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Ng (1998) has shown that ribbon knots generate

an index 2 subgroup of G

Polynomiality and Gauss Sums

Goussarov (1998) (see also Goussarov–Polyak–Viro

(2001)) found an intriguing way to compute finite-

type invariants from a Gauss diagram presentation of

a knot, showing in particular that finite-type invar-

iants grow as polynomials in the number of crossings

n and can be computed in polynomial time in n

(though actual computer programs are still missing!).

Gauss diagrams are obtained from knot diagrams

in much of the same way as Chord diagrams are

obtained from singular knots, except all crossings

are counted and not just the double points, and

certain over/under and sign information is asso-

ciated with each crossing/chord so that the knot

diagram can be recovered from its Gaus s diagram.

In the example below (Figure 8), we also dashed a

subdiagram of the Gauss diagram equivalent to the

chord diagram shown in Figure 7.

If G is a Gauss diagram and D is a chord diagram,

then let hD, Gi be the number of subdiagrams of

G equivalent to D, counted with appropriate signs

(to be preci se, we also need to base the diagrams

involved and count subdiagrams that respect the

basing).

Theorem 16 (Goussarov 1998, Goussarov et al.

2000). If V is a type m invariant, then there are

finitely many (based) cho rd diagrams D

with at

most m chords and rational numbers 

so that

V(K) =



, Gi whenever G is a Gauss dia-

gram representing a knot K.

Computing the Kontsevich Integral

While the Kontsevich integral Z

is a cornerstone of

the theory of finite-type invariants, it has been

computed for surprisingly few knots. Even for the

unknot, the result is nontrivial:

Theorem 17 (‘ ‘Wheels,’’ Bar-Natan et al. 2000,

2003). The framed Kontsevich integral of the unknot,

(), expressed in terms of diagrams in

B, is given

by = exp

[

n = 1

, where the ‘‘modified Ber-

noulli numbers’’ b

are defined by the power series

expansion

n = 0

= (1=2) log (sinh x=2)=ðx=2Þ,

the ‘‘2n-wheel’’ !

is the free Jacobi diagram made

of a 2n-gonwith2nlegs(so, e.g., !

= ), and where

exp

[

means ‘‘exponential in the disjoint union sense.’’

Closed-form formulas have also been given for the

Kontsevich integral of framed unknots, the Hopf

link and Hopf chains.

Theorem 17 has a companion that utilizes the same

element , the ‘‘wheeling’’ theorem (Bar-Natan et al.

2000, 2003). The wheeling theorem ‘‘upgrades’’ the

vector space isomorphism  : B!A to an algebra

isomorphism and is related to the Duflo isomorphism

of the theory of Lie algebras. It is amusing to note that

the wheeling theorem (and hence Duflo’s theorem in

the metrized case) follows using finite-type techniques

from the ‘‘1 þ 1 = 2onanabacus’’identity(Figure 9).

Taming the Kontsevich Integral

While explicit calculations are rare, there is a nice

structure theorem for the values of the Kontsevich

integral, saying that for a knot K and up to any fixed

number of loops in the Jacobi diagrams, 

1

(K)can

be described by finitely many rational functions (with

denominators powers of the Alexander polynomial)

which dictate the placement of the legs. This structure

theorem was conjectured in Rozansky (2003), proven

in Kricker (2000), and partially generalized to links in

Garoufalidis and Kricker (2004).

The Rozansky–Witten Theory

One way to construct linear functionals on A (and

hence finite-type invariants) is using Lie algebras

and representations as discussed earlier; much of our

insight about A comes this way. But there is another

construction for such functionals (and hence invar-

iants), due to Rozansky and Witten (1997), using

contractions of curvature tensors on hyper-Ka¨ hler

Figure 7 A chord diagram.

–

Figure 8 A knot and its Gauss diagram.

Figure 9 A knot theoretic 1 þ1 = 2.

346 Finite-Type Invariants

manifolds. Very little is known about the Rozansky–

Witten approach; in particular, it is not known if it

is stronger or weaker than the Lie algebraic

approach. For an application of the Rozansky–

Witten theory back to hyper-Ka¨ hler geometry

check Hitchin and Sawon (2001), and for a

unification of the Rozansky–Witten approach with

the Lie algebraic approach (albeit at a categorical

level) check Roberts and Wille ton (in preparation).

The Melvin–Morton Conjecture and the

Volume Conjecture

The Melvin–Morton conjecture (stated Melvin and

Morton (1995), proven Bar-Natan and Garoufalidis

(1996)) says that the Alexander polynomial can be read

off certain coefficients of the colored Jones polynomial.

The Kashaev–Murakami–Murakami volume conjec-

ture (stated Kashaev (1997) and J Murakami and H

Murakami (2001), unproven) says that a certain

asymptotic growth rate of the colored Jones polyno-

mial is the hyperbolic volume of the knot complement.

Both conjectures are not directly about finite-type

invariants but both have ramifications to the theory

of finite-type invariants. The Melvin–Morton con-

jecture was first proven using finite-type invariants

and several later proofs and generalizations (see

(Bar-Natan)) also involve finite-type invariants. The

volume conjecture would imply, in particular, that

the hyperbolic volume of a knot complement can be

read from that knot’s finite-type invariants, and

hence finite-type invariants would be at least as

strong as the volume invariant.

