Daniel Moskovich- A Kontsevich invariant for coloured knots

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A KONTSEVICH INVARIANT FOR COLOURED KNOTS

DANIEL MOSKOVICH

Abstract. Using the non-commutative surgery presentation of a knot in [29],we construct a non-commutative version of the rational Kontsevich invariant

for p–coloured knots as a D2p–equivariant invariant of their irregular dihedralcovering spaces. As part of the construction we prove a non-commutative

analogue of the Kirby theorem for untying links of p–coloured knots.

1. Introduction

A knot is a smooth embedding K : S 1

S 3. A coloured knot is a pair Ô

K, ρÕ

consisting of a knot and an epimorphism ρ : π1 Ô

S 3¡

K Õ

։ G onto a fixed discretegroup G called a colouring . We may obtain invariants of coloured knots as G–

equivariant invariants of the covering space determined by ρ of S 3 branched overK .

When G

Z, ρ is uniquely determined. A fruitful approach in this case has beento first replace the knot K with a surgery presentation of itself — a unit–framedlink L made up of unknotted components in ker ρ in S 3 ¡ O (the complement of the unknot) such that surgery by L on Ô

S 3,OÕ recovers Ô

S 3, K Õ . We may recover

invariants of K from invariants of L by surgery formulae. The link L lifts to a linkL in the infinite cyclic covering of the complement of the unknot (which is S 3),

such that surgery on L recovers C

Ô

K Õ the infinite cyclic covering of S 3

¡

K . The

link L is called a surgery presentation of C

Ô

K Õ . We may recover Z–equivariant

invariants of C

Ô

K Õ from invariants of L by surgery formulae.

Garoufalidis and Kricker used these ideas in a series of papers [14, 15, 16, 19, 20]

in which they construct a Z–equivariant invariant of the C

Ô

K Õ

which they call arational Kontsevich invariant of K . The loop expansion property allows us to viewthe rational Kontsevich invariant as a Z–equivariant LMO invariant [36], in otherwords as a universal LMO invariant for all cyclic coverings of the knot complement.Its 1–loop part may be expressed as a rational function depending on the Alexanderpolynomial of the knot as conjectured by Rozansky [34].

The above approach can be generalized to any class of coloured knots whosesurgery presentations are known. In [29] we discovered a surgery presentation forthe next simplest class of coloured knots, those with G

D2 p with pÈ Ø 3, 5 Ù —

the dihedral groups of order 6 and 10. Using this result, in the present paper weconstruct a D2 p–equivariant invariant of a dihedral covering space which gives anon-commutative analogue of the rational Kontsevich invariant.

Coloured knots Ô

K, ρÕ withρ : π1 Ô

S 3¡

K Õ ։ D2 p are called p–coloured knots and

ρ is called a p–colouring of K . The following are commutative diagrams illustratingthe process of translating between invariants of coloured knots and G–equivariantinvariants of covering spaces in the case G

Z and in the case G

D2 p with p

È Ø 3, 5 Ù :1



2 DANIEL MOSKOVICH

C

Ô

K Õ

surg Ô LÕ

S 3É

L

π1

ï

ï

î

ï

ï

î

π2

K È

S 3

surgÔ

LÕ

T É

L

M Ô K,ρ Õ

surg Ô LÕ

S 3É

L

πρ

ï

ï

î

ï

ï

î

πρ1

Ô

K, ρÕ È

S 3

surgÔ

LÕ

S 3¡

OÔ

K,ρÕ

É

L

(1.1)

In the right diagram Ô

K, ρÕ is a p–coloured knot with coloured untying invariant

n (Section 2) and OÔ K,ρ Õ

is a connect-sum of n copies of Ô

p1, ρ1 Õ (see Figure 2.1).By abuse of notation we also denote by ρ1 the p–colouring of the connect-sum. Thespace M

Ô K,ρ Õ

is the irregular dihedral cover of S 3 branched over K associated to ρ,

and S 3 is the irregular branched dihedral cover of S 3 branched over OÔ

K,ρÕ

associ-ated to ρ1. The associated degree p covering maps are πρ and πρ1 correspondingly.

We begin by replacing a p–coloured knot Ô

K, ρÕ with a 1–framed link L

S 3¡

OÔ K,ρ Õ

whose components are unknotted and represent homotopy classes inker ρ1. Such a link is called an untying link for Ô

K, ρÕ since surgery by L on

Ô

S 3,Ô

OÔ K,ρ Õ

, ρ1 Õ recovers Ô

S 3,Ô

K, ρÕ Õ .

In order to obtain invariants of Ô K, ρ Õ as invariants of L, we must know how torelate two untying links for the same p–coloured knots. In Section 2.2 we explain

that the non-commutative surgery presentation of a knot is unique up to a certainset of moves analogous to the Kirby moves of the abelian case (this is proved inSection 6.1).

The next stage is to lift the surgery presentation to S 3 the irregular brancheddihedral cover of Ô

S 3,Ô

OÔ K,ρ Õ

, ρ1 Õ Õ . In Section 3 we recover L from L. Taking adiagram-valued Kontsevich integral of L in Section 4 and diagrammatically inte-grating it in Section 5, we obtain a D2 p-equivariant invariant of M

Ô K,ρ Õ

which maybe seen as a p–coloured analogue of the rational Kontsevich invariant (that this isindeed a 3–manifold invariant is proved in Section 6.2). In the case that M

Ô

K,ρÕ

is an integral homology 3–sphere the definition of the above invariant simplifies toyield a p–coloured diagrammatic counterpart to Gaussian integration (Section 5.2).

During our construction we use many spaces of links and diagrams, the mainones of which are listed in Table 1 for easy reference.

1.1. Notations and Conventions.

- We work over the rