Marathe K. Topics in Physical Mathematics

Подождите немного. Документ загружается.

358 11 Knot and Link Invariant s

Alexander–Conway polynomial, or simply the Conway polynomial. The

Alexander–Conway polynomial of an oriented link L is denoted by ∇

(z), or

simply by ∇(z)whenL is ﬁxed. By changing a link diagram at one crossing

we can obtain three diagrams corresponding to links L

, L

−

,andL

,which

are identical except for this crossing (see Figure 11.4).

−

Fig. 11.4 Altering a link at a crossing

We denote the corresponding polynomials of L

, L

−

,andL

by ∇

, ∇

−

and ∇

respectively. The Alexander–Conway polynomial is uniquely deter-

mined by the following simple set of axioms.

AC1. Let L and L



be two oriented links that are ambient isotopic. Then

∇



(z)=∇

(z). (11.3)

AC2. Let S

be the standard unknotted circle embedded in S

. It is usually

referred to as the unknot and is denoted by O.Then

∇

(z)=1. (11.4)

AC3. The polynomial satisﬁes the skein relation

∇

(z) −∇

−

(z)=z∇

(z). (11.5)

We note that the original Alexander polynomial Δ

is related to the

Alexander–Conway polynomial of an oriented link L by the relation

(t)=∇

1/2

− t

−1/2

Despite these and other major advances in knot theory, the Tait conjec-

tures remained unsettled for more than a century after their formulation.

Then, in the 1980s, Jones discovered his polynomial invariant V

(t), called

the Jones polynomial, while studying von Neumann algebras [210]and

gave its interpretation in terms of statistical mechanics. A state model for

11.2 Invariants of Knots and Links 359

the Jones polynomial was then given by Kauﬀman using his bracket poly-

nomial. These new polynomial invariants led to the proofs of most of the

Tait conjectures. As with the earlier invariants, Jones’ deﬁnition of his poly-

nomial invariant is algebraic and combinatorial in nature and was based on

representations of the braid groups and related Hecke algebras.

The Jones polynomial V

(t)ofκ is a Laurent polynomial in t,whichis

uniquely determined by a simple set of properties similar to the axioms for

the Alexander–Conway polynomial. More generally, the Jones polynomial

can be deﬁned for any oriented link L as a Laurent polynomial in t

1/2

that reversing the orientation of all components of L leaves V

unchanged.

In particular, V

does not depend on the orientation of the knot κ.Fora

ﬁxed link, we denote the Jones polynomial simply by V . Recall that there

are three standard ways to change a link diagram at a crossing point. The

Jones polynomials of the corresponding links are denoted by V

−

and

respectively. Then the Jones polynomial is characterized by the following

properties:

JO1. Let L and L



be two oriented links that are ambient isotopic. Then



(t)=V

(t). (11.6)

JO2. Let O denote the unknot. Then

(t)=1. (11.7)

JO3. The polynomial satisﬁes the skein relation

−1

− tV

−

=(t

1/2

− t

−1/2

. (11.8)

An important property of the Jones polynomial 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 κ

the mirror image of the knot κ.Then

(t)=V

−1

). (11.9)

Since the Jones polynomial is not symmetric in t and t

−1

, it follows that in

general

(t) = V

(t). (11.10)

We note that a knot is called amphichiral (achiral in biochemistry) if

it is equivalent to its mirror image. We shall use the simpler biochemistry

terminology, so a knot that is not equivalent to its mirror image is called

chiral. The condition expressed by (11.10) is suﬃcient but not necessary

for chirality of a knot. The Jones polynomial did not resolve the following

conjecture by Tait concerning chirality.

360 11 Knot and Link Invariant s

The chirality conjecture: If the crossing number of a knot is odd, then it

is chiral.

A 15-crossing knot which provides a counterexample to the chirality conjec-

ture is given in [195].

There was an interval of nearly 60 years between the discovery of the

Alexander polynomial and the Jones polynomial. Since then a number of

polynomial and other invariants of knots and links have been found. A par-

ticularly interesting one is the two-variable polynomial generalizing the Jones

polynomial V which was deﬁned in [211]. This polynomial is called the HOM-

FLY polynomial (its name formed from the initials of authors of the article

[140]) and is denoted by P . The HOMFLY polynomial P (α, z) satisﬁes the

skein relation

−1

− αP

−

= zP

. (11.11)

Both the Jones polynomial V

and the Alexander–Conway polynomial ∇

are special cases of the HOMFLY polynomial. The precise relations are given

by the following theorem.

Theorem 11.5 Let L be an oriented link. Then the polynomials P

, V

,and

∇

satisfy the relations:

(t)=P

(t, t

1/2

− t

−1/2

) and ∇

(z)=P

(1,z).

After deﬁning his polynomial invariant, Jones also established the relation

of some knot invariants with statistical mechanical models [208]. Since then

this has become a very active area of research. We now recall the construction

of a typical statistical mechanics model. Let X denote the conﬁguration space

of the model and let S denote the set (usually with some additional structure)

of internal symmetries. The set S is also called the spin space. A state of the

statistical system (X, S)isanelements ∈F(X,S). The energy E

of the

system (X,S) is a functional

: F(X, S) → R,k∈ K

where the subscript k ∈ K indicates the dependence of energy on the set

K of auxiliary parameters, such as temperature, pressure, etc. For example,

in the simplest lattice models the energy is often taken to depend only on

the nearest neighboring states and on the ambient temperature, and the spin

space is taken to be S = Z

, corresponding to the up and down directions.

The weighted partition function of the system is deﬁned by



(s)w(s),

where w : F(X, S) → R is a weight function and the sum is taken over

all states s ∈F(X, S). The partition functions corresponding to diﬀerent

weights are expected to reﬂect the properties of the system as a whole. Cal-

culation of the partition functions remains one of the most diﬃcult problems

11.2 Invariants of Knots and Links 361

in statistical mechanics. In special models the calculation can be carried out

by auxiliary relations satisﬁed by some subsets of the conﬁguration space.

The star-triangle relations or the corresponding Yang–Baxter equations are

examples of such relations. 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 gives the corresponding polynomial.

However, these statistical models did not provide a geometrical or topo-

logical interpretation of the polynomial invariants. Such an interpretation

was provided by Witten [405] by applying ideas from quantum ﬁeld theory

(QFT) to the Chern–Simons Lagrangian. In fact, Witten’s model allows us

to consider the knot and link invariants in any compact 3-manifold M .Wit-

ten’s ideas led to the creation of a new area—which we discussed in Chapter

7—called topological quantum ﬁeld theory. TQFT, at least formally, allows

us to express topological invariants of manifolds by considering a QFT with a

suitable Lagrangian. An excellent account of several aspects of the geometry

and physics of knots may be found in the books by Atiyah [14] and Kauﬀman

[218].

We conclude this section by discussing a knot invariant that can be deﬁned

for a special class of knots. In 1978 Bill Thurston [373] created the ﬁeld of

hyperbolic 3-manifolds. A hyp erbolic manifold is a manifold that admits

a metric of constant negative curvature or equivalently a metric of constant

curvature −1. The application of hyperbolic 3-manifolds to knot theory arises

as follows. A knot κ is called hyperbolic if the knot complement S

\ κ is a

hyperbolic 3-manifold. It can be shown that the knot complement S

\ κ

of the hyperbolic knot κ has ﬁnite hyperbolic volume v(κ). The number

v(κ) is an invariant of the knot κ and can be computed to any degree of

accuracy; however the arithmetic nature of v(κ) is not known. This result also

extends to links. It is known that torus knots are not hyperbolic. The ﬁgure

eight knot is the knot with the smallest crossing number that is hyperbolic.

Thurston made a conjecture that eﬀectively states that almost every knot is

hyperbolic. Recently Hoste and Weeks made a table of knots with crossing

number 16 or less by making essential use of hyperbolic geometry. Their

table has more than 1.7 million knots, all but 32 of which are hyperbolic.

Thistlethwaite has obtained the same table without using any hyperbolic

invariants. A fascinating account of their work is given in [195]. We would

like to add that there is a vast body of work on the topology and geometry

of 3-manifolds which was initiated by Thurston. At present the relation of

this work to the methods and results of the gauge theory, quantum groups,

or statistical mechanics approaches to the study of 3-manifolds remains a

mystery. There are some conjectures relating invariants obtained by diﬀerent

methods. For example, the volume conjecture (due to R. Kashaev, H.

Murakami, and J. Murakami, see [306]) states that the invariant v(κ)ofthe

hyperbolic knot κ is equal to the limit of a certain function of the colored

Jones polynomial of the knot κ. There is also an extension of the volume

conjecture to links and to non-hyperbolic knots. This extension has been

362 11 Knot and Link Invariant s

veriﬁed for special knots and links such as the ﬁgure eight knot and the

Borromean rings.