A particularly noteworthy result and direction for

further research is Gukov’s (preprint) recent unifica-

tion of these two conjectures under the Chern–

Simons umbrella (along with some relations to

three-dimensional quantum gravity).

Beyond Knots

Forlackofspace,wehaverestrictedourselvesheretoa

discussion of finite-type invariants of knots. But the

basic ‘‘differentiation’’ idea of the first section calls for

generalization, and indeed it has been generalized

extensively. We will only make a few quick comments.

Finite-type invariants of homotopy links (links

where each component is allowed to move across

itself freely) and of braids are extremely well

behaved. They separate, they all come from Lie

algebraic constructions and in the case of braids,

step-by-step integration as discussed previously works

(for homotopy links the issue was not studied).

Finite-type invariants of 3-manifolds and especially

of integral and rational homology spheres have been

studied extensively and the picture is nearly a

complete parallel of the picture for knots. There are

several competing definitions of finite-type invar-

iants, and they all agree up to regrading. There are

weight systems and they are linear funct ionals on a

space A(;) which is a close cousin of A and B and is

related to Lie algebras and hyper-Ka¨ hler manifolds in

a similar way. There is a notion of a ‘‘universal’’

invariant, and there are several con structions; they all

agree or are conjectured to agree, and they are related

to the Chern–Simons–Witten theory.

Finite-type invariants were studied for several

other types of topological objects, including knots

within other manifolds, higher-dimensional knots,

virtual knots, plane curves and doodles and more

(see Bar-Natan).

Acknowledgmnts

This work was partially supported by NSERC

grant RGPIN 262178. Electronic versions: http://

www.math.toronto.edu/drorbn/papers/EMP/ and

arXiv:math.GT/0408182. Copyright is reta ined by

the author. The author wishes to thank O Dasbach

and A Savage for comments and corrections,

especially pertaining to Problem 2.

See also: Finite-Type Invariants of 3-Manifolds; Knot

Invariants and Quantum Gravity; Kontsevich Integral;

Mathematical Knot Theory; von Neumann Algebras:

Introduction, Modular Theory, and Classification Theory.

Further Reading

The first reference is an extensive bibliography of finite invariants,

listing more than 550 references. To save space, references for this

article are only cited above stating the authors’ full last name(s)

and approximate year of publication.

Bar-Natan D Bibliography of Vassiliev invariants, http://

www.math.toronto.edu/drorbn/VasBib.

Bar-Natan D (1998) On associators and the Grothendieck–

Teichmuller group I. Selecta Mathematica 4: 183–212.

Bar-Natan D and Thurston D Algebraic Structures on knotted objects

and universal finite type invariants (in prepartion), http://

www.math.toronto.edu/drorbn/papers/AlgebraicStructures.

Drinfel’d VG (1990) Quasi-Hopf algebras. Leningrad Mathema-

tical Journal 1: 1419–1457.

Drinfel’d VG (1991) On quasitriangular Quasi-Hopf algebras and

a group closely connected with Gal(Q=Q). Leningrad Math-

ematical Journal 2: 829–860.

Gukov S Three-dimensional quantum gravity, Chern–Simons

theory, and the A-polynomial. Preprint, arXiv:hep-th/0306165.

Hitchin N and Sawon J (2001) Curvature and characteristic

numbers of hyper-Ka¨hler manifolds. Duke Mathematical

Journal 106–3: 599–615, arXiv:math.DG/9908114.

Kashaev RM (1997) The hyperbolic volume of knots from

quantum dilogarithm. Letters in Mathematical Physics 39:

269–275, arXiv:q-alg/9601025.

Finite-Type Invariants 347

Knizhnik VG and Zamolodchikov AB (1984) Current algebra and

Wess–Zumino model in two dimensions. Nuclear Physics B

247: 83–103.

Murakami J and Murakami H (2001) The colored Jones

polynomials and the simplicial volume of a knot. Acta

Mathematica 186(1): 85–104, arXiv:math.GT/9905075.

Reshetikhin Yu N and Turaev VG (1990) Ribbon graphs and

their invariants derived from quantum groups. Communica-

tions in Mathematical Physics 127: 1–26.

Witten E (1989) Quantum field theory and the Jones polynomial.

Communications in Mathematical Physics 121: 351–399.

Finite-Type Invariants of 3-Manifolds

TTQLeˆ , Georgia Institute of Technology, Atlanta,

GA, USA

Introduction

Physics Background and Motivation

Suppose G is a s emisimple compact Lie group and

M aclosedoriented3-manifold.Witten (1989)

defined quantum invariants by the path integral

over all G-connections A:

ZðM; G ; kÞ :¼

expð

ﬃﬃﬃﬃﬃﬃﬃ

1

k CSðAÞÞDA

where k is an integer and CS(A) is the Chern–Simons

functional,

CSðAÞ¼

4



A ^ dA þ



The path integral is not mathematically rigorous.

