Marathe K. Topics in Physical Mathematics

Подождите немного. Документ загружается.

368 11 Knot and Link Invariant s

V =



and V

⊂ V

··· .

We also have the following theorem due to Kontsevich.

Theorem 11.7 The set V

is non-empty and forms a real vector space. Fur-

thermore the quotient space V

n−1

is isomorphic to the space of linear

functions on the n-chord diagrams modulo certain relations (one and four

term relations).

There are several open questions about Vassiliev invariants. For example:

What can one say about d

=dimV

and the quantum dimension dim

V =



of the real vector spaces V ?

11.5 Self-linking Invariants of Knots

As we indicated in Chapter 7, one of the earliest investigations in combi-

natorial knot theory is contained in several unpublished notes written by

Gauss between 1825 and 1844 and published posthumously. In obtaining a

topological invariant by using a physical ﬁeld theory, Gauss had anticipated

topological ﬁeld theory by almost 150 years. Even the term “topology” was

not used then. Gauss’s linking number formula can also be interpreted as

the equality of topological and analytic degree of a suitable function. Start-

ing with this a far-reaching generalization of the Gauss integral to higher

self-linking integrals was obtained by Bott, Taubes, and Cattaneo. We now

discuss their work.

Let us recall the relevant deﬁnitions. Let X, Y be two closed oriented n-

manifolds. Let q ∈ Y be a regular value of a smooth function f : X → Y .

Then f

−1

(q) has ﬁnitely many points p

,...,p

.Foreachi,1≤ i ≤ j,

deﬁne σ

=1(resp.,σ

= −1) if the diﬀerential Df : TX → TY restricted to

the tangent space at p

is orientation-preserving (resp., reversing). Then the

diﬀerential topological deﬁnition of the mapping degree,orsimplydegree,

of f is given by

deg(f ):=



i=1

. (11.29)

For the analytic deﬁnition we choose a volume form v on Y and deﬁne

deg(f ):=



∗



. (11.30)

In the analytic deﬁnition one often takes a normalized volume form so that



v = 1. This gives a simpler formula for the degree. It follows from the well

known de Rham’s theorem that the topological and analytic deﬁnitions give

the same result. To apply this result to deduce the Gauss formula, denote

the two curves by C, C



. Then the map

11.5 Self-linking Invariants of Knots 369

λ : C ×C



→ S

deﬁned by

λ(r, r



):=

(r −r



)

|r −r



, ∀(r, r



) ∈ C ×C



is well-deﬁned by the disjointness of C and C



.Ifω denotes the standard

volume form on S

,thenwehave

ω =

xdy ∧ dz + ydz ∧ dx + zdx ∧ dy

+ y

+ z

)

3/2

The pull-back λ

∗

(ω)ofω to C × C



is precisely the integrand in the Gauss

formula and



ω =4π. It is easy to check that the topological degree of λ

equals the linking number m. Let us deﬁne the Gauss form φ on C ×C



φ :=

4π

∗

(ω).

Then the Gauss formula for the linking number can be rewritten as



φ = m.

Now the map λ is easily seen to extend to the 6-dimensional space C

)

deﬁned by

):=R

× R

\{(x, x) | x ∈ R

} = { (x

) ∈ R

× R

| x

= x

The space C

) is called the conﬁguration space of two distinct points

in R

.Denotingbyλ

the extension of λ to the conﬁguration space we can

deﬁne the Gauss form φ

on the space C

)by

4π

∗

(ω). (11.31)

The deﬁnition of the space C

) extends naturally to deﬁne C

(X), the

conﬁguration space of n distinct points in the manifold X as follows:

(X):={(x

,...,x

) ∈ X

| x

= x

for i = j, 1 ≤ i, j ≤ n}.