11.3 TQFT Approach to Knot Invariants

In this section we discuss a surprising application of the Chern–Simons the-

ory to the calculation of some invariants of knots and links via TQFT. We

begin with some historical observations. The earliest example of TQFT is the

“derivation” of the Ray–Singer analytic torsion [323,324] in terms of Chern–

Simons theory given in [341], although TQFT was not introduced until later.

The analytic torsion is deﬁned by using determinants of Laplacians on forms

that are regularized by zeta function regularization. It is an anlogue of the

classical Reidemeister torsion of 3-manifolds as generalized to arbitrary man-

ifolds by Franz and de Rham. Reidemeister torsion was the ﬁrst topological

invariant that could distinguish between spaces that are homotopic but not

homeomorphic. It can be used to classify lens spaces. Ray and Singer conjec-

tured that these two invariants should be the same for compact Riemannian

manifolds. This conjecture was proved independently by Cheeger and M¨uller

in the late 1970s. See the book [382] by Turaev for further details.

Quantization of classical ﬁelds is an area of fundamental importance in

modern mathematical physics. Although there is no satisfactory mathemati-

cal theory of quantization of classical dynamical systems or ﬁelds, physicists

have developed several methods of quantization that can be applied to speciﬁc

problems. Most important among these is Feynman’s path integral method

of quantization, which has been applied with great success in QED (quan-

tum electrodynamics), the theory of quantization of electromagnetic ﬁelds.

On the other hand the recently developed TQFT has been very useful in

deﬁning, interpreting, and calculating new invariants of manifolds. We note

that at present TQFT cannot be considered a mathematical theory and our

presentation is based on a development of the inﬁnite-dimensional calcula-

tions by formal analogy with ﬁnite dimensional results. Nevertheless, TQFT

hasprovideduswithnewresultsaswell as a fresh perspective on invariants

of low-dimensional manifolds. For example, at this time a geometric inter-

pretation of polynomial invariants of knots and links in 3-manifolds such as

the Jones polynomial can be given only in the context of TQFT.

Recall from Chapter 7 that a quantum ﬁeld theory may be considered

an assignment of the quantum expectation Φ

to each gauge-invariant

function (i.e., a quantum observable) Φ : A(M) → R.IntheFeynmanpath

integral approach to quantization the quantum expectation Φ

of an ob-

servable is expressed as a ratio of partition functions as follows:

Φ

(Φ)

(1)

. (11.12)

11.3 TQFT Approach to Knot Invariants 363

There are several examples of gauge-invariant functions. For example, pri-

mary characteristic classes evaluated on suitable homology cycles give an

important family of gauge-invariant functions. The instanton number k of

P (M, G) belongs to this family, as it corresponds to the second Chern class

evaluated on the fundamental cycle of M representing the fundamental class

[M]. The pointwise norm |F

of the gauge ﬁeld at x ∈ M,theabsolute

value |k| of the instanton number k, and the Yang–Mills action are also

gauge-invariant functions. Another important example of a quantum observ-

able is given by the Wilson loop functional, which we will use later in

this section. It is gauge-invariant and hence deﬁnes a quantum observable.

Regarding a knot κ as a loop we get a quantum observable W

ρ,κ

associated

to the knot. For a link L with ordered components κ

,κ

,...,κ

and corre-

sponding representations ρ

,ρ

,...,ρ

of G we deﬁne the Wilson functional

(ω):=W

,κ

(ω)W

,κ

(ω) ...W

,κ

(ω), ∀ω ∈A

When M is 3-dimensional, P is trivial (in a non-canonical way). We ﬁx a

trivialization to write P (M, G)=M ×G and write A

for A

P (M,G)

. Then the

group of gauge transformations G

can be identiﬁed with the group of smooth

functions from M to G and we denote it simply by G

. The gauge theory

used by Witten in his work is the Chern–Simons theory on a 3-manifold with

gauge group SU(n). The Chern–Simons Lagrangian L

is deﬁned by

4π

tr(A ∧ F −

A ∧A ∧ A)=

4π

tr(A ∧ dA +

A ∧ A ∧ A). (11.13)