According to the stationary-phase approximation in

quantum field theory, in the limit k !1the path

integral decomposes as a sum of contributions from

the flat connections:

ZðM; G; kÞ

flat connections f

ðf Þ

ðM; G; kÞ as k !1

Each contri bution is exp(2

ﬃﬃﬃﬃﬃﬃﬃ

1

k CS(f )) times a

power series in 1=k. The contribution from the

trivial connection is important, especially for

rational homology 3-spheres, and the coefficients

of the powers (1=k)

, calculated using (n þ 1)-loop

Feynman diagrams by quantum field theory techni-

ques, are known as perturbative invar iants.

Mathematical Theories

A mathematically rigorous theory of quantum

invariants Z(M, G ; k) was pioneered by Reshetikhin

and Turaev in 1990 (see Turaev (1994) ).

A number-theoretical expansion of the quantum

invariants into power series that should correspond

to the perturbative invariants was given by Ohtsuki

(in the case of sl

, and general simple Lie algebras

by the author) in 1994. This led him to introducing

finite-type invariant (FTI) theory for 3-manifolds. A

universal perturbative invariant was constructed by

Le–Murakami–Ohtsuki (LMO) in 1995; it is uni-

versal for both finite-type invariants and quantum

invariants, at least for homology 3-spheres.

Rozansky in 1996 defined perturbative invariants

using Gaussian integral, very close in the spirit to

the original physics point of view. Later Habiro

(for sl

and Habiro and the author for all simple

Lie algebras) found a finer expansion of quantum

invariants, known as the cyclotomic expansion, but

no physics origin is known for the cyclotomic

expansion. The cyclotomic expansion helps to show

that the LMO invariant dominates all quantum

invariants for homology 3-spheres.

The purpose of this article is to give an overview

of the mathematical theory of finite-type and

perturbative invariants of 3-manifolds.

Conventions and Notations

All vector spaces are assumed to be over the ground

field Q of rational numbers, unless otherwise stated.

For a graded space A, let Gr

A be the su bspace of

grading n and Gr

n

A the subspace of grading n.

For x 2 A, let Gr

x and Gr

n

x be the projections of

x onto, respectively, Gr

A and Gr

n

All 3-manifolds are supposed to be closed and

oriented. A 3-manifold M is an integral homology

3-sphere (ZHS) if H

(M, Z) = 0; it is a rational

homology 3-sphere (QHS) if H

(M, Q) = 0. For a

framed link L in a 3-manifold M denote M

the

3-manifold obtained from M by surgery along L (see

e.g., Turaev (1994)).

Finite-Type Invariants

After its introduction by Ohtsuki in 1994, the theory

of FTIs of 3-manifolds has been developed rapidly

by many authors. Later Goussarov and Habiro

independently introduced clasper calculus, or

Y-surgery, which provides a powerful geometric

technique and deep insight in the theory. Y-surgery,

corresponding to the commutator in group theory,

naturally gives rise to 3-valent graph s.

348 Finite-Type Invariants of 3-Manifolds

Generality on FTIs

Decreasing filtration In a theory of FTIs, one

considers a class of objects, and a ‘‘good’’ decreasing

filtration F

F

 on the vector space

F = F

spanned by these objects. An invariant of

the objects with values in a vector space is of order

less than or equal to n if its restriction to F

nþ1

is 0;

it is of finite type if it is of order  n for some n.An

invariant has order n if it is of order  n but not

 n  1. Good here means at least the space of FTI

of each order is finite dimensional. It is desirable to

have an algorithm of polynomial time to calculate

every FTI. In addition, one wants the set of FTIs to

separate the objects (completeness).

The space of invariants of order  n can be

identified with the dual space of F

nþ1

; its

subspace F

nþ1

is isomorphic to the space of

invariants of order  n modulo the space of invar-

iants of order  n  1. Informally, one can say that

nþ1

is more or less the set of invariants of

order n.

Elementary moves, the knot case Usually the

filtrations are defined using ‘‘independent elemen-

tary moves.’’ For the class of knots the elementary

move is given by crossing change. Any two knots

can be connected by a finite sequenc e of such moves.

The idea is if K, K

, the nth term of the

filtration, then K  K

nþ1

, where K

is obtained

from K by an elementary move. Formal definition is

as follows. Suppose S is a set of double points of a

knot diagram D.Let

½D; S¼

S

ð1Þ

where the sum is over all subsets S

of S, including

the empty set, D

is the knot obtained by changing

the crossing at every point in S

, and #S

is the

number of elements of S

. Then F

is the vector

space spanned by all elements of the form [D, S]

with #S = n. For the knot case, the Kontsevich

integral is an invariant that is universal for all FTIs

(see Bar-Natan (1995)).

Ohtsuki’s Definition of FTIs for ZHS

An el ement ary move here is a surgery along a

knot: M ! M

,whereK is a framed knot in a

ZHS M. A collection of m oves corresponds to

surgery on a framed li nk. To always remain in the

class of ZHS we need to restrict ourselves to unit-

framed and algebraically split links, that is, framed

links i n ZHS each component of which has

framing 1 and the linking number of every two

components is 0. It is easy to prove that a link L

in a ZHS M is unit-framed and algebrai cally s plit

if and only if M

is a ZHS for every sublink L

L. For a unit-framed, algebraically split link L in a

ZHS M define

½M; L¼

L

ð1Þ

j#L

which is an element in the vector space M freely

spanned by ZHS.

