integral for framed knots is

defined in the same way, using the hump H with

zero writhe number.

Is Universal Vassiliev Invariant Universal?

At present, it is not known whether the Kontsevich

integral separates knots, or even if it can tell the

orientation of a knot. However, the corresponding

problem is solved, in the affirmative, in the case of

braids and string links (theorem of Kohno–

Bar-Natan (Bar-Natan 1995b , Kohno 1987).

Omitting Long Chords

We will state a technical lemma which is highly

important in the study of the Kontsevich integral. It

is used in the proof of the multiplicativity, in the

combinatorial construction, etc.

Suppose we have a Morse knot K with a

distinguished tangle T ( Figure 6). Let m and M be

the maximal and minimal values of t on the tangle T.

In the horizontal planes between the levels m and M,

we can distinguish two kinds of chords: ‘ ‘short’’

chords that lie either inside T or inside K nT,and

‘ ‘long’’ chords that connect a point in T with a point

in K nT.DenotebyZ

(K) the expression defined by

thesameformulaastheKontsevichintegralZ(K)

where only short chords are taken into consideration.

More exactly, if C is a connected component of the

integration domain whose projection on the coordi-

nate axis t

is entirely contained in the segment [m, M],

then in the sum over the pairings P we include only

those pairings that include short chords.

Lemma ‘‘Long’’ chords can be omitted when

computing the Kontsevich integral: Z

(K) = Z(K).

Kontsevich’s Integral and Operations on Knots

The Kontsevich integral behaves in a nice way with

respect to the natural operations on knots, such as

mirror reflection, changing the orientation of the

knot, mutation of knots (see Chmutov and Duzhin

(2001)), cabling (see Willerton (2002)). We give

some details regarding the first two items.

Fact 1 Let R be the operation that sends a knot

to its mirror image. Define the corresponding

operation R on chord diagrams as multiplication

by (1)

, where n is the order of the diagram. Then

the Kontsevich integral commutes with the opera-

tion R: Z(R(K)) = R(Z(K)), where by R(Z(K)) we

mean simultaneous application of R to all the chord

diagrams participating in Z(K).

Corollary The Kontsevich integral Z(K) and the

universal Vassiliev invariant I(K) of an amphicheiral

knot K consist only of even order terms. (A knot K is

called ‘‘amphicheiral,’’ if it is equivalent to its mirror

image: K = R(K).)

Fact 2 Let S be the operation on knots which

inverts their orientation. The same letter will also

denote the analogous operat ion on chord diagrams

(inverting the orientation of the outer circle or,

which is the same thing, axial symmetry of the

diagram). Then the Kontsevich integral commutes

with the operation S of inverting the orientation:

Z(S(K)) = S(Z(K)).

Corollary The following two assertions are

equivalent:

(i) Vassiliev invariants do not distinguish the

orientation of knots and

(ii) all chord diagrams are symmetric: D = S(D)

modulo four-term relations

The calculations of Kneissler (1997) show that up

to order 12 all chord diagrams are symmetric. For

bigger orders, the problem is still open.

Multiplicative Properties

The Kontsevich integral for tangles is multiplicative:

ZðT

ÞZðT

Þ¼ZðT

 T

whenever the product T

 T

, defined by vertical

concatenation of tangles, exists. Here, the product

short

long

Figure 6 Short and long chords.

Kontsevich Integral 235

on the left-hand side is understood as the image of

the element Z(T

)  Z(T

) under the natural map

A(T

) A( T

) !A(T

 T

This simple fact has two important corollaries:

1. For any knot K, the Kontsevich integral Z(K)is

a group-like element of the Hopf algebra

that is,

ðZðK ÞÞ ¼ ZðKÞZðKÞ

where  is the comultiplication in A defined

above.

2. The final Kontsevich integral, taken in a different

normalization

ðKÞ¼ZðHÞIðKÞ¼

ZðKÞ

ZðHÞ

c=21

is multiplicative with respect to the connected

sum of knots:

ðK

Þ¼I

ðK

ÞI

ðK

Arithmetical Properties

For any knot K the coefficients in the expansion of

Z(K) over an arbitrary basis consisting of chord

diagrams are rational (see Kontsevich (1993), Le

and Murakami (1996), and below).

Combinatorial Construction of the

Kontsevich Integral

Sliced Presentation of Knots

The idea i s to cut the knot into a number of

standard simple tangles, compute the Kontsevich

integralforeachofthem,andthenrecoverthe

integral of the whole knot from these simple

pieces.

More exactly, we represent t he knot by a family

of plane diagrams continuously depending on a

parameter "2(0, "

) and cut by horizontal planes

into a number of slices with the following

properties.

1. At every boundary level of a slice (dashed lines

in the pictures below), the dist ances betwee n

various strings are asymptotically pro-

portional to d ifferent whole powers of th e

parameter ".

2. Every slice contains exactly one special event

and several strictly vertical strings which

are farther away (at lower powers of ")from

any string participating in the e vent than its

width.

3. There are three types of special events:

associativity:

= A

–

braiding:

–

min/max:

m =

M =

where, in the two last cases , the strings may be

replaced by bunches of parallel strings which

are closer to each other than the width of this

event.

Recipe of Computation of the Kontsevich Integral

Given such a sliced representation of a knot, the

combinatorial algorithm to compute its Kontsevich

integral consists in the following:

1. Replace each special event by a series of chord

diagrams supported on the correspond ing tangle

according to the rule

m; M 7!1

7!R; B

7!R

1

7!; A



7!

1

where

R ¼

 exp



2  2

3!  2

þ

 ¼ 1 

ð2Þ

ð2iÞ

½a; b



ð3 Þ

ð2iÞ

ð½a; ½a; b þ ½b; ½a; bÞ þ 

( 2

A(3) is the Knizhnik–Zamolodchikov

Drinfeld associator defined be low; it is an infinite

series in two variables a =

, b = ).

2. Compute the product of all these series from

top to bottom taking into account the connec-

tion of the strands of different tangles, thus

obtaining an element of the algebra

To accomplish the algorithm, we need two

auxiliary operations on chord diagrams:

1. S

: A(p)!A(p) defined as multiplication by

(1)

on a chord diagram containing k end-

points of chords on the string number i.Thisis

the correction term in the computation of R

and  in the case when the tangle contains

236 Kontsevich Integral