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Isomonodromic Deformations 177

The Jones Polynomial

V F R Jones, University of California at Berkeley,

Berkeley, CA, USA

ª 2006 Published by Elsevier Ltd.

Introduction

A ‘‘link’’ is a finite family of disjoint, smooth,

oriented or unoriented, closed curves in R

equivalently S

. A ‘‘knot’’ is a link with one

component. The ‘‘Jones polynomial’’ V

(t)isa

Laurent polynomial in the variable

ﬃﬃ

which is

defined for every oriented link L but depends on

that link only up to orientat ion-preserving diffeo-

morphism, or equivalently isotopy, of R

. Links can

be represented by diagrams in the plane and the

Jones polynomials of the simplest links are given

below.

= – +

√t

= t + t

– t

=–

(1 + t

)

√t

– + 1– t + t

The Jones polynomial of a knot (and generally a

link with an odd number of components) is a

Laurent polynomial in t.

The most elementary ways to calculate V

(t)

use the ‘‘linear skein theory’’ ideas of Conway

(1970). Indeed, it is not hard to see by induction

that V

(t) is defined by its invariance under

isotopy, the normalization V

(t) = 1 and the skein

formula

 tV



ﬃﬃ



ﬃﬃ



which holds for any three oriented links having

diagrams which are identical except near one crossing

where they differ as below.

–

As such the Jones polynomial resembles the

Alexander (1928) polynomial 

(t) which can be

calculated in exactly the same manner as V

(t)

except that the skein relation becomes



 



ﬃﬃ



ﬃﬃ





A two-variable generalization P

of both 

and

, sometimes called the HOMFLYPT polynomial,

was found in Freyd et al. (1985) and Przytycki and

Traczyk (1988). It satisfies the most general skein

relation

þ yP



þ zP

¼ 0

for homogeneous variables x, y, and z.

The other skein-like definition of V

was found in

Kauffman (1987). Begin with unoriented link dia-

grams up to planar istotopy. The Kauffman bracket

hLi of such a diagram is calculated using

〈〉 〉

= A 〈

〉

+ A

–1

〈

where the hi notation means that the relation may

be applied to that part of the link diagrams inside

the bracket, the rest of the diagrams being identical.

If hLi were to be an invariant of three-dimensional

isotopy it is easy to see that

〈〉

= –

– A

–2

which further implies

〈〉

= A

–3

〈

〉

Thus, hLi cannot be a three-dimensional isotopy

invariant as such. However, if L is given an

orientation (then called

L), a simple renormalization

solves the problem and it is true that

ðÞ V

ðA

Þ¼A

3 writhe ð

LÞ

hLi

where writhe (

L) is the sum over the crossings of L

of þ1 for a positive crossing

and 1 for a

negative crossing

The formula () is readily proved by induction but

a more structural proof will be discussed later on,

connected with physics. If the crossings in a link

alternate between over and under as one follows the

string around , the highest and lowest degree terms in

the Kauffman bracket can readily be located. This

led to the proof of some old conjectures about

alternating knots in Murasugi (1987), Kauffman

(1987), and Thistlethwaite (1987).

The Kauffman two-variable polynomial F

(a, x)is

defined in Kauffman (1990) by considering the

linear skein relation involving all four possibilities

at a crossing:

–

∞

This polynomial contains V

(T) as a specializa-

tion but not the Alexander polynomial.

The above polynomials are quite powerful at

distinguishing links one from another, including

links from their mirror images, which corresponds

for the Jones polynomial to replacing t by t

1

. More

power can be added to the polynomials if simple

geometric operations are allowed. ‘‘Cabling’’ entails

replacing a singl e strand with several parallel copies

and the polynomials of cables of a link are also

isotopy invariants if attention is paid to the writhe

of a diagram.

The following problem, however, is open at the

time of writing this article: ‘‘Does there exist a knot

in R

, different from the unknot , whose Jones

polynomial is equal to 1?’’

For links with more than one component, it is

known (Thistlethwaite 2001, Eliahou et al. 2003)

that the answer to the corresponding question is yes,

the simplest example being:

One of the reasons that the question above has

not been answered is presumably that, unlike with

the Alexander polynomial, we have little intuitive

understanding of the meaning of the ‘‘t’’ in V

(t).

Perhaps, the most promising theory in this context is

in Khovanov (2000) where a complex is constructed

whose Euler characteristic, in an appropriately

graded sense, is the Jones polynomial. The homol-

ogy of the complex is a finer invariant of links

known as ‘‘Khovanov homology.’’