For a non-negative integer n let F

be the

subspace of M span ned by [M, L]with#L = n.

Then the descending filtration M = F

F



 defines a theory of FTIs on the class of

ZHS.

Theorem 1

(i) (Ohtsuki) The dimension of F

(M) is finite for

every n.

(ii) (Garoufalidis–Ohtsuki) One has F

3nþ1

(M) =

3nþ2

(M) = F

3nþ3

(M).

The orders of FTIs in this theory are multiples of 3.

The first nontrivial invariant, which is the only (up

to scalar) invariant of degree 3, is the Casson

invariant.

The Goussarov–Habiro Definition

Y-surgery or clasper surgery Consider the standard

Y-graph Y and a small neighborhood N(Y)ofitin

the standard R

(see Figure 1). Denote by L(Y)the

six-component framed link diagram in N(Y)  R

each component of which has framing 0 in R

(see Figure 1).

A framed Y-graph C in a 3-manifold M is the

image of an embedding of N(Y) into M. The surgery

of M along the image of the six-component link

L(Y) is called a Y-surgery along C, den oted by M

If one of the leaves bounds a disk in M whose

interior is disjoint from the graph, then M

homeomorphic to M.

Matveev in 1987 proved that two 3-manifolds M

and M

are related by a finite sequence of

Y-surgeries if and only if there is an isomorphism

from H

(M, Z) onto H

, Z) preserving the

Y-

raph Sur

ery link L (Y )Its nei

hborhood N (Y )

Figure 1 Y-graph.

Finite-Type Invariants of 3-Manifolds 349

linking form on the torsion group. It is natural to

partition the class of 3-manifolds into subclasse s of

the same H

and the same linking form.

Goussarov–Habiro filtrations For a 3-manifold M

denote by M(M) the vector space spanned by all

3-manifolds with H

and linking form the same as

those of M. Define, for a set S of Y-graphs in M,

[M, S] =

S

(1)

,andF

M(M) the vector

space spanned by all [N, S] such that N is in M(M)

and #S = n. The following theorem of Goussarov

and Habiro (Goussarov 1999, Garoufalidis et al.

2001, Habiro 2000) shows that the FTI theory

based on Y-surgery is the same as the one of Ohtsuki

in the case of ZHS.

Theorem 2 For the case M= M(S

), one has

2n1

= F

The Fundamental Theorem of FTIs of ZHS

Jacobi diagrams A closed Jacobi diagram is

a vertex-or iented triv alent graph, that is, a graph

for which the degree of each vertex is equal to 3 and

a cyclic order of the three half-edges at every vertex

is fixed. Here, multiple edges and self-loops are

allowed. In pictures, the orientation at a vertex is

the clockwise orientation, unless otherwise stated.

The ‘‘degree’’ of Jacobi diagram is half the number

of its vertices.

Let Gr

A(;), n  0, be the vector space spanned

by all closed Jacobi diagrams of degree n, modulo

the antisymmetry (AS) and Jacobi (IHX ) relations

(see Figure 2).

The universal weight map W Suppose D is a closed

Jacobi diagram of degree n.EmbeddingD into R

 S

arbitrarily and then projecting down onto R

general position, one can describe D by a diagram,

with over/under-crossing information at every dou-

ble point just as in the case of a link diagram. We

can assume that the orientation at every vertex of D

is given by a clockwise cyclic order. From the image

of D, construct a set G of 2nY-graphs as in

Figure 3. Here only the cores of a Y-graph are

drawn, with the convention that each framed

Y-graph is a small neighborhood of its core in R

If G

is a proper subset of G , then in G

there is a

Y-graph, one of the leaves of which bounds a disk,

hence S

= S

. Thus, W(D):= [S

, G] = S

 S

.By

definition, W(D) 2F

; it might depend on the

embedding of D into R

, but one can show that

W(D) is well defined in F

2nþ1

. The map W was

first constructed by Garoufalidis and Ohtsuki in the

framework of F

Fundamental theorem

Theorem 3 (Leˆ et al. 1998, Leˆ 1997). The map W

descends to a well-defined linear map

W :Gr

A(;) !F

2nþ1

and moreover, is an

isomorphism between the vector spaces Gr

A(;)

and F

2nþ1

, for M= M(S

The theorem essentially says that the set of

invariants of degree 2n is dual to the space of closed

Jacobi diagram Gr

A(;). The proof is based on the

LMO invariant (see the next section).

A Q-valued invariant I of order 2n restricts to a

linear map from F

2nþ1

to Q. The composition

of I and W is a functional on Gr

A(;) called the

‘‘weight system’’ of I. The theorem shows that every

linear functional on Gr

n

A(;) is the weight of an

invariant of order 2n.

Relation to knot invariants Under the map that

sends an (unframed) knot K  S

to the ZHS

obtained by surgery along K with framing 1, an

invariant of degree 2n (in the F

theory) of ZHS

pulls back to an invariant of order 2n of knots.