Recall that in this case the Chern–Simons action A

takes the form



4π



tr(A ∧ dA +

A ∧ A ∧A), (11.14)

where k ∈ R is a coupling constant, A denotes the pull-back to M of the

gauge potential (connection) ω by a section of P ,andF = F

= d

A is the

gauge ﬁeld (curvature of ω)onM corresponding to the gauge potential A.A

local expression for (11.14)isgivenby

4π





αβγ

tr(A

∂

), (11.15)

where A

= A

are the components of the gauge potential with respect

to the local coordinates {x

}, {T

} is a basis of the Lie algebra su(n)in

the fundamental representation, and 

αβγ

is the totally skew-symmetric Levi-

Civita symbol with 

123

=1.Letg ∈G

be a gauge transformation regarded

as a function from M to SU(n) and deﬁne the 1-form θ by

θ := g

−1

dg = g

−1

∂

gdx

364 11 Knot and Link Invariant s

Then the gauge transformation A

of A by g has (local) components

= g

−1

g + g

−1

∂

g, 1 ≤ μ ≤ 3. (11.16)

In the physics literature the connected component of the identity G

⊂G

is called the group of small gauge transformations. A gauge transformation

not belonging to G

is called a large gauge transformation. By a direct calcu-

lation, one can show that the Chern–Simons action is invariant under small

gauge transformations, i.e.,

)=A

(A), ∀g ∈G

Under a large gauge transformation g the action (11.15) transforms as follows:

)=A

(A)+2πkA

, (11.17)

where

24π





αβγ

tr(θ

) (11.18)

is the Wess–Zumino action functional. It can be shown that the Wess–Zumino

functional is integer-valued and hence, if the Chern–Simons coupling constant

k istakentobeaninteger,thenwehave

)

= e

(A)

The integer k is called the level of the corresponding Chern–Simons the-

ory. The action enters the Feynman pathintegralinthisexponentialform.

It follows that the path integral quantization of the Chern–Simons model is

gauge-invariant. This conclusion holds more generally for any compact sim-

ple group G if the coupling constant c(G) is chosen appropriately. The action

is manifestly covariant since the integral involved in its deﬁnition is inde-

pendentofthemetriconM and this implies that the Chern–Simons theory

is a topological ﬁeld theory. It is this aspect of the Chern–Simons theory

that plays a fundamental role in our study of knot and link invariants. For

k ∈ N, the transformation law (11.17) implies that the Chern–Simons action

descends to the quotient B

= A

as a function with values in R/Z.

is called the moduli space of gauge equivalence classes of connections.

We denote this function by f

, i.e.,

: B

→ R/Z

is deﬁned by

[ω] →A

(ω), ∀[ω]=ωG

∈B

. (11.19)

The ﬁeld equations of the Chern–Simons theory are obtained by setting the

ﬁrst variation of the action to zero as

11.3 TQFT Approach to Knot Invariants 365

δA

=0.

The ﬁeld equations are given by

∗ F

= 0 or, equivalently, F

=0. (11.20)

The calculations leading to the ﬁeld equations (11.20) also show that the

gradient vector ﬁeld of the function f

is given by

grad f

2π

∗ F. (11.21)

The gradient ﬂow of f

plays a fundamental role in the deﬁnition of Floer

homology. A discussion of Floer homology and its extensions is given in Chap-

ter 10, following the article [275]. The solutions of the ﬁeld equations (11.20)

are called the Chern–Simons connections. They are precisely the ﬂat con-

nections.

We take the state space of the Chern–Simons theory to be the moduli

space of gauge potentials B

. The partition function Z

of the theory is

deﬁned by

(Φ):=



−iA

(ω)

Φ(ω)DA,

where Φ : A

→ R is a quantum observable (i.e., a gauge-invariant function)

of the theory and A

is deﬁned by (11.14). Gauge invariance implies that

Φ deﬁnes a function on B

, and we denote this function by the same letter.

The expectation value Φ

of the observable Φ is given by

Φ

(Φ)

(1)



−iA

(ω)

Φ(ω)DA



−iA

(ω)

If Z

(1) exists, it provides a numerical invariant of M . For example, for

M = S

and G = SU(2), using the action (11.14) Witten obtains the follow-

ing expression for this partition function as a function of the level k:

(1) =



k +2

sin



k +2



. (11.22)

This result is a special case of the WRT invariants that we discussed in Chap-

ter 10. These new invariants of 3-manifolds arose as a byproduct of Witten’s

TQFT interpretation of the Jones polynomial as shown next. Taking for Φ the

Wilson loop functional W

ρ,κ

,whereρ is the fundamental representation of G

and κ is the knot under consideration, leads to the following interpretation

of the Jones polynomial (up to a phase factor):

Φ

= V

(q), where q = e

2πi/(k+2)

366 11 Knot and Link Invariant s

For a framed link L,wedenotebyL the expectation value of the corre-

sponding Wilson loop functional for the Chern–Simons theory of level k and

gauge group SU(n)andwithρ

the fundamental representation for all i.