Braids

Abraid(seeBirman (1974))onn strings is a

collection of curves in R

joining n points in a

horizontal plane to the n points directly below

them on another horizontal plane. If the end-

points of the braid are on a straight line, the

braid can be dr awn as i n the example below

(where n = 4).

The crucial property of a braid is that the tangent

vector to the curves can never be horizontal. Braids

are considered up to isotopies which are supported

between the top and bottom planes.

Braids on n strings form a group, called B

, under

concatenation (plus some isotopy) as below:

αβ

Let 

, 

, ..., 

n1

be the braids below:

=,,

n–1

. .

. .. . .

180 The Jones Polynomial

Artin’s presentation (Birman 1974) of the braid

group is on the generators 

, 

, ..., 

n1

with the

relations



iþ1



¼ 

iþ1



iþ1

for 1  i  n  2



¼ 



if ji  jj2

Thus, to find linear representations of B

,itsuffices

to find matrices 

, 

, ..., 

n1

satisfying the above

relations (with  replaced by ). One such representa-

tion (of dimension n) called the (nonreduced) Burau

representation is given by the row-stochastic matrices



1 tt00...

1000...

0010...

000... 1



1000...

01tt 0 ...

0100...

0001...

000... 1

; ...

...;

n1

10 ... 0

01 ... 0

0 ... 1 tt

0 ... 10

This representation is known not to be faithful for

n  5 but faithful for n  3. The case n = 4 remains

open. (See Moody (1991), Long and Paton (1993),

and Bigelow (1999)).

Braids can be viewed in several ways, which lead to

several generalizations. For instance, identifying the

vertical axis for a braid with time and taking the

intersection of horizontal planes with the braids shows

that elements of B

can be thought of as motions of n

distinct points in the plane. Thus, it is natural that

ﬃ 

ðfC

ng=S

when  is the set {(z

, ..., z

)jz

= z

for some i 6¼ j}

and the symmetric gro up S

acts freely on C

n by

permuting coordinates. But  is the zero-set of the

frequently encountered function

i<j

ðz

 z

so the braid group may naturally be generalized as

the fundamental group of C

minus the singular

set of some algebraic function (Birman 1974). Or,

motions of points can be extended to motions of the

whole plane and a braid defines a diffeomorphism of

the plane minus n points. Thus, the braid group may

be generalized as the ‘‘mapping clas s group’’ of a

surface with marked points (Birman 1974).

The Temperley–Lieb Algebra

If  2 C one may define the algebra TL(n, ) with

identity 1 and generators e

, e

, ..., e

n1

subject to

the following relations:

¼ e

i1

¼ e

¼ e

if ji  jj2

Counting reduced words on the e

’s shows that

dimfTLðn;Þg 

n þ 1



and in Jones (1983) it is shown that these numbers,

the Catalan numbers, are indeed the dimensions of

the Temperley–Lieb algebras. In the obvious way,

TL(n, )  TL(n þ 1, ). If 

1

is not in the set

{4 cos

q; q 2 Q}, TL(n, ) is semisimple and its

structure is given by the following Bratteli diagram:

where the integers on each row are the dimensions

of the irreduci ble representations of TL(n, ) and the

diagonal lines give the restriction of representations

of TL(n, )toTL(n  1, ). These representations

are naturally indexed by Young diagrams with n

boxes and at most two rows:

with the

diagonal lines in the Bratteli diagram corresponding

to removal/addition of a box. The dimension of the

representation corresponding to the diagram whose

second row has r boxes (r  n)is





r 1



The Jones Polynomial 181

One may attempt to make TL(n, ) into a



-algebra and look for Hilbert space represen-

tations (with e

6¼ 0), by imposing e



= e

. From

(Wenzl 1987), this is only possible (for all n) when

1.  2 R,0< 1=4, or

2. 

1

2 {4 cos

=m, m = 3, 4, 5, ...}.

The proof uses the fact that f

, inductively defined by

nþ1

¼ f



½2

½n þ 1

½n þ 2

nþ1

must be an orthogonal projection with e

= f

= 0

for i  n. These f

are sometimes called Jones–Wenzl

idempotents. (Here 

1

= 2 þ q

þ q

2

and for this

and later formulas we define the quantum integer

[n]

= (q

 q

n

)=(q  q

1

)).

