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

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If G acts on M freely, then

ðX; AÞ¼

½de

S

CMR

½X;;A

½33

satisfies [30]. The factor (X, A)in[30] is called

‘‘projection’’ in Cordes et al. (1996).

Let E ! M be a G-equivariant vector bundle with

a fixed G-invariant connection r, moment , and

an invariant section s. Consider the supers pace

action

½X; ; ; A¼S

½X; ; AþS

CMR

½X; ; A

In the Wess–Zumino gauge and after the Gaussian

integral over f, it becomes the Atiyah–Jeffrey action

½x; ;;;;

¼ S

½x; ;;þS

CMR

½x; ;;;½34

If s intersect the zero section transversely and G acts

on s

1

(0) freely, then s

1

(0)=G is smooth and

1

ð0Þ=G



! ¼

½dx½d ½d½d½d½d

O

ðx; ;Þe

S

½x;;;;;

½35

for any closed equivariant form ! on M. Equation

[35] is the formula of Atiyah and Jeffrey (1990) and

of Witten (1988a) in an infinite-dimensional setting.

When s

1

(0)=G is not smooth, the right-hand side of

[35] can be regarded as a definition of the left-hand

side.

It is often convenient to add to S

another term

S½X; ; A¼

0j1

ð½

F; ; DÞ

ﬃﬃﬃﬃﬃﬃﬃ

1

ð; ½; Þ þ

ð½; ; ½; Þ ½36

Since [36] is -exact and no new field is added, the

integral [35] does not change if S is added to S

Applications to Cohomological

Field Theories

We now apply the Mathai–Quillen construction

formally to a number of cases in which both the

rank of the vector bundle and the dimension of the

base space are infinite. Thus, the (bosonic and

fermionic) integrals in [24] or [35] become path

integrals in quantum mechanics or quantum field

theory.

Supersymmetric Quantum Mechanics

Let (M, g) be a Riemannian manifold and LM =

Map(S

, M), the loop space. At each point u 2 LM,

which is a map u : S

! M, the tangent space is

LM =(u



TM). In particular,

u = du=dt, where t

is a parameter on S

, is a tangent vect or at u and

u 7!

u is a vector field on LM. For any Morse

function h on M, s(u) =

u þ (grad h)  u is another

vector field on LM.

Vector fields on LM can be identified as sections of

the bundle ev



TM ! S

 LM,whereev:S



LM ! M is the evaluation map. The Levi-Civita

connection r on TM pulls back to a connection on



TM and the covariant derivatives along LM define

a natural connection r

on T(LM). For example,

for any tangent vector V 2 T

LM =(u



TM), we

have r

s(u) = r

V þ (r

grad h)  u,wherer

the pullback connection on u



TM.TheRiemann

curvature tensor R on M determines that on LM.

The (infinite-dimensional) analog of [22] is

½du½d ½dexp 

dtL½u; ;



½37

where ,  2 T

LM =(u



TM) are fermionic and

L½u; ;¼

gð

u þ grad h;

u þ grad hÞ



ﬃﬃﬃﬃﬃﬃﬃ

1

gð; r

þr

grad hÞ



gð; R ð ; ÞÞ½38

Here and below, factors of

ﬃﬃﬃﬃﬃﬃ

1

and 2 in [22] are

absorbed in the path-integral measure. [38] is, up to

a total derivative, the Lagrangian of the Euc lidean

N = 2 supersymmetric quantum mechanics on M.

The partition function [37] is equal to Euler

characteristic number of LM or M, which can

be confirmed by an (exact) stationary-phase

calculation.

Topological Sigma Model

Let  be a Riemann surface with complex structure

" and let (M, !) be a symplectic manifold with a

compatible almost-complex structure J. Let E be a

vector bundle over Map(, M) so that the fiber over

u is E

=(u



TM  T



). For any u 2 Map(, M),

du 2E

and u 7!du is a section of E. The pullback

of the Levi-Civita connection on TM, tensored with

a connection on T



, defines a connection on E.

ThevectorbundletowhichweapplytheMathai–

Quillen formalism is the antiholomorphic part E

of E.

The fiber over u 2 Map(, M)isE

=((u



TM 



)

). The sub-bundle E

has a connection r

via

projection from E. E

has a natural section

s : u 7!@

u = (1=2)(du þ J  du  "). Solutions to the

equation



u = 0 are pseudoholomorphic (or

J-h olomorphic) curves; let M= s

1

(0) be the space of

such curves. Its (virtual) dimension is

dim M¼

ðÞdim M þ 2c

ðu



TMÞ½39

Mathai–Quillen Formalism 397

Along any V 2 T

Map(, M ) =( u



TM), the covar-

iant derivative of s =



is calculated in Wu (1995):



Þ¼

ðr

V þ J r

V  "Þ

J ðdu  " þ J  duÞ½40

where r

is the pullback connection on u



TM.

To write the Mathai–Quillen formalism for the

bundle E

! Map(, M), we let 2 (u



TM) and

 2 ((u



TM  T



)

) be fermionic fields. Equa-

tion [23] becomes the Lagrangian

L½u; ;¼

kduk

ðdu; J  du  "Þ



ﬃﬃﬃﬃﬃﬃﬃ

1

ð; r

þðr

JÞdu  "Þ



ð; ðRð ; Þ

ðr

JÞ

ÞÞ½41

It is precisely the Lagrangian of the topolog ical

sigma model of Witten (1988b). Here, the pairing

( , ) is induced by the Riemannian metric !( , J )

on M and a metric on  that is compatible with ".

The second term in [41], integrated over , is equal



! = h[!], u



[]i.

For any differential form  2 

(M), let O



(u, )

be the observable obtained from ev



 2 

( 

Map(, M )) by identifying 



(Map(, M)) with

C(Map(R

0 j1

, Map(, M))). If  is closed and

 2 H

() is a homology cycle, then W

,

(u, ) =





(u, ) is identified with a closed (p  q)-form

on Map(, M). For closed 

2 

(M) and



2 H

()(1  i  r), the expectation values

i¼1



;

½du½d ½d

i¼1



;

ðu; Þe

S½u; ;

½42

are the Gromov–Witten invariants of (M, !). More-

over, [42] is nonzero only if

i = 1

 q

) = dim M.

Topological Gauge Theory

Let M be a compact, oriented 4-manifold, G,a

compact, semisimple Lie group, and P ! M,a

principal G-bundle. Denote by A the space of

connections on P and G, the group of gauge

transformations. The Lie algebra of G is Lie(G) =

(ad P) =

(M,ad P). At A 2A, the tangent space is

A=

(M,ad P). Both spaces have inner products

if we choose an invariant inner product ( , )ontheLie

algebra g of G and a Riemannian metric g on M.The

infinitesimal action of G on A is C = r

Lie(G) ! T

With a Riemannian metric, any 2-form on M

decomposes into self-dual and anti-self-dual parts:



(M) =

(M)  



(M). We consider a trivial

vector bundle E!Awhose fiber is 

(M,ad P).

G acts on E and the bundle is G-equivariant. The

trivial connection on E is G-invariant; the moment is

given by  2 (ad P): 2 

(M,ad P) 7![, ]. The

bundle E has a natural section s : A 2A7!F

, the

self-dual part of the curvature. Its derivative along

V 2 

(M,ad P) = T

A is L

s = (r

. The sec-

tion s is G-invariant, the zero set s

1

(0) is the space

of anti-self-dual connections, and the quotient

M= s

1

(0)=G is the instanton moduli space. Its

(virtual) dimension is

dim M¼4



hðgÞkðPÞ

dim GððMÞþðMÞÞ

where



h(g) is the dual Coxeter number of g and

kðPÞ¼



hðgÞ

ðAdPÞ; ½Mi 2 Z

is the instanton number of P.

We proceed with the Mathai–Quillen interpretation

of Atiyah and Jeffrey (1990).Let 2 

(M,ad P),

 2 

(M,ad P),  2 (ad P) be fermionic fields and

,  2 (ad P), bosonic fields. The combination of

[34] and [36] is given by the Lagrangian

L½A; ;;;;

þð; r

Þ



ﬃﬃﬃﬃﬃﬃﬃ

1

ð; r

Þ

ﬃﬃﬃﬃﬃﬃﬃ

1

ð; r



ﬃﬃﬃﬃﬃﬃﬃ

1

ð; ½ ; Þ

ﬃﬃﬃﬃﬃﬃﬃ

1

ð; ½; þ½; Þ

k½; k

½43

Here, ( , ) is the pairing induced by a Riemannian

metric on M and an invariant inner product on g.

With an additional topological term proportional to

, ^ F

), [43] is the Lagrangian of topological

gauge theory of Witten (1988a).

There is a tautological connection on the

G-bundle AP !AM. It is invariant under the

G-action. Identifying 



(A) with C(Map(R

0 j1

, A))

and using the Cartan model, the G-equivariant

curvature is F = F

ﬃﬃﬃﬃﬃﬃ

1

þ . For any homology

cycle  2 H

(M),



ðA; ;Þ¼



hðgÞ



ðF; ^FÞ ½ 44

corresponds to a closed G-equivariant form on A.

For 

2 H

(M)(1  i  r), the expectation values

i¼1



volðGÞ

½dA½d ½d½d½ d½d



i¼1



ðA; ;Þe

S½A; ;;;;

½45

398 Mathai–Quillen Formalism

are, up to a factor of jZ(G)j, Donaldson invariants

of M. Moreover, [45] is nonzero only if

i = 1

(4  q

) = dim M .

Other cohomological field theories can also be

understood or constructed by the Mathai–Quillen

formalism. Of such we mention only the topological

field theories of abelian and nonabelian monopoles

in Labastida and Marin˜ o (1995), which are related

to the Seiberg–Witten invariants.

See also: Characteristic Classes; Donaldson–Witten

Theory; Equivariant Cohomology and the Cartan Model;

K-Theory; Topological Quantum Field Theory: Overview;

Topological Sigma Models.

Further Reading

Atiyah MF and Jeffrey LC (1990) Topological Lagrangians and

cohomology. Journal of Geometry and Physics 7: 119–138.

Austin DM and Braam PJ (1995) Equivariant homology.

Mathematical Proceedings of the Cambridge Philosophical

Society 118: 125–139.

Berline N, Getzler E, and Vergne M (1992) Heat Kernels and

Dirac Operators. Berlin: Springer.