To verify the deﬁning relations for the Jones polynomial of a link L in S

Witten starts by considering the Wilson loop functionals for the associated

links L

−

and uses TQFT with Chern–Simons Lagrangian to obtain

the relation

αL

 + βL

 + γL

−

 =0, (11.23)

where the coeﬃcients α, β, γ are given by the expressions

α = −exp



2πi

n(n + k)



, (11.24)

β = −exp



πi(2 − n −n

)

n(n + k)



+exp



πi(2 + n − n

)

n(n + k)



, (11.25)

γ =exp(

2πi(1 − n

)

n(n + k)

). (11.26)

We note that the calculation of the coeﬃcients α, β, γ is closely related to

the Verlinde fusion rules [393]and2d conformal ﬁeld theories. Substituting

the values of α, β, γ into equation (11.23) and canceling a common factor

exp



πi(2−n

)

n(n+k)



,weget

− t

n/2

L

+(t

1/2

− t

−1/2

)L

 + t

−n/2

L

−

 =0, (11.27)

wherewehaveput

t =exp



2πi

n + k



The Laurent polynomial determined by the skein relation (11.27) is called

the Jones–Witten polynomial for gauge group SU(n). For SU(2) Chern–

Simons theory equation (11.27), under the transformation

√

t →−1/

√

t,goes

over into equation (11.8), which is the skein relation characterizing the Jones

polynomial. The signiﬁcance of this transformation is related to the choice of

a square root for t. It is more transperent when the quantum group deﬁnition

of the Jones polynomial is used. We note that recently the Alexander–Conway

polynomial has also been obtained by the TQFT methods in [146].

If V

(n)

denotes the Jones–Witten polynomial corresponding to the skein

relation (11.27), then the family of polynomials {V

(n)

} can be shown to be

equivalent to the two variable HOMFLY polynomial P (α, z).

We remark that the vacuum expectation values of Wilson loop observ-

ables in the Chern–Simons theory have been computed recently up to sec-

ond order of the inverse of the coupling constant. These calculations have

provided a quantum ﬁeld theoretic deﬁnition of certain invariants of knots

and links in 3-manifolds [87, 178]. A geometric formulation of the quan-

11.4 Vassiliev Invariants of Singular Knots 367

tization of Chern–Simons theory is given in [25]. Another important ap-

proach to link invariants is via solutions of the Yang–Baxter equations

and representations of the corresponding quantum groups (see, for exam-

ple, [221, 223, 294, 317, 326, 380, 383]). For relations between link invari-

ants, conformal ﬁeld theories and 3-dimensional topology see, for example,

[88,230, 328].

11.4 Vassiliev Invariants of Singular Knots

We deﬁne a singular knot s asaknotwithatmostaﬁnitenumbern

of double point singularities. Each such singularity can be resolved in two

ways by deforming one of the two crossing threads to an overcrossing or an

undercrossing. We ﬁx a double point p and use orientation to deﬁne a knot

(resp., s

−

) by overcrossing the ﬁrst (resp., second) time the path passes

the point p. This gives us two singular knots with n − 1 singular points. Let

v denote a knot invariant of a regular (i.e., non-singular) knot. We deﬁne a

Vassiliev invariant v of the singular knot s with n singular points inductively

by the formula

v(s):=v(s

) −v(s

−

) . (11.28)

This deﬁnition allows us to express v as a ﬁnite sum of the invariant v on

a set of regular knots obtained by the complete resolution of the singular

knot s into regular knots. The deﬁnition can be extended to oriented links

by applying it to each component knot.

Deﬁnition 11.2 We say that a Vassiliev invariant v is of order at most n

if v is zero on any singular knot with more than n double points. The order

of a Vassiliev invariant v denoted by ord(v) is the largest natural number m

such that v is not zero on some singular knot with exactly m double points.

The existence of nonzero Vassiliev invariants of a given ﬁnite order was

establishedin[391]. In fact, knot polynomials such as the Jones polynomial

lead to Vassiliev invariants as follows:

Theorem 11.6 Let V

(q) be the Jones polynomial of the knot K.Putq = e

and expand V

as a power series in x:

∞



n=0

Then the coeﬃcient v

of x

in the series expansion is a Vassiliev invariant

of order n.

The set V of Vassiliev invariants has a natural structure of an inﬁnite-

dimensional real vector space. It is ﬁltered by a sequence of ﬁnite-dimensional

vector spaces V

:= {v ∈ V | ord(v) ≤ n}, n ≥ 0. That is