When 

1

= 4 cos

(=m), the Hilbert space repre-

sentations decompose according to Bratteli diagrams

obtained by trunca ting – eliminating the 1 on the

mth row, and all representations below and to the

right of it, so that for m = 7 we would obtain

In terms of Young diagrams, this corresponds to

only taking those diagrams whose row lengths differ

by at most m  2. The ex istence of these Hilbert

space representations is from Jones (1983).

The Temperley–Lieb algebras arose in Jones (1983)

as orthogonal projections onto subfactors of II

factors.

As such the Hilbert space structure was manifest. The

trace on a II

factor also yielded a trace on the TL(n, ).

To be precise, there is for each m a unique linear

map tr : TL(n, ) !C with:

1. tr(1) = 1

2. tr(ab) = tr(ba)

3. tr(xe

nþ1

) = tr(x) for x 2 TL(n þ 1, ).

This trace may be calculated either from (1), (2),

and (3), or us ing the repre sentations, as a weighted

sum of ordinary matrix traces. The weight for the

representation of TL(n, ), the second row of whose

Young diagram has r boxes, is

½n  r þ 1

ð½2

Thus, if x 2 TL(n, ) and 

is the





r1



dimensional irreducible representation, then

trðxÞ¼

ðq þ q

1

½n=2

r¼0

½n  r þ 1

trace ð

ðxÞÞ

One also has

trðf

Þ¼

½n þ 2

ð½2

nþ1

so that the disappearance of the ‘‘1’’ from the

Bratteli diagram is mirrored by the vanishing of the

trace of the corresponding proje ction.

Positivity of tr, tr(a



a)  0, is responsible for all the

Hilbert space structures. To explicitly construct the

Hilbert space representations, one may use the GNS

construction: take the quotient of the -algebra by the

kernel of the form ha, bi= tr(b



a) which makes this

quotient a Hilbert space on which TL(n, )willact

with the e

’s as orthogonal projections. Explicit bases

can be obtained easily if desired, using paths on the

Bratteli diagram, or Young tableaux.

A useful diagrammatic presentation of TL(n, )

was discovered in Kauffman (1987 ). A (Kauffman)

TL diagram (for non-negative integers m and n)isa

rectangle with n marked points on the top and m on

the bottom with nonintersecting smooth curves

inside the rectangle connecting the boundary points

as illustrated below.

A (5, 7)-diagram

Two Kauffman TL diagrams are considered the same

if they connect the same pairs of boundary points.

The vector space TL(m, n, ) with basis the set of

(m, n) diagrams, and  2 C, becomes a category with

this concatenation together with the rule that closed

curves may be removed, each one counting a

(multiplicative) factor of . We illustrate their

product in TL(m, n, ) below:

182 The Jones Polynomial

Of special interest is the algebra TL(n, n, ). If we

define E

to be the diagram below:

12 ii + 1

12 ii

+ 1

then E

= E

, E

i1

= E

, and E

= E

for

ji  jj2. Thus, provided  6¼ 0, we have an

isomorphism between TL(n, 

2

)andTL(n, n, )by

mapping e

to (1=)E

One of the nicest features of the Kauffm an

diagrams is that they yield simple explicit bases for

the irreducible representations. To see this, call a

curve in a diagr am a ‘‘through-string’’ if it connects

the top of the rectangle to the bottom . Then all

(m, n) diagrams are filtered by the number of

through-strings and if we let TL(m, n, k, )be

the span of (m, n) diagrams with at most k

through-strings, we have TL(k, n, )TL(n, m, k, ) 

TL(k, m, k, ). Thus, V

n, m

= TL(n, m, m, )=TL(n, m,

m  1, )isaTL( n, 

2

)-module, a basis of which is

given by (m, n)-diagrams with m through-strings

(m  n). The number of such diagrams is





m1



and it follows from Jones (1983) that all these

representations are irreducible for ‘‘generic’’  (i.e.,

 62 {2 cos Q}) and that they may be identified with

those indexed by Young diagrams as below:

n, m

n –

The invariant inner product on V

n, m

is defined by

hv, wi= w



v for the natural identification of V

m, m

with C (



is the obvious involution from (m, n)

diagrams to (n, m) diagrams.).

The Original Definition of V

(t)

Given a braid  2 B

one may form an oriented link

 called the closure of  by tying the top of the braid

to the bottom as illustrated below:

β =

All oriented links occur in this way (Birman 1974)

but if  2 B

, 

1

and 

1

(in B

nþ1

) have the

same closure.