In [149] it is shown how to obtain a functorial compactiﬁcation C

(X)of

the conﬁguration space C

(X) in the algebraic geometry setting. A detailed

account of the geometry and topology of conﬁguration spaces of points on a

manifold with special attention to R

and S

is given by Fadell and Husseini

in [121]. In his lecture at the Geometry and Physics Workshop at MSRI in

Berkeley (January 1994), Bott explained how the conﬁguration spaces enter

in the study of embedding problems and, in particular, in the calculation of

embedding invariants. Let f : X→ Y be an embedding. Then f induces

370 11 Knot and Link Invariant s

embeddings of Cartesian products f

: X

→ Y

,n∈ N. The maps f

give

embeddings of conﬁguration spaces C

(X) → C

(Y ),n ∈ N by restriction.

These maps in turn extend to the compactiﬁcations giving a family of maps

: C

(X) → C

(Y ),n∈ N.

It is these maps C

that play a fundamental role in the study of embedding

invariants. As we have seen above, the Gauss formula for the linking number

is an example of such a calculation. These ideas are used in [54]toobtain

self-linking invariants of knots. Conﬁguration spaces have also been used in

the work of Bott and Cattaneo [52, 53] to study integral invariants of 3-

manifolds. The ﬁrst step is to observe that the λ

deﬁned in (11.31)can

be deﬁned on any two factors in the conﬁguration space C

)toobtaina

family of maps λ

andtheseinturncanbeusedtodeﬁnetheGaussforms

,i= j, 1 ≤ i, j ≤ n,

4π

∗

(ω), where λ

,...,x

):=

− x

∈ S

. (11.32)

Let K

denote the parametrized knot

f : S

→ R

with



=1, ∀t ∈ S

Then we can use C

to pull back forms φ

to C

) as well as to the spaces

n,m

)ofn + m distinct points in R

of which only the ﬁrst n are on S

These forms extend to the compactiﬁcations of the respective spaces and we

continue to denote them by the same symbols. Integrals of forms obtained by

products of the φ

over suitable spaces are called the self-linking integrals. In

the physics literature self-linking integrals and invariants for the case n =4

have appeared in the study of perturbative aspects of the Chern–Simons ﬁeld

theory in [30, 31, 177, 178]. A detailed study of the Chern–Simons perturba-

tion theory from a geometric and topological point of view may be found in

[25, 24]. The self-linking invariant for n = 4 can be obtained by using the

Gauss forms φ

as follows: Let K denote the space of all parametrized knots.

Then the Gauss forms pull back to the product K×C

), which ﬁbers over

K by the projection π

on the ﬁrst factor. Let α denote the result of inte-

grating the 4-form φ

∧φ

along the ﬁbers of π

. While α is well-deﬁned, it

is not locally constant (i.e., dα = 0) and hence does not deﬁne a knot invari-

ant. The necessary correction term β is obtained by integrating the 6-form

∧ φ

over the space C

3,1

). In [54]itisshownthatα/4 − β/3

is locally constant on K and hence deﬁnes a knot invariant. It turns out that

this invariant belongs to a family of knot invariants, called ﬁnite-type in-

variants, deﬁned by Vassiliev [391], which we discussed in Section 11.4 (see

also [13]). In [235] Gauss forms with diﬀerent normalization are used in the

formula for this invariant and it is stated that the invariant is an integer equal

11.6 Categoriﬁcation of the Jones Polynomial 371

to the second coeﬃcient of the Alexander–Conway polynomial of the knot.

Kontsevich views the self-linking invariant formula as forming a small part of

a very broad program to relate the invariants of low-dimensional manifolds,

homotopical algebras, and non-commutative geometry with topological ﬁeld

theories and the calculus of Feynman diagrams. It seems that the full real-

ization of this program will require the best eﬀorts of mathematicians and

physicists in the new millennium.

11.6 Categoriﬁcation of the Jones Polynomial

As we discussed earlier, the discovery of the Jones polynomial invariant of

links renewed interest and greatly increased research activity towards ﬁnding

new invariants of links. Witten’s work gave a new interpretation in terms

of topological quantum ﬁeld theory and in the process led to new invariants

of 3-manifolds. Reshetikhin and Turaev gave a precise mathematical deﬁni-

tion of these invariants (now called WRT invariants) in terms of representa-

tions of the quantum group sl

(2, C) (a Hopf algebra deformation of the Lie

algebra sl(2, C)). Quantum groups were discovered independently by Drin-

feld and Jimbo. By the early 1990s a number of invariants were constructed

starting with the pair (L, g), where L is a link with components colored by

representations of the complex simple Lie algebra g. Many (but not all) of

these invariants are expressible as Laurent polynomials in a formal variable q.

These polynomial invariants have representation-theoretic interpretation in

terms of intertwiners between tensor products of irreducible representations

of the quantum group U

(g) (a Hopf algebra deformation of the universal

enveloping algebra of g). These invariants are often referred to as quantum

invariants of links and 3-manifolds. They form part of a new (rather loosely

deﬁned) area of mathematics called quantum topology.

In modern mathematics the language of category theory is often used

to discuss properties of diﬀerent mathematical structures in a uniﬁed way.

In recent years category theory and categorical constructions have found

applications in other branches of mathematics and also in theoretical physics.

This has developed into an extensive area of research. In fact, as we siaw

in Chapter 7, the axiomatic formulation of TQFT is given by the use of

cobordism categories. We will use a special case of it in our discussion of

Khovanov homology.

We begin by recalling that a categoriﬁcation of an invariant I is the con-

struction of a suitable (co)homology H

∗

such that its Euler characteristic

χ(H

∗

) (the alternating sum of the ranks of (co)homology groups) equals I.

Historically, the Euler characteristic was deﬁned and understood well before

the advent of algebraic topology. Theorema egregium of Gauss and the closely

related Gauss–Bonnet theorem and its generalization by Chern give a geomet-

ric interpretation of the Euler characteristic χ(M)ofamanifoldM.They

372 11 Knot and Link Invariant s

can be regarded as precursors of Chern–Weil theory as well as index the-

ory. Categoriﬁcation χ(H

∗

(M)) of this Euler characteristic χ(M)byvarious

(co)homology theories H

∗

(M) came much later. A well-known recent exam-

ple that we have discussed is the categoriﬁcation of the Casson invariant by

the Fukaya–Floer homology. Categoriﬁcation of quantum invariants such as

knot polynomials requires the use of quantum Euler characteristic and multi-

graded knot homologies. Khovanov [219] has obtained a categoriﬁcation of

the Jones polynomial V

(q) by constructing a bi-graded sl(2)-homology H

i,j

determined by the link L. It is called the Khovanov homology of the link L

and is denoted by KH(L). The Khovanov polynomial Kh

(t, q) is deﬁned

(t, q)=



i,j

dim H

i,j

. (11.33)

It can be thought of as a two-variable generalization of the Poincar´e polyno-

mial. The quantum or graded Euler characteristic of the Khovanov homology

equals the non-normalized Jones polynomial. That is,

(q)=χ

(KH(L)) =



i,j

(−1)

dim H

i,j

Khovanov’s construction follows Kauﬀman’s state-sum model of the link

L and his alternative deﬁnition of the Jones polynomial. Let

L be a regular

projection of L with n = n

+ n

−

labeled crossings. At each crossing we

can deﬁne two resolutions or states, the vertical or 1-state and horizontal or

0-state. Thus, there are 2

total resolutions of

L which can be put into a one-

to-one correspondence with the vertices of an n-dimensional unit cube. For

each vertex x let |x| be the sum of its coordinates and let c(x)bethenumber

of disjoint circles in the resolution

L determined by x. Kauﬀman’s

state-sum expression for the non-normalized Jones polynomial

V (L)canbe

written as follows:

(q)=(−1)

−

−2n

−

)



(−q)

|x|

(q + q

−1

)

c(x)

, (11.34)

where the sum is taken over all the vertices x of the cube. Dividing this by the

unknot value (q + q

−1

) gives the usual normalized Jones polynomial V (L).