Blau M and Thompson G (1997) Aspects of N

 2 topological

gauge theories and D-branes. Nuclear Physics B 492: 545–590.

Cordes S, Moore M, and Ramgoolam S (1996) Lectures on 2D

Yang–Mills Theory, Equivariant Cohomology and Topologi-

cal Field Theories (Les Houches, 1994), pp. 505–682.

Amsterdam: North-Holland.

Guillemin VW and Sternberg S (1999) Supersymmetry and

Equivariant de Rham Theory. Berlin: Springer.

Kalkman J (1993) BRST model for equivariant cohomology and

representatives for the equivariant Thom class. Communica-

tions in Mathematical Physics 153: 447–463.

Kumar S and Vergne M (1993) Equivariant cohomology with

generalized coefficients. Aste´risque 215: 109–204.

Labastida JMF and Marin˜ o M (1995a) A topological Lagrangian

for monopoles on four-manifolds. Physics Letters B 351:

146–152.

Labastida JMF and Marin˜ o M (1995b) Non-abelian monopoles

on four-manifolds. Nuclear Physics B 448: 373–395.

Mathai V and Quillen D (1986) Superconnections, Thom classes,

and equivariant differential forms. Topology 25: 85–100.

Witten E (1988a) Topological quantum field theory. Commu-

nications in Mathematical Physics 117: 353–386.

Witten E (1988b) Topological sigma models. Communications in

Mathematical Physics 118: 411–449.

Wu S (1995) On the Mathai–Quillen formalism of t opological

sigma models. Journal of Geometry and Physics 17:

299–309.

Zhang W (2001) Lectures on Chern–Weil Theory and Witten

Deformations. Nankai Tracts in Mathematics, vol. 4.

Singapore: World Scientific.

Mathematical Knot Theory

L Boi, EHESS, Paris, France

Fundamental Concepts of the

Topological Theory of Knots and Links

The first known discovery relating to knots as

mathematical objects was made by Gauss around

1833 in a note that refers to the knotting together of

closed curves. This investigation originated in his work

on electromagnetic theory that led him to compute

inductance in a system of two linked circular wires. In

this note he had given an analytic formula for the

linking number of a pair of knotted curves. This

number is a combinatorial topological invariant (it is

an integer number). Moreover, one can now show that

this number is invariant under Reidemeister moves

(discussed in a later section). The linking coefficient

can be generalized for the case of p-andq-dimensional

manifolds in R

pþqþ1

. The formula for the parametrized

curves 

(t)and

(t) with radius vectors r

(t), r

(t)is

given by the following formula:

lkð

;

Þ¼

4

1

2

ðr

 r

; dr

 r

½1

The linking coefficient allows us to distinguish some

two component links. Another approach to the link

coefficient is that involving Seifert surfaces. (On this

subject, see the section ‘‘Isotopies, Reidemeister

moves, torus knots, and the linking number .’’)

A systematic study of knots in R

, however, was

only begun in the second half of the nineteenth

century by Tait and his followers. They were

motivated by Kelvin’s theory of atoms modeled on

knotted vortex tubes of ether. It was expected that

physical and chemical properties of various atoms

could be expressed in terms of properties of knots

such as the knot invariants. Even though Kelvin’s

theory did not work, the theory of knots grew as a

subfield of combinatorial and algebraic topology.

Recently, new invariants of knots have been

discovered and they have led to the solution of

long-standing problems in knot theory. Surprising

connections between the theory of knots and

statistical mechanics, quantum groups, and quantum

field theory are emerging. Moreover, knot theory

has been shown to be intimately connected with

many problems in physics, chemistry, and biology.

Tait classified the knots in terms of the crossing

number of a regular projection. A regular projection

of a knot on a plane is an orthogonal projection of

Mathematical Knot Theory 399

the knot such that, at any crossing in the projection,

exactly two strands intersect transv ersely. He made

a number of observations about some general

properties of knots which have come to be known

as the ‘‘Tait conjectures.’’ In its simplest form, the

classification problem for knots can be stated as

follows. Given a projection of a knot, is it possible

to decide in finitely many steps if it is equivalent to

an unknot. This question was answered affirma-

tively by W Haken in 1961. (For details, see Burde

and Zieschang (1985)).

General Notions and Definitions

Let M be a closed orientable 3-manifold. A smooth

embedding of S

in M is called a knot in M. A link

in M is a finite collection of disjoint knots. The

number of disjoint knots in a link is called the

number of components of the link. Thus, a knot can

be considered as a link with one component. Two

links L, L

in M are said to be equivalent if there

exists a smooth orientation-preserving automorph-

ism f : M !M such that f (L) = L

. For links with

two or more components, we require f to preserve a

fixed given ordering of the components. Such a

function f is called an ambient isotopy and L and L

are called ambient isotopic. Here, we shall take M to

be S

ﬃ R

[ {1} and simply write ‘‘a link’’ instead

of ‘‘a link in S

.’’ The diagrams of links are drawn as

links in R

. A link diagram of L is a plane projection

with crossings marked as over or under. The

simplest combinatorial invariant of a knot K is the

crossing number c(K ). It is defined as the minimum

number of crossings in any projection of the knot K.