This was conjectured by Garoufalidis and proved by

Habegger.

Other classes of rational homology 3-spheres

Actually, the theorem was first proved in the frame-

work of F

. Clasper surgery theory allows Habiro

(2000) to generalize the fundamental theorem to QHS:

for M a QHS, the universal weight map W :Gr

A(;) !F

M(M)=F

2nþ1

M(M), defined similarly as

+ = 0AS

+ + = 0

IHX

(Jacobi)

Figure 2 The AS and IHX relations.

Figure 3 The weight map.

350 Finite-Type Invariants of 3-Manifolds

in the case of ZHS, is an isomorphism, and

2n1

M(M) = F

M(M).

Other filtrations and approaches Other equiva lent

filtrations were introduced (and compared) by

Garoufalidis, Garoufalidis and Levine (1997), and

Garoufalidis–Goussarov–Polyak (2001). Of impor-

tance is the one using subgroups of mapping class

groups in Garoufalidis and Levine (1997). A theory

of n-equivalence was constructed by Goussarov and

Habiro that encompasses many geometric aspects of

FTIs of 3-manifolds (Habiro 2000, Goussarov

1999). Cochran and Melvin (2000) extended the

original Ohtsuk i definition to manifolds with

homology, using algebraically split links, but the

filtrations are different from those of Goussarov–

Habiro.

The Le–Murakami–Ohtsuki Invariant

Jacobi Diagrams

An open Jacobi diagram is a vertex-oriented uni–

trivalent graph, that is, a graph with univalent and

trivalent vertices together with a cyclic ordering of

the edges incident to the trivalent vertices. A

univalent vertex is also called ‘‘a leg.’’ The degree

of an open Jacobi diagram is half the number of

vertices (trivalent and univalent). A Jacobi diagram

based on X, a compact oriented 1-manifold, is a

graph D together with a decomposition D = X [ ,

such that D is the result of gluing all the legs of an

open Jacobi diagram  to distinct interior points of

X. The degree of D, by definition, is the degree of .

In Figure 4 X is depicted by bold lines. Let A

(X)be

the space of Jacobi diagrams based on X modulo the

usual antisymmetry, Jacobi and the new STU

relations. The completion of A

(X) with respect to

degree is denoted by A(X).

When X is a set of m-ordered oriented intervals,

denote A(X)byP

, which has a natural algebra

structure where the product DD

of two Jacobi

diagrams is defined by stacking D on top of D

(concatenating the corresponding oriented intervals).

When X is a set of m-ordered oriented circles,

denote A(X)byA

. By identifying the two

endpoints of each interval, one gets a map pr : P

, which is an isomorphism if m = 1 (see

Bar-Natan (1995)).

For x 2A

and y 2A

, the connected sum is

defined by x#

y := pr((pr

1

x)(pr

1

m

), where

(pr

1

m

is the element in P

with pr

1

y on each

oriented interval.

Symmetrization maps Let B

be the vector space

spanned by open Jacobi diagrams whose legs are

labeled by elements of {1, 2, ..., m}, modulo the

antisymmetry and Jacobi relations. One can define

an analog of the Poincare–Birkhoff–Witt isomorph-

ism  : B

as follows. For a diagram D, (D)

is obtained by taking the average over all possible

ways of ordering the legs labeled by j and attaching

them to the jth oriented interval. It is known that 

is a vector space isomorphism (Bar-Natan (1995)).

The Framed Kontsevich Integral of Links

For an m-component framed link L  R

, the

(framed version of the) Kontsevich integral Z(L)is

an invariant taking values in A

(see, e.g., Ohtsuki

(2002)). Let  := Z(K), when K is the unknot with

framing 0, and



Z(L):= Z(L)#

. An explicit for-

mula for  is given in Bar-Natan et al. (2003).

Removing Solid Loops: The Maps 

Suppose x 2B

is an open Jacobi diagram with legs

labeled by {1, ..., m}. If the number of vertices of

any label is different from 2n, or if the degree of

D > (m þ 1)n, we set 

(D) = 0. Otherwise, parti-

tioning the 2n vertices of each label into n pairs and

identifying points in each pair, from x we get a

trivalent graph which may contain some isolated loops

(no vertices) and which depends on the partition.

Replacing each isolated loop by a factor 2n,

and summing up over all partitions, we get



(D) 2 Gr

n

A(;).

For x 2A

, choose y 2P

such that pr( y ) = x.

Using the isomorphism  we pull back 

1

y 2B

Define 

(x):= 

(

1

y). One can prove that 

(x)

does not depend on the choice of the preimage y of x.

Note that 

lowers the degree by nm.

Definition of the Le–Murakami–Ohtsuki

Invariant Z

LMO

In A(;):=

n = 0

A(;) let the produ ct of two

Jacobi diagrams be their disjoint union. In addition,

define the coproduct (D) = 1  D þ D  1 for D a

connected Jacobi diagram. Then A(;) is a commu-

tative cocommutative graded Hopf algebra.

For the unknot U



with framing 1, one has



(



Z(U



)) = (1)

þ (terms of degree  1); hence,

their inverses exist. Suppose the linking matrix of an

=-STU

Figure 4 The STU relation.

Finite-Type Invariants of 3-Manifolds 351

oriented framed link L  R

has 

positive eigen-

values and 



negative eigenvalues. Define



ðLÞ¼





ZðLÞÞ

ð



ZðU

ÞÞÞ



ð



ZðU



ÞÞÞ





2 Grad

n

ðAð;ÞÞ ½1

Theorem 4 (Leˆ et al. 1998). 

(L) is an invariant

of the 3-manifold M = S

We can combine all the 

to get a better

invariant:

LMO

ðMÞ :¼ 1 þ Grad

ð

ðMÞÞ

þþGrad

ð

ðMÞÞþ2Að;Þ

For M a QHS, we also define

LMO

ðMÞ :¼ 1 þ

Grad

ð

ðMÞÞ

dðMÞ

þþ

Grad

ð

ðMÞÞ

dðMÞ

þ

where d(M) is the cardinality of H

(M, Z).

Proposition 1 (Leˆ et al. 1998). Both Z

LMO

(M) and

LMO

(M)(when defined) are group-like elements,

that is,

ðZ

LMO

ðMÞÞ ¼ Z

LMO

ðMÞZ

LMO

ðMÞ

ð

LMO

ðMÞÞ ¼

LMO

ðMÞ

LMO

ðMÞ

Moreover,

LMO

) =

LMO

) 

LMO

Universality Properties of the LMO Invariant

Let us restrict ourselves to the case of ZHS.

Theorem 5 (Leˆ 1997). The less than or equal to n

degree part Gr

n

LMO

is an invariant of degree 2n.

Any invariant of degree  2n is a compo-

sition w(Gr

n

LMO

), where w :Gr

n

A(;) ! Q is a

linear map.

Clasper calculus (or Y-surgery) theory allows

Habiro to extend the theorem to rational homology

3-spheres.

The Arhus Integral

The Arhus integral (ca. 1998) of Bar-Natan,

Garoufalidis, Rozansky and Thurston, based on a

theory of formal integration, calc ulates the LMO

invariant of rational homology 3-spheres. The

formal integration theory has a conceptual flavor

and helps to relate the LMO invariant to perturba-

tive expansions of quantum invariants. We give here

the definition for the case when one does surgery on

a knot K with nonzero framing b. The link case is

similar (see Bar-Natan et al. (2002a, b)).

When K is a knot,



Z(K) is an element of A



B

. Note that B

is an algebra where the

product is the disjoint union t. Since the framing is

b, one has



ZðKÞ¼exp

ðbw

=2ÞtY

where w

is the ‘‘dashed interval’’ (the only

connected open Jacob i diagram without trivalent

vertex), and Y is an element in B every term of

which must have at least one trivalent vertex. For

uni–trivalent graphs C, D 2B

let

hC; Di¼

0 if the numbers of legs of C; D

are different

sum of all ways to glue legs of C and D

together

One defines



Z(K):= hexp

( w

=2b), Yi.Then



ZðKÞ¼

n¼0

ð



ZðKÞÞ

ðbÞ

Hence,

LMO

ðS

Þ¼



ZðKÞ



ZðU

signðbÞ

Other Approaches

Another construction of a universal perturbative

invariant based on integrations over configuration

spaces, closer to the original physics approach but

harder to calculate because of the lack of a surgery

formula, was developed by Axelrod and Singer,

Kontsevich, Bott and Cattaneo, Kuperberg and

Thurston (see Axelred and Singer (1992), Bott and

Cattaneo (1998)).

Quantum Invariants and Perturbative

Expansion

Fix a simple (complex) Lie algebra g of finite

dimension. Using the quantized enveloping algebra

of g one can define quantum link and 3-manifold

invariants. We recall here the definition, adapted for

the case of roots lattice (projective group case).

Here our q is equal to q

in the text book (Jantzen

1995). Fix a root system of g.LetX, X

, Y denote

respectively the weight lattice, the set of dominant

weights, and the root lattice. We normalize the

invariant scalar product in the real vector space of

the weight lattice so that the length of any short root

ﬃﬃﬃ

352 Finite-Type Invariants of 3-Manifolds

Quantum Link Invariants

Suppose L is a framed oriented link with m-ordered

components, then the quantum invariant

(

, ..., 

) is a Laurent polynomial in q

1=2D

where 

, ..., 

are dominant weights, standing

for the simple g-modules of highest weights



, ..., 

, and D is the determinant of the Cartan

matrix of g (see, e.g., Turaev (1994) and Leˆ (1996)).

The Jones polynomial is the case when g = sl

and

all the 

’s are the highest weights of the funda-

mental representation. For the unknot U with zero

framing, one has (here  is the half-sum of all

positive roots)

ðÞ¼

positive roots 

ðþjÞ=2

 q

ðþjÞ=2

ðjÞ=2

 q

ðjÞ=2

We will also use another normalization of the

quantum invariant:

ð

; ...;

Þ :¼ J

ð

; ...;

Þ

j¼1

ð

This definition is good only for 

2 X

. Note

that each  2 X is either fixed by an element of the

Weyl group under the dot action (see Humphreys

(1978)) or can be moved to X

by the dot action.