The Khovanov complex can be constructed using 2-dimensional TQFT and

related Frobenius algebra as follows: Let V be a graded vector space over

a ﬁxed ground ﬁeld K, generated by two basis vectors v

with respective

degrees ±1. The total resolution associates to each vertex x a 1-dimensional

manifold M

consisting of c(x) disjoint circles. We can construct a (1 +

1)-dimensional TQFT (along the lines of Atiyah–Segal axioms discussed in

Chapter 7) by associating to each edge of the cube a cobordism as follows:

If xy is an edge of the cube we can get a pair of pants cobordism from M

to M

by noting that a circle at x can split into two at y or two circles at x

can fuse into one at y. If a circle goes to a circle then the cylinder provides

11.6 Categoriﬁcation of the Jones Polynomial 373

the cobordism. To the manifold M

at each vertex x we associate the graded

vector space

(L):=V

⊗c(x)

{|x|}, (11.35)

where {k} is the degree shift by k. We deﬁne the Frobenius structure (see

the book [227] by Kock for Frobenius algebras and their relation to TQFT)

on V as follows. Multiplication m : V ⊗ V → V is deﬁned by

m(v

⊗ v

)=v

,m(v

⊗ v

−

)=v

−

m(v

−

⊗ v

)=v

−

,m(v

−

⊗ v

−

)=0.

Co-multiplication Δ : V → V ⊗ V is deﬁned by

Δ(v

)=v

⊗ v

−

+ v

−

⊗ v

,Δ(v

−

)=v

−

⊗ v

−

Thus, v

is the unit. The co-unit δ ∈ V

∗

is deﬁned by mapping v

to 0

and v

−

to 1 in the base ﬁeld. The rth chain group C

(L) in the Khovanov

complex is the direct sum of all vector spaces V

(L), where |x| = r, and the

diﬀerential is deﬁned by the Frobenius structure. Thus,

(L):=⊕

|x|=r

(L). (11.36)

We remark that the TQFT used here corresponds to the Frobenius alge-

bra structure on V deﬁned above. The rth homology group of the Khovanov

complex is denoted by KH

. Khovanov proved that the homology is inde-

pendent of the various choices made in deﬁning it. Thus, we have the next

theorem.

Theorem 11.8 The homology groups KH

are link invariants. In particular,

the Khovanov polynomial

(t, q)=



dim

(KH

)

is a link invariant that specializes to the non-normalized Jones polynomial.

The Khovanov polynomial is strictly stronger than the Jones polynomial.

We note that the knots 9

and 10

125

are chiral. Their chirality is de-

tected by the Khovanov polynomial but not by the Jones polynomial. Also,

there are several pairs of knots with the same Jones polynomials but dif-

ferent Khovanov polynomials. For example (5

, 10

132

) is such a pair. Using

equations (11.35)and (11.36) and the algebra structure on V ,wecanre-

duce the calculation of the Khovanov complex to an algorithm. A computer

program implementing such an algorithm is discussed in [32]. A table of Kho-

vanov polynomials for knots and links up to 11 crossings is also given there.

We now illustrate Khovanov’s categoriﬁcation of the Jones polynomial of the

right-handed trefoil knot 3

374 11 Knot and Link Invariant s

Example 11.1 (Categoriﬁcation of V (3

)) For the standard diagram of the

trefoil 3

, we have n = n

=3and n

−

=0. The quantum dimensions of the

non-zero terms of the Khovanov complex with the shift factor included are

given by

=(q + q

−1

)

=3q(q + q

−1

),C

=3q

(q + q

−1

)

= q

(q + q

−1

)

and C

=0for all other i ∈ Z. The non-normalized Jones polynomial can be

obtained by using the above chain complex (with diﬀerential induced by the

Frobenius algebra deﬁned earlier) or directly from (11.34) giving

(q)=(q + q

+ q

− q

). (11.37)

The normalized or standard Jones polynomial is then given by

(q)=

q + q

+ q

− q

q + q

−1

= q

+ q

− q

By direct computation or using the program in [32] we obtain the following

formula for the Khovanov polynomial of the trefoil

Kh(t, q)=q + q

+ t

,Kh(−1,q)=χ

(q).

Based on computations using the program described in [32], Khovanov,

Garoufalidis, and Bar-Natan (BKG) have formulated some conjectures on

the structure of Khovanov polynomials over diﬀerent base ﬁelds. We now

state these conjectures.