The classification of knots up to crossing number 17

is now known. The crossing numbers of some

special families of knots are known; however, the

question of finding the crossing number of an

arbitrary knot is still unanswered. Another combi-

natorial invar iant of a knot K that is easy to define is

the unknotting number u(K). It is defined as the

minimum number of crossing changes in any

projection of the knot K which makes it into a

projection of the unknot. Upper and lower bounds

for u(K) are known for any knot K. An explicit

formula for u(K) for a family of knots called torus

knots, conjectured by Milnor nearly 40 years ago,

has been proved recently by a number of different

methods. The 3-manifold S

nK is called the knot

complement of K. The fundamental group 

nK)

of the knot compl ement is an invariant of the knot

K. It is called the fundamental group of the knot and

is denoted by 

(K). Equivalent knots have homeo-

morphic complements and conversely. However,

this result does not extend to links. (For details

and a proof, see Manturov (2004), chapter 4).

The Fundamental Group of Knots and

Its Role in Topology

For a better understanding of the above consider-

ations, we need to introduce briefly the important

concept of fundamental group in topology. The

fundamental group plays an essential role in

topology; it is involved in the entire technical

apparatus of the subject, and likewise in all

applications of topological methods. In fact, for

low-dimensional manifolds (i.e., of dimension 2 or

3) the fundamental group underlies all nontrivial

topological facts.

Classical knot theory is concerned with the space

nK = M, an open 3-manifold. There is a natural

embedding of the torus T

in M, namely as the

boundary of small tubular neighborhood of the knot

K. Similarly, for a link we obtain a disjoint union of

2-tori in M. The principal topological invariant of a

knot K is the fundamental group 

(M) of the

complement M of K, with distinguished subgroup

the natural image of 

), T 2 M

, with the

obvious standard basis. The classical theorem of

Papakyriakopoulos of the 1950s asserts that a knot

is equivalent to the trivial one if and only if 

(M)is

abelian. It was known by Haken in the early 1960s

that there is an algorithm for deciding whether or

not any knot is equivalent to the trivial knot.

However, while it appears to have been established

(by Waldhausen and others in the 1960s and 1970s)

that two knots are topologically equivalent if and

only if the corresponding fundamental groups with

labeled abelian subgroups are isomorphic, the

existence of an appropriate algorithm for deciding

such equivalence remains an open question. The

complexity of the knot group 

(M) has led to the

search for more effectively computable invariants to

distinguish knots and links. (On this subject, see the

section ‘‘Polynomial invariants of knots and links.’’)

Starting with the oriented diagram of the knot or

link K on the plane, one calculates in the standar d

manner (see Crowell and Fox (1963) and Neuwirth

(1965)) a presentation of the group 

(M) of the

knot (M = S

nK), obtaining one generator for the

edge of the diagram of a trefoil knot and a pair of

relations for each crossing. Since one relation of

each such pair simply equates the pair of generators

corresponding to the edges forming the upper

branch of the crossing, the presentation reduces

immediately to the standard one involving the same

number of generators and relations. The 2-complex

400 Mathematical Knot Theory

L with exactly one 0-cell, and with 1-cells labeled by

generators an d 2-cells labeled by the relations, is

then a deformation retract of M. Lifting to the

universal cover we obtain a boundary operator on a

complex of free Z[

]-modules, which takes the form

of a square matrix with entries from this group ring,

and it is this matrix that is related to some

differentiation as follows. Denoting the generators by

and relators by r

, one defines the operator @

ða

Þ¼

ðbcÞ¼@

ðbÞþb@

ðcÞ

the matrix in question then has entries q

given by

¼ @

ðr

Mapping each generator a

to t, we obtain a

complex of modules over the ring of integer Laurent

polynomials, with boundary operator the corre-

sponding square matrix now with Laurent poly-

nomials as entries. The determinant of this matrix

turns out to be zero, and the highest common factor

of its cofactors, after multipli cation by a suitable

power of t, turns out to be just the Alexander

polynomials A(t).

Let us say a bit more using a little different

notation on this question. Let A

(K) and J

(K)be

the Alexander polynomial and the Jones polynomial,

respectively. One of the earliest problems in knot

theory was: to what extent does the topological type X

of the complementary space X = S

nK and/or the

isomorphism class G of its fundamental group

G(K) = 

(X, x

) suffice to classify knots? The trefoil

knot is the simplest example of nontrivial knot, so it

seems remarkable that, not long after the discovery

of the fundamental group of a topological space,

Max Dehn (1914) succeeded in proving that the

trefoil knot and its mirror image had isomorphic

groups, but their knot types were distinct. Dehn’s

(ingenious) proof was the beginning of a long story,

with many contributions which reduced repeatedl y

the number of distinct knot types that could have

homeomorphic complements and/or isomorphic

groups, until it was finally proved, quite recently,

that (1) X determines K and (2) if K is prime, then G

determines K up to unoriented equivalence. Thus,

there are at most four distinct oriented prime knot

types which have the same knot group.

The knot group G is finitely prese nted; however,

it is infinite, torsion-free, and (if K is not the unknot)

nonabelian. Its isomorphism class is in general not

easily understood via a direct attack on the problem.