We define Q

(

, ..., 

) for arbitrary 

2 X by

requiring that Q

(

, ..., 

) = 0 if one of the 

’s is

fixed by an element of the Weyl group, and that

(

, ..., 

) is component-wise invariant under

the dot action of the Weyl group, that is, for every

, ..., w

in the Weyl group,

ðw

 

; ...; w

 

Þ¼Q

ð

...;

Proposition 2 (Leˆ 1996). Suppose 

, ..., 

are in

the root lattice Y.

(i) (Integrality) Then Q

(

, ..., 

) 2 Z[q

1

], (no

fractional power).

(ii) (Periodicity) When q is an rth root of 1, then

(

, ..., 

) is invariant under the action of

the lattice group rY, that is, for y

, ..., y

2 Y,

(

, ..., 

) = Q

(

þ ry

, ..., 

þ ry

Quantum 3-Manifold Invariants

Although the infinite sum



(

, ..., 

)

does not have a meaning, heuristic ideas show that

it is invariant under the second Kirby move, and

hence almost defines a 3-manifold invariant. The

problem is to regularize the infinite sum. One

solution is based on the fact that at rth roots of

unity, Q

(

, ..., 

) is periodic, so we should use

the sum with 

’s run over a fundamental set P

the action of rY, where

:¼fx ¼ c



þþc

‘



‘

j 0  c

; ...; c

‘

< rg

Here 

, ..., 

‘

are basis roots. For a root  of

unity of order r, let

ðÞ¼



2ðP

\YÞ

ð

; ...;

Þj

q¼

If F



() 6¼ 0, define



ðÞ :¼

ðÞ

ðF

ðÞÞ



ðF



ðÞÞ





Recall that D is the determinant of the Cartan

matrix. Let d be the maximum of the absolute values

of entries of the Cartan matrix outside the diagonal.

Theorem 6 (Leˆ 2003)

(i) If the order r of  is coprime with dD, then



() 6¼ 0.

(ii) If F



( ) 6¼ 0 then 

( ):= 

( ) is an invariant

of the 3-manifold M = S

Remark 1 The version presented here corresponds to

projective groups. It was defined by Kirby and Melvin

for sl

, Kohno and Takata for sl

,andbyLeˆ (2003) for

arbitrary simple Lie algebra. When r is coprime with

dD, there is also an associated modular category that

generates a topological quantum field theory. In most

texts in literature, say Kirillov (1996) and Turaev

(1994), another version 

was defined. The reason we

choose 

is: it has nice integrality and eventually

perturbative expansion. For relations between the

version 

and the usual 

,seeLeˆ (2003).

Examples When M is the Poincare´ sphere and

g = sl



Psl

ðqÞ¼

1  q

n¼0

ð1  q

nþ1

ð1  q

nþ2

Þ...ð1  q

2nþ1

Here q is a root of unity, and the sum is easily

seen to be finite.

Integrality The following theorem was proved for

g = sl

by Murakami (1995) and for g = sl

Takata–Yokota and Masbaum–Wenzl (using ideas

of J Roberts) and for arbitrary simple Lie algebras

by Leˆ (2003).

Theorem 7 Suppose the order r of  is a prime big

enough, then 

( ) is in Z[] = Z[ exp (2i=r)].

Finite-Type Invariants of 3-Manifolds 353

Perturbative Expansion

Unlike the link case, qua ntum 3-manifold invar iants

can be defined only at certain roots of unity. In

general, there is no analytic extension of the

function 

around q = 1. In perturbative theory,

we want to expand the function 

around q = 1

into power series. For QHS, Ohtsuki (for g = sl

)

and then the present author (for all other simple Lie

algebras) showed that there is a number-theoretical

expansion of 

around q = 1 in the following sense.

Suppose r is a big enough prime, a nd

 = exp(2i=r). By the integrality (Theorem 7),



ðÞ2Z ½¼Z½q=ð1 þ q þ q

þþq

r1

Choose a representative f (q) 2 Z[q]of

( ).

Formally substitute q = (q  1) þ 1inf(q):

f ðqÞ¼c

r;0

þ c

r;1

ðq  1Þþþc

r;n2

ðq  1Þ

n2

The integers c

r, n

depend on r and the representative

f(q). It is easy to see that c

r, n

(mod r) does not depend

on the representative f(q) and hence is an invariant of

QHS. The dependence on r is a big drawback. The

theorem below says that there is a rational number c

not depending on r,suchthatc

r, n

(mod r)isthe

reduction of either c

or c

modulo r, for sufficiently

large prime r. It is easy to see that if such c

exists, it

must be unique. Let s be the number of positive roots

of g. Recall that ‘ is the rank of g.

Theorem 8 For every QHS M, there is a sequence

of numbers

2 Z

ð2n þ 2sÞ!jH

ðM; ZÞj



such that for sufficiently large prime r

r;n



ðM; ZÞj



‘

ðmod rÞ

where

ðM; ZÞj



¼1

is the Legendre symbol. Moreover, c

is an invariant

of order  2 n.