Theorem 1 (Markov) (Birman 1974). Let  be the

equivalence relation on

‘

n = 1

(all braids on any

number of strings) generated by the two ‘‘moves’’

  

1

and   

1

. Then 

 

if and only

if the links



and



are the same.

It is easily checked that, if 1, e

, e

, ... satisfy

the TL rel ations of the sect ion ‘‘The Temper ley–Lieb

algeb ra,’’ then send ing 

to (t þ 1)e

1 (with 

1

2 þ t þ t

1

) defines a representation 

of B

inside

TL(n, ) for each n. The representation is unitary for

the C



-algebra structure when 

1

= 4 cos

=n,

n = 3, 4, 5, ... (and t = e

2i=n

). It is an open question

whether 

is faithful for all n. It contains the Burau

representation as a direct summand.

Combining the properties of the trace tr defined

on TL with Markov’s theorem, one obtains imme-

diately that, for  2 B

, the following function of t

depends only on

:



ﬃﬃ



ﬃﬃ



n1

ﬃﬃ

e

trð

ðÞÞ

(here e 2 Z is the ‘‘exponent sum’’ of  as a word on



, 

, ..., 

n1

A simple check using the (oriente d) skein-theoretic

definition of the Jones polynomial shows that this

function of t is precisely V

^

(t). This is how V

(t)

was first discovered in Jones (1985).

Although less elementary, this approach to V

(t)

does have some advantages. Let us mention a few.

1. One may use representation theory to do calcula-

tions. For instance, using the weighted sum of

ordinary traces to calculate tr as in the section

‘‘The Temperley–Lieb algebra,’’ one obtains read-

ily the Jones polynomial of a torus knot (i.e.,



where  = (





p1

)

2 B

if p and q are

relatively prime). It is

ðp1Þðq1Þ=2

1  t

ð1  t

pþ1

 t

qþ1

 t

pþq

2. If one restricts attention to links realizable as

 for

 2 B

for fixed n, the computation of V



(t)canbe

performed in polynomial time as a function of the

number of crossings in

. Thus, one has computa-

tional access to rather complicated families of links.

3. Unitarity of the representation when t = e



2i

can

be used to bound the size of jV

(t)j. For instance,

if  2 B

and V



(t) = (

ﬃﬃ

 (1=

ﬃﬃ

))

k1

, then 

is in the kernel of 

, and jV



2i=n

)j

(2 cos =n)

k1

for any other  2 B

The Jones Polynomial 183

The representation of the braid group inside the TL

algebra should be thought of as an extension of the

Jones polynomial to ‘‘special knots with boundary.’’

The coefficients of the words in the e

’s (or equivalently

the Kauffman TL diagrams) are all invariants of the

braid. We can further remove the braid restriction and

consider arbitrary knots and links with boundary,

known as ‘‘tangles’’ (Conway 1970).

A 3-tangle

Tangles may be oriented or not and their

invariants may be evaluated either by reduction to

a system of elementary tangles using skein relations

or by organi zing the tangle and representing it in an

algebra. See Turaev (1994).

A similar algebraic approach is available for the

HOMFLYPT and Kauffman two-variable polyno-

mials. The algebra playing the role of the TL algebra

is the Hecke algebra for HOMFLYPT (Freyd et al.

1985, Jones 1987) and the BMW algebra (Birman and

Wenzl 1989, Murakami 1990) for the Kauffman

polynomial. The BMW algebra was discovered after

the Kauffman polynomial in order to provide an

analog of the TL and Hecke algebras. For detailed

analysis of the Hilbert space and other structures for

both Hecke and BMW algebras, see Wenzl (1988) and

Wenzl (1990).

Connections with Statistical Mechanics

One might say that turning a knot into a braid

organizes the knot by ‘‘putting it on a lattice,’’

thereby creat ing a physical model with the crossings

of the knot as interactions. Taking the trace of the

braid is evaluating the partition function with

periodic (vertical) boundary conditions.

This is more than wishful thinking. The Temperley–

Lieb algebra arose from transfer matrices in both

the Potts and ice-type models in two dimensions

(Temperley and Lieb 1971) and each ‘‘e

’ ’ implements

the addition of one more interaction to the system.

(The same e

’s as in the ice-type models were

rediscovered in the subfactor context in Pimsner and

Popa (1986)). Thus, the Jones polynomial of a closed

braid is the partition function for a statistical mecha-

nical model on the braid. In Jones (1983),itisobserved

that knowledge of the Jones polynomial for a family of

links called French sinnets would constitute a solution

of the Potts model in two dimensions.