In such circumstances, the obvious thing to do is to

pass to the abelianized group, but unfortunately

G=[G, G] ﬃ H

(X; Z) is infinite cyclic for all knots,

so it is of no use in distinguishing knots. Passing to

the covering space X that belongs to [G, G], we note

that there is a natural action of the cyclic group

G=[G, G]on



X via covering translations. The

action makes the homology group H

(



X; Z) into a

Z[q, q

1

]-module, where q is the generator of

G=[G, G]. This module turns out to be finitely

generated. It is the famous Alexander module. While

the ring Z[q, q

1

] is not a principal ideal domain

(PID), relevant aspects of the theory of modules over

a PID apply to H

(



X; Z). In particular, it splits as a

direct sum of cyclic module, the first nontrivial one

being Z[q, q

1

]=A

(K). Thus, A

(K) is the generator

of the ‘‘order ideal,’’ and the smallest nontri vial

torsion coefficient in the module H

(



X). In

particular, A

(K) is very clearly an invariant of the

knot group.

We remark that when a knot is replaced by its

mirror image (i.e., the orientation on S

is reversed),

the Alexander and Jones polynomials A

(K) and

(K) go over to A

q1

(K) and J

q1

(K), respectively.

As noted earlier, A

(K) is invariant under such a

change, but from the simplest example, the trefoil

knot, we see that J

(K) is not. Now recall that G

does not change under changes in the orientation of

. This simple argument shows that J

(K) cannot be

a group invariant! Thus, it seems interesting indeed

to ask about the underlying topology behind the

Jones polynomial.

Isotopies, Reidemeister Moves, Torus

Knots, and the Linking Number

Because each knot is a smooth embedding of S

, it can be arbitrarily closely approximated by an

embedding of a closed broken line in R

. Here we

mean a good approximation such that after a very

small smoothing (in the neighborhood of all ver-

tices) we obtain a knot from the same isotopy class.

However, generally this might not be the case.

Definition 1 An embedding of a disjoint union of

n closed broken lines in R

is called a polygonal

n-component link. A polygonal knot is a polygonal

one-component link.

Definition 2 A link is called tame if it is isotop ic to

a polygonal link and wild otherwise.

All C

-smooth knots are tame. In the sequel, all

knots are taken to be smooth, hence, tame.

Definition 3 Two polygonal links are isotopic if

one of them can be transformed to the other by

means of an iterated sequence of elementary

isotopies and reverse transformations. The

Mathematical Knot Theory 401

elementary isotopy, generally, is assumed to be a

replacement of an edge with two edges provided

that the triangle has no intersection points with

other edges of the link.

It can be proved that the isotopy of smooth links

corresponds to that of polygonal links; the proof is

technically complicated. Like smooth links, poly-

gonal links admit planar diagrams with overcross-

ings and undercrossings, having such a diagram one

can restore the link up to isotopy.

Definition 4 By a planar isotopy of a smooth-link

planar diagram we mean a diffeomorphism of the

plane onto itself not changing the combinatorial

structure of the diagram.

Obviously, planar isotopy is an isotopy, that is, it

does not change the link isotopy type in R

Theorem 1 (Reidemeister) Two diagrams D

and

of smooth links generate isotopic links if and

only if D

can be transformed into D

by using a

finite sequence of planar isotopy and the three

Reidemeister moves W

, W

Theorem 2 Suppose that D and D

are regular

diagrams of two knots (or links) K and K

respectively. Then K  K

, D  D

We may conclude from the above theorems that

the problem of equivalence of knots, in essence, is

just a problem of the equivalence of regular

diagrams. Therefore, a knot (or link) invariant may

be thought of as a quantity that remains unchanged

when we apply any one of the Reidemeister moves

to a regular diagram.

Knots and links embedded in R

can be consid-

ered as curves (families of curves) in 2-surfaces,

where the latter surfaces are standardly embedded in

. In this section we shall briefly show that all

knots and links can be obtained in this manner.

Consider a handle surface S

standardly embedded

in R

and a curve (knot) K in it. We can now ask the

following question: which knot isotopy classes can

appear for a fixed g? First, let us note that for g = 0

there exists only one knot embeddable in S

,namely

the unknot. The case g = 1 (torus, torus knots) gives

us some interesting information. Consider the torus

as a Cartesian product S

 S

with coordinates

, ’ 2 [0, 2 ], where 2 is identified with 0. In two

dimensions, the torus can be illustrated as a square

with opposite sides identified. Let us embed this torus

standardly in R

;moreprecisely,

ð; ’Þ!ððR þ r cos ’Þcos ;

ðR þ r cos ’Þsin ; r sin ’Þ½2

Here R is the outer radius of the torus, r the small

radius (r<R),  the longitude, and ’ the meridian.

For the classification of torus knots we shall need

the classification of isotopy classes of nonintersect-

ing curves in T

: obviously, two curves isotopic in

are isotopic in R

. Without loss of generality, we

can assume the considered closed curve to pass

through the point (0, 0) = (2,2). It can intersect

the edges of the square several times. In addition,

assume all these intersections to be transverse. Let us

calculate separately the algebraic number of inter-

sections with horizontal edges and those with

vertical edges. Here, passing through the right edge

or through the upper edge is said to be positive; that

through the left or the lower edge is negative. Thus,

for each curve of such type we obtain a pair of

integer numbers. So, each torus knot passes p times

the longitude of the torus, and q times its meridian,

where GCD(p, q) = 1. It is easy to see that for any

coprime p and q such a curve exists: one can just

take the geodesic line {q  p’ = 0 (mod 2)}. Let us

denote the torus knot by T(p, q). So, in order to

classify torus knots, one should consider pairs of

coprime numbers p, q and see which of them can be

isotopic in the ambient space R

. The simplest case

is when either p or q equals 1. The next simplest

example of a pair of coprime number s is p = 3, q = 2

(or p = 2, q = 3). In each of these cases we obtain the

trefoil knot. Let us state the following important

result.