The BKG Conjectures: For any prime knot κ there exists an even integer

s = s(κ) and a polynomial Kh



(t, q) with only non-negative coeﬃcients such

that

1. over the base ﬁeld K = Q,

(t, q)=q

s−1

[1 + q

+(1+tq

)Kh



(t, q)]

2. over the base ﬁeld K = Z

(t, q)=q

s−1

(1 + q

)[1 + (1 + tq

)Kh



(t, q)]

3. moreover, if the knot κ is alternating, then s(κ) is the signature of the

knot and Kh



(t, q)containsonlypowersoftq

The conjectured results are in agreement with all the known values of the

Khovanov polynomials.

If S ⊂ R

is an oriented surface cobordism between links L

and L

then it

induces a homomorphism of Khovanov homologies of links L

and L

.These

homomorphisms deﬁne a functor from the category of link cobordisms to

the category of bi-graded Abelian groups [205]. Khovanov homology extends

11.6 Categoriﬁcation of the Jones Polynomial 375

to colored links (i.e., oriented links with components labeled by irreducible

ﬁnite-dimensional representations of sl(2)) to give a categoriﬁcation of the

colored Jones polynomial. Khovanov and Rozansky have deﬁned an sl(n)-

homology for links colored by either the deﬁning representation or its dual.

This gives categoriﬁcation of the specialization of the HOMFLY polynomial

P (α, q)witha = q

. The sequence of such specializations for n ∈ N would

categorify the two variable HOMFLY polynomial P (α, q). For n =0the

theory coincides with the Heegaard–Floer homology of Ozsv´ath and Szabo

[309].

In the 1990s Reshetikhin, Turaev, and other mathematicians obtained sev-

eral quantum invariants of triples (g,L,M), where g is a simple Lie algebra,

L ⊂ M is an oriented, framed link with components labeled by irreducible

representations of g,andM is a 2-framed 3-manifold. In particular, there are

polynomial invariants L that take values in Z[q

−1

,q]. Khovanov has conjec-

tured that at least for some classes of Lie algebras (e.g., simply laced) there

exists a bi-graded homology theory of labeled links such that the polynomial

invariant <L>is the quantum Euler characteristic of this homology. It

should deﬁne a functor from the category of framed link cobordisms to the

category of bigraded Abelian groups. In particular, the homology of the un-

knot labeled by an irreducible representation U of g should be a Frobenius

algebra of dim(U).

Epilogue

It is well known that the roots of “physical mathematics” go back to the

very beginning of human attempts to understand nature. The abstraction of

observations in the motion of heavenly bodies led to the early developments

in mathematics. Indeed, mathematics was an integral part of natural philos-

ophy. Rapid growth of the physical sciences aided by technological progress

and increasing abstraction in mathematical research caused a separation of

the sciences and mathematics in the twentieth century. Physicists’ methods

were often rejected by mathematicians as imprecise, and mathematicians’

approaches to physical theories were not understood by physicists. We have

already given many examples of this. However, theoretical physics did inﬂu-

ence development of some areas of mathematics. Two fundamental physical

theories, relativity and quantum theory, now over a century old, sustained

interest in geometry and functional analysis and group theory. Yang–Mills

theory, now over half a century old, was abandoned for many years before

its relation to the theory of connections in a ﬁber bundle was found. It has

paid rich dividends to the geometric topology of low-dimensional manifolds

in the last quarter century. Secondary characteristic classes were given less

than secondary attention when they were introduced. These classes turn out

to be the starting point of Chern–Simons theory.

A major conference celebrating 20 years of Chern–Simons theory orga-

nized by the Max Planck and the Hausdorﬀ institutes in Bonn was held in

August 2009. It drew research workers in mathematics and theoretical physics

from around the world. The work presented under the umbrella of this sin-

gle topic shows that the 30 year old marriage between theoretical physics

and mathematics is still going strong. Many areas such as statistical mechan-

ics, conformal ﬁeld theory, and string theory not included in this work have

already led to new developments in mathematics.

The scope of physical mathematics continues to expand rapidly. Even for

the topics that we have considered in this book a number of new results are

appearing and new connections between old results are emerging. In fact, the

377