In Temperley and Lieb (1971), the TL relations

are used to establish the mathematical equivalence

of the Potts and ice-type (six-vertex) models. In

Baxter (1982, chapter 12), this equivalence is shown

for Potts models on an arbitrary planar graph. In

view of this, it is not surprising that statistical

mechanical models can be defined directly on link

diagrams to give explicit formulas for V

(t) (and

other invariants) as partition functions. This works

most easily for the Q-state Potts model.

Given an unoriented link diagram D, shade the

regions of the plane black and white and form the

planar graph  whose vertices are the black regions

and whose edges are the crossings as below:

Assign þ and  to each edge according to the

following scheme:

–

Fix Q 2 N and two symmetric mat rices w



(a, b)

for 1  a, b  Q. The partition function of the

diagram is then

states

edges of 



ð; 

where a ‘‘state’’ is a function from the vertices of 

to {1, 2, ..., Q} and, given an edge of  and a state,

 and 

denote the values of the state at the ends of

that edge (w

and w



are used according to the sign

of the edge).

The ‘‘Potts model’’ is defined by the property that

the ‘‘Boltzmann weights’’ w



(, 

) depend only on

whether  = 

or not. It is a miracle that the choice

(with Q = 2 þ t þ t

1

)



ð; 

Þ¼

1

if  ¼ 

1 otherwise



184 The Jones Polynomial

gives the Jones polynomial of the link defined by D

as its partition function (up to a simple normal-

ization). See Jones (1989) for details.

It is natural to look for other choices of w



which

give knot invariants. The Fateev–Zamolodchikov

(1982) model gives a classical knot invariant but

besides that (and some variants on the Jones

polynomial) there is only one other known choice of

any interest, discovered in Jaeger (1992).Inthiscase,

Q = 100 and the Boltzmann weights are symmetric

under the action of the Higman–Sims group on the

Higman–Sims graph with 100 vertices. The knot

invariant is a special value of the Kauffman two-

variable polynomial.

The other side of Temperley–Lieb equivalence is

the ‘‘ice-type’’ model which is a ‘‘vertex model .’’

That is to say the ‘‘spins’’ reside on the edges of a

graph and the interactions occur at the vertices. To

use vertex models in knot theory, the knot projec-

tion D itself is the (4-valent) graph. The ice-type

model has two spin states per edge so that a state of

the system is a function from the edges of the graph

to the set {}; the Boltzmann weights are given by

two 4  4 matrices w



(

, 

) where the ’s

are 1andw

and w



are the contributions of

and

to the partition function, respectively. Furthermore,

we may think of a state as a locally constant

function  on D so for any f :{1} !R we may

form the term

f ()d corresponding to interac-

tion with an external field (d is the curvature or

change of angle form on D). Then the partition

function is

states

crossings of D



ð

;

f ðÞd

A (nonphysical) specialization of the six-vertex

model yields values of f and w



for which Z

is a

link invariant equal to V

(t). See Jones (1989).

As with the Potts model, one may try to generalize

to more general w



and f.Thisismuchmore

successful for these ‘‘vertex’’ models than it was for

models like the Potts model. The theory of quantum

groups (Jimbo 1986, Drinfeld 1987, Rosso 1988)

allows one to obtain link invariants (as partition

functions for vertex models) for each simple finite-

dimensional Lie algebra A andeachassignmentofan

irreducible representation of A to the components of

the link. The images of the braid generators 

in the

corresponding braid group representations are called

‘‘R-matrices.’’ It is the Yang–Baxter equation that

gives isotopy invariance of the partition function. In

this way, one obtains (by an infinite family of one-

variable specializations) the HOMFLYPT polynomial

(sl

) and the Kauffman polynomial (orthogonal and

symplectic algebras) and more polynomials. The

geometric operation of cabling corresponds to the

tensor product of representations.

Connections with Quantum Field Theory

Conformal Field Theory

If ’ is a (multicomponent) field in one chiral half of

a two-dimensional conformal field theory (CFT), the

correlation functions

h’ðz

Þ’ðz

Þ’ðz

Þi

(where z

2 C) are expected to be singular if z

= z

for some i 6¼ j, holomorphic otherwise and satisfy a

linear differential equation. Thus, analytic con tinua-

tion should determine a unitary monodromy repre-

sentation of 

n{(z

, z

, ..., z

)jz

= z

for some

i 6¼ j}) on the vector space of solutions to the

differential eq uation near a point. In Tsuchiya and

Kanie (1988), these representations were calculated

for the SU(2) WZW (Wess–Zumino–Witten) model,

where the differential equation is known as the

Khniznik–Zamolodchikov equation. The corre-

sponding braid group representations were shown

to be those obtained in the section ‘‘The original

definition of V

(t)’’ and cablings thereof.