Theorem 3 For an y coprime integers p and q, the

tori (p, q) and (q, p) are isotopic.

Proof For a proof of this theorem, see Rolfsen

(1990). Note that the (p, q) torus knot in one full

torus is just the (q, p) torus knot in the other one.

Thus, mapping one full torus to the other one, we

obtain an isotopy of (p, q) and (q, p) torus knots.

This homotopy of full tori can be expressed as a

continuous process in S

. Indeed, torus knots of type

(p, q) can be represented by a series of planar

diagrams. Moreover, it is possible to demonstrate a

way of coding a knot (link) as a (p-strand) braid

closure.

Analogously to the case of torus knots, one can

define torus links which are links embedded into the

torus standardly embedded in R

. We know the

construction of torus knots. So, in order to draw a

torus link, one should take a torus knot K  T (one

can assume that it is represented by a straight linear

curve defined by the equation q  p’ = 0 (mod 2)

and add to the torus T some closed nonintersecting

simple curves; each curve should be nonintersecting

and should not intersect K. Thus, these curves

should be embedded in TnK, that is, in the open

402 Mathematical Knot Theory

cylinder. Each curve on the cylinder is either

contractible or passes the longitude of the cylinder

once. So, each curve in T nK is either contractible

inside TnK, or ‘‘parallel’’ to K inside T, that is,

isotopic to the curve given by the equation q 

p’ = " (mod 2) inside TnK. Thus, the following

theorem holds.

Theorem 4 Each torus knot is isotopic to the

disconnected sum of a trivial link and a link that is

represented by a set of parallel torus knots of the

same type (p, q).

As we already know, a link invariant is a function

defined on links that is invariant under isotopies. We

shall represent links by using their planar diagrams.

According to the Reidemeister theorem, in order to

prove the invariance of some function on links, it is

sufficient to check this invariance under the three

Reidemeister moves. First, let us consider the

simplest integer- valued invariant of two-component

links. Let L be a link consisting of two oriented

components A and B and let L

be the planar

diagram of L. Consider those crossings of the

diagram L

where the component A goes over the

component B. There are two possible types of such

crossings with respect to the orientation. For each

positive crossing we assign the number (þ1), for

each negative crossing we assign the number (1).

Let us summarize these numbers along all crossings

where the component A goes over the component B.

Thus, we obtain some integer number and, in fact,

this number is invariant under Reidemeister moves.

The so-obtained link invariant is called linking

coefficient.

Polynomial Invariants of Knots and Links

By changing a link diagram at one crossing we can

obtain three diagrams corresponding to links

, L



, and L

which are identical except for this

crossing. In the 1920s, Alexander gave an algorithm

for computing a polynomial invariant 

(t)

(a Laurent polynomial in t) of a knot K, called the

Alexander polynomial, by using its projection on a

plane. He also gave its topological interpretation as

an annihilator of a certain cohomology module

associated to the knot K. In the 1960s, Conway

defined his polynomial invariant and gave its

relation to the Alexander polynomial. This poly-

nomial is called the Alexander–Conway polynomial.

The Alexander–Conway polynomial of an oriented

link L is denoted by r

(z) or simply by r(z) when L

is fixed. We denote the corresponding polynomials

of L

, L



, and L

by r

, r



, and r

, respectively.

The Alexander–Conway polynomial is uniquely

determined by the following axioms.

Axiom 1 Let L and L

be two oriented links which

are ambient isotopic. Then

ðzÞ¼r

ðzÞ½3

Axiom 2 Let S

be the standard unknotted circle

embedded in S

. It is usually referred to as the

unknot and is denoted by O. Then

ðzÞ¼1 ½4

Axiom 3 The polynomial satisfies the following

skein relation:

ðzÞr



ðzÞ¼zr

ðzÞ½5

We note that the original Alexander polynomial



is related to the Alexander–Conway polynomial

of an oriented link L by the relation



ðtÞ¼r

ðt

1=2

 t

1=2

Þ½6

In the 1980s, Jones discovered his polynomial

invariant V

(t), called the Jones polynomial, while

studying von Neumann algebras a nd gave its

interpretation in terms of statistical mechanics. A

state model for the Jones polynomial was then

given by Kauffman (1987) using his bracket

polynomial. These new polynomial invariants have

led to the proofs of most of the Tait conjectures.