The series t

(q  1) :=

n = 0

(q  1)

, called

the Ohtsuki series, can be considered as the

perturbative expansion of the function 

at q = 1.

For actual calculation of t

(q  1), see Leˆ (2003),

Ohtsuki (2002), and Rozansky (1997).

Recovery from the LMO invariant It is known that

for any metrized Lie algebra g, there is a linear map

:Gr

A(;) ! Q (see Bar-Natan (1995)).

Theorem 9 One has

n¼0

ðGr

LMO

Þh

¼ t

ðq  1Þj

q¼e

This shows that the Ohtsuki series t

(q  1) can

be recovered from, and hence totally determined by,

the LMO invariant. The theorem was proved by

Ohtsuki for sl

. For other simple Lie algebras, the

theorem follows from the Arhus integral (see Bar-

Natan et al. (2002a, b) and Ohtsuki (2002)).

Rozansky’s Gaussian Integral

Rozansky (1997) gave a definition of the Ohtsuki

series using formal Gaussian integral in the impor-

tant work. The work is only for sl

, but can be

generalized to other Lie algebras; it is closer to the

original physics ideas of perturbative invariants.

Cyclotomic Expansion

The Habiro Ring

Let us define the Habiro ring

Z[q]by

Z½q :¼ lim

Z½q=ðð1  qÞð1  q

Þ...ð1  q

ÞÞ

Habiro (2002) called it the cyclotomic completion

of Z[q]. Formally,

Z[q] is the set of all series of the

form

f ðqÞ¼

n¼0

ðqÞð1  qÞð1  q

Þ...ð1  q

ðqÞ2Z½q

Suppose U is the set of roots of 1. If  2 U then

(1  )(1  

) (1  

) = 0ifn is big enough;

hence, one can define f()forf 2

Z[q]. One can

consider every f 2

Z[q] as a function with domain U.

Note that f () 2 Z[] is always an algebraic integer.

It turns out that

Z[q] has remarkable properties,

and plays an important role in quantum topology.

Note that the formal derivative of (1  q)

(1  q

) ...(1  q

) is divisible by (1  q)(1

) ...(1  q

)withk > (n  1)=2. This means every

element f 2

Z[q] has a derivative f

Z[q], and hence

derivatives of all orders in

Z[q]. One can then

associate to f 2

Z[q] its Taylor series at a root  of 1:



ðf Þ :¼

n¼0

ðnÞ

ðÞ

ðq  Þ

which can also be obtained by noticing that (1  q)

(1  q

) ...(1  q

) is divisible by (q  )

if n is

bigger than k times the order of . Thus, one has a

map T



Z[q] ! Z[][[q  ]].

354 Finite-Type Invariants of 3-Manifolds

Theorem 10 (Habiro 2004)

(i) For each root of unity  , the map T



is injective,

that is, a function in

Z[q] is determined by its

Taylor expansion at a point in the domain U.

(ii) if f () = g() at infinitely many roots  of prime

power orders, then f = gin

Z[q].

One important consequence is that

Z[q]isan

integral domain, since we have the embedding

Z[q] ,!Z[[q  1]].

In general the Taylor series T

f has 0 convergence

radius. However, one can speak about p-adic

convergence to f () in the following sense. Suppose

the order r of  is a power of prime, r = p

. Then it is

known that (  1)

is divisible by p

if n > mk.

Hence, T

f () converges in the p-adic topology, and

it can be easily shown that the limit is exactly f ().

The above properties suggest considering

Z[q]as

a class of ‘‘analytic functions’’ with domain U.

Quantum Invariants as an Element of

Z[q]

It was proved, by Habiro for sl

and by Habiro with

the present author for general simple Lie algebras,

that quantum invariants of ZHSs belong to

Z[q]

and thus have remarkable integrality properties:

Theorem 11

(i) For every ZHS M, there is an invariant I

Z[q] such that if  is a root of unity for which

the quantum invariant 

( ) can be defined,

then I

() = 

().

(ii) The Ohtsuki series is equal to the Taylor series

of I

at 1.

Corollary 1 Suppose M is a ZHS.

(i) For every root of unity , the quantum invariant

at  is an algebraic integer, 

() 2 Z[]. (No

restriction on the order of  is required.)

(ii) The Ohtsuki series t

(q  1) has integer coeffi-

cients. If  isarootoforderr= p

, where p is

prime, then the Ohtsuki series at  converges

p-adically to the quantum invariant at .

(iii) The quantum invariant 

is determined by

values at infinitely many roots of prime power

orders and also determined by its Ohtsuki series.

(iv) The LMO invariant totally determines the

quantum invariants 

Part (ii) was conjectured by R Lawrence for sl

and first proved by Rozansky (also for sl

). Part (iv)

follows from the fact that the LMO invariant

determines the Ohtsuki series; it exhibits another

universality property of the LMO invar iant.

See also: Finite-Type Invariants; Knot Invariants and

Quantum Gravity; Lie Groups: General Theory; Quantum

3-Manifold Invariants.