Topological Quantum Field Theory

In Witten (1989), the following formula appears:

ðe

2i=ðkþ2Þ

exp

h

trðA ^ dA þ 2=3 A ^ A ^ AÞ





tr Pexp

½DA

where A ranges over all functions from S

to the Lie

algebra su(2), modulo the action of the gauge group

SU(2). Also h = =k and j runs over the components

of the link L, to each of which is assigned an

irreducible representation of SU(2). Parallel trans-

port around a component j using A yields the linear

map Pexp

A whose trace is constant modulo gauge

transformations. And [DA] is a fictitious diffeo-

morphism invariant measure on all A’s modulo

gauge transformation.

The Jones Polynomial 185

There are at least two ways to interpret this

formula.

1. As a solvable topological quantum field theory

(TQFT) in 2 þ 1 dimensions, according to Witten

(1988) and Atiyah (1988, 1989). One is then

obliged to expand the context and conclude that

2i=n

) is defined for (possibly empty) links in

an arbitrary 3-manifold. The TQFT axioms then

provide an explicit formula for the invariant if the

3-manifold is obtained from surgery on a link. In

particular, the invariant of a 3-manifold without a

link is a statistical mechanics type sum over

assignments of irreducible representations of

SU(2) to the components of the surgery link. The

key condition making this sum finite is that only

representations up to a certain dimension (deter-

mined by n) are allowed. This is the vanishing of

the Jones–Wenzl idempotent of the section ‘‘The

Temperley–Lieb algebra.’’ This explicit formula

was rigourously shown to be a manifold invariant

in Reshetikhin and Turaev (1991).Foramore

simple treatment, see Lickorish (1997) and for the

whole TQFT treatment, see Blanchet et al. (1995).

2. As a perturbative QFT. The stationary-phase

Feynman diagram technique may be applied to

obtain the coefficients of the expansion of Witten’s

formula in powers of h or equivalently 1=n. These

coefficients are known to be ‘‘finite type’’ or

Vassiliev invariants and have expressions as

integrals over configurations of points on the link,

see Vassiliev (1990) and Bar-Natan (1995).

Algebraic Quantum Field Theory

In the Haag–Kastler operator algebraic framework

of quantum field theory (Haag 1996), statis tics of

quantum systems were interpreted in Doplicher

et al. (1971, 1974) (DHR) in terms of certain

representations of the symmetric group correspond-

ing to permuting regions of spacetime. To obtain the

symmetric group, the dimension of spacetime needs

to be sufficiently large. It was proposed in

Fredenhagen et al. (1989) that the DHR theory

should also work in low dimensions with the braid

group replacing the symmetric group, and that

unitary braid group representations defined above

should be the ones occurring in quantum field

theory. The ‘‘statistical dimension’’ of the DHR

theory turns up as the square root of the index of a

subfactor (this connection was clearly established in

Longo (1989, 1980)). The mathematical issue of the

existence of quantum fields with braid statistics was

established in Wassermann (1998) using the language

of loop group representations. Actual physical systems

with nonabelian braid statistics have not yet been

found but have been proposed in Freedman (2003)

as a mechanism for quantum computing.

Acknowledgments

The author would like to thank Tsou Sheung Tsun

for her help in preparing this article.

The work of this author was supported in part by

NSF Grant DMS 0401734, the NZIMA, and the

Swiss National Science Found ation.

See also: Braided and Modular Tensor Categories;

-Algebras and their Classification; Hopf Algebras

and q-Deformation Quantum Groups; Knot Homologies;

Knot Invariants and Quantum Gravity; Knot Theory

and Physics; Large-N and Topological Strings;

Mathematical Knot Theory; Schwarz-Type Topological

Quantum Field Theory; String Field Theory; Topological

Knot Theory and Macroscopic Physics; Topological

Quantum Field Theory: Overview; von Neumann

Algebras: Introduction, Modular Theory, and

Classification Theory; von Neumann Algebras:

Subfactor Theory; Yang–Baxter Equations.