The Jones polynomial V

(t)ofK is a Laurent

polynomial in t, which is uniquely de termined by a

simple set of properties similar to the axioms for

the Alexander–Conway polynomials. More gener-

ally, the Jones polynomial can be defined for any

oriented link L as a Laurent polynomial in t

1=2

,so

that reversing the orientation of all components of

L leaves V

unchanged. In particular, V

does not

depend on the orientation of the knot K. For a

fixed link, we denote the Jones polynomial simply

by V. Recall that there are three standar d ways to

change a link diagram at a crossing point. The

Jones polynomial is characterized by the following

properties:

1. Let L and L

be two orien ted links which are

ambient isotopic. Then

ðtÞ¼V

ðtÞ½7

2. Let O denote the unknot. Then

ðtÞ¼1 ½8

3. The polynomial satisfies the following skein

relation:

1

 tV



¼ðt

1=2

 t

1=2

ÞV

½9

Mathematical Knot Theory 403

An important property of the Jones polynomial that

is not shared by the Alexander–Conway polynomial

is its ability to distinguish between a knot and its

mirror image. More precisely, we have the following

result. Let K

be the mirror image of the knot K.

Then

ðtÞ¼V

ðt  1Þ½10

Since the Jones polynomial is not symmetric in t and

1

, it follows that in general

ðtÞ 6¼ V

ðtÞ½11

We note that a knot is called amphicheiral (achiral

in biochemistry) if it is equivalent to its mirror

image. We shall use the simpler biochemistry term.

So, a knot that is not equivalent to its mirror image

is called chiral. The condition expressed by [11 ] is

sufficient but not necessary for chirality of a knot.

The Jones polynomial did not resolve the following

conjecture by Tait concerning chirality: if the cross-

ing number of a knot is odd, then it is chiral.

However, it has been demonstrated recently that a

15-crossing knot provide s a counterexample to the

chirality conjecture.

New Invariants and Their Applications

in Mathematical Physics

There was an interval of nearly 60 years between the

discovery of the Alexander polynomial and the Jones

polynomial. Since then a number of polynomials

and other invariants of knots and links have been

found. A particularly interesting one is the two-

variable polynom ial generalizing V, called the

HOMFLY polynomial (name formed from the

initials of authors of the article (Freyd et al. 1985)

and denoted by P. The HOMFLY polynomial

P(, z) satisfies the following skein relation:



1

 P



¼ zP

½12

Both the Jones polynomial V and the Alexander–

Conway polynomial r

are special cases of the

HOMFLY polynomial. The precise relati ons are

given by the following theorem.

Theorem 5 Let L be an oriented link. Then the

polynomials P

, V

, and r

satisfy the following

relations:

ðtÞ¼P

ðt; t

1=2

t

1=2

Þ and r

ðzÞ¼P

ð1;zÞ½13

After defining his polynomial invariant, Jones also

established the relation of some knot invariants with

statistical mechanical models. Since then this has

become a very active area of research. By

constructing a typical statistical mechanics model –

the star–triangle relations of the Yang–Baxter

equations are an example of such model – one

obtains a state model for the Alexander or the Jones

polynomial of a knot, by associating to the knot a

statistical system, whose partition function

:¼

ðsÞ!ðsÞ½14

gives the corresponding polynomial. (For details, see

Jones (1989)). In the function above, ! = F(X, S) !R

is a weight funct ion and the sum is taken over all

states s 2 F(X, S). The energy E

of the system (X, S)

is a functional,

: FðX; SÞ!R; k 2 K ½15

where the subscript k 2 K indicates the dependence

of energy on the set K of auxiliary parameters, such

as temperature, pressure, etc.

However, these statistical models did not provide

a geome trical or topological interpretation of the

polynomial invariant. Such an interpretation was

provided by Witten (1989) by applying ideas from

quantum field theory to the Chern–Simons Lagran-

gian. In fact, Witten’s model allows us to consider

the knot and link invariants in any compact

3-manifold M.

Vassiliev Invariants and the Space

of All Knots: New Generalizations

of Knot Theory

An entirely new collection of knot invariants,

which arose out of techniques pioneered by Arnold

in singularity theory, has been introduced by V A

Vassiliev in t he 1990s. The knot invariants, like

the Alexander polynomial, associate a knot with

some sort of mathematical quantity. A Vassiliev

invariant, on the other hand, is an invariant that

satisfies a set of conditions. In this sense, all the

invariants introduced above – the Jones polyno-

mial, the HOMFLY and the Kauffman polyno-

mial, the Conway polynomial, and the Alexander

polynomial – can all be shown to be Vas siliev

invariants. However, not all the knot invariants are

Vassiliev invariant s, for instance, the signature of a

knot is not a Vassiliev invariant. The new Vassiliev

invariants have a solid basis in a very interesting

new topology, where one studies not a single knot,

but a space of all knots. Vassiliev’s knot invariants

are rational numbers. They lie in vector space V

dimension d

, i = 1, 2, 3, ..., with invariants in V

having ‘‘order’’ i. These invariants are built from

different families of crossing changes.

404 Mathematical Knot Theory

Considering that Vassiliev’s invariants require

introducing an important conceptual change, shift-

ing our attention from the knot K, which is the

image of S

under an embedding  : S

, to the

embedding  itself. A knot type K thus becomes an

equivalence class {} of embeddings of S

into S

The space of all such equivalence classes of embed-

dings is disconnected, with a component for each

smooth knot type. In this way, one passes from

embeddings to smooth maps, thereby admitting

maps which have various types of singularities. Let



M be the space of all smooth maps from S

to S

This space is connected and contains all knot types.

Our space will remain connected and will contain all

knot types if we place two mild restri ctions on our

maps. Let M denote the collection of all  2



such that (S

) passes through a fixed point  and is

tangent to a fixed direction at . The space M has

some interesting properties, the main one being that

it can be approximated by certain affine spaces, and

these affine spaces contain representatives of all

knot types. The walls between distinct chambers in

M constitute the discriminant , that is, ={ 2

M j} has a multiple point or a place where its

derivative vanishes or other singularities. The space

M   is our space of all knots.

The additive properties of the Alexander and

Jones polynomials have a very attractive interpreta-

tion in terms of Vassiliev invariants. By a result of

Bar-Natan, all coefficients of the Alexander poly-

nomial are Vassiliev invariants (see Bar-Natan

(1995)). The same can be said of the Jones

polynomial, as proved by a theorem of Birman and

Lin (1993). There is an attractive formula due to

Kontsevich expressing all Vassiliev invariants ana-

lytically in terms of multiple integrals, assuming that

the knot or link diagram comes with some generic

Morse function (e.g., the projection of the planar

diagram on the y-axis). Moreover, from the work of

Kontsevich it follows that it is possible to give a

purely combinatorial characterization of all Vassi-

liev invariants (other than the one mentioned above)

by associating to an oriented knot K in R

(given via

coordinates z = z(t)(= x(t) þ iy(t)), t ) a chord diagram,

which is just a circle with 2k distinct points labeled

, Q

, j = 1, 2, ..., k,markedonit,andbyimposing

certain relations on the free abelian group freely

generated by all chord diagrams.

Theorem 6 Let V

(t) be the Jones polynomial of a

knot K. Let V

(q) be the infinite series obtained

from V

(t) by substituting e

(= 1 þ q þ q

=2! þ =

n = 0

=n!) for t. So we may write

ðqÞ¼b

þ b

þ

Then J

(K) = b

is a Vassiliev invariant induced by

the Jones polynomial of order (at most) m .

The structure and significance of the HOMFLY and

Kauffman polynomials can be interpreted in the

language of Vassiliev invariants, which are invariants

of finite type. The notion of finite type is of

extraordinary significance in studying these invariants.

One reason for this is the following basic lemma:

Lemma 7 If a graph G (an embedded 4-valent

graph) has exactly k nodes, then the value of a

Vassiliev invariant v

of type k on G, v

(G), is

independent of the embedding of G .

Let us show briefly this important result. Suppose

V is any invariant of oriented links taking values in

some abelian group. This V can be extended to be

an invariant of singular links in the following way

(Kauffman 2001): a singular link is an immersion

of simple closed curves in S

with finitely many

transverse double-points. These self-intersections are

required to remain transverse in any isotopy

demonstrating the equivalence of such singular

links. If the definition of V has been extended over

singular links with n  1 double points, define it on

a singular link L



with n singularities by

VðL

Þ¼VðL

ÞVðL



where V(L



), V(L

), and V(L



) are identical except

near a point where they form a node. Note that

V(L

)andV(L



) each has n  1 double points.

Then V is called a Vassiliev invariant of order n,or

an invariant of finite type n,ifV(L) = 0 for every

L with n þ 1 or more singularities. Recall the

Alexander–Conway polynomial invariant, r

(z) 2

Z[z], of oriented links defined by r

unknot

(z) = 1and

ðzÞr



ðzÞ¼zr

ðzÞ

Extend this over singular links by the above method.

Then if L



is a link with r singularities, r



(z) =

(z), where L

is a link with r  1 singularities.

Thus, by induction on r,ifL has r singularities then

(z) has a factor of z

. This implies at once that the

coefficient of z

in the Conway polynomial of a link

is a Vassiliev invariant of order n. Now suppose one

considers the HOMFLY polynomial and makes the

substitution (l, m) = (it

N =2

,i(t

1=2

 t

1=2

)). The char-

acterizing skein relation becomes

N=2

PðL

Þt

N=2

PðL



Þ¼ðt

1=2

 t

1=2

ÞPðL

Note that this becomes the Jones polynomial when

N = 2. Now make the further substitution t = exp x.

Here exp x should be thought of as the classical

power series expansion. Of course, exp (x=2) and

Mathematical Knot Theory 405

exp(x=2) have power series expansi ons; the power

series can be multiplied and added to give another

power series. Thus, P(L) has a power series

expansion in powers of x. It follows immediately

that P (L

)P(L



) = xS(x) for some power series

S(x). Hence, the proof used for the Conway

polynomial shows at once that the coefficient of x

in the power series expansion of P(L) is a Vassiliev

invariant of order n.

All present studies of Vassiliev invariants clearly

indicate a major role of these invariants in the future

developments of knot theory and topological quan-

tum field theories. Many questions in knot theory

remain open, nevertheless, in future it will, very likely

be one of the most fruitful and beautiful subjects of

research in mathematics and in mathematical physics.

Knot theory a lso attract s attention from the fact that

it is revealing new astounding and profound links

between geometry, algebra, and topology.

See also: Finite-Type Invariants; The Jones Polynomial;

Knot Invariants and Quantum Gravity; Knot Theory and

Physics; Kontsevich Integral; String Topology: Homotopy

and Geometric Perspectives; Topological Knot Theory

and Macroscopic Physics; Topological Quantum Field

Theory: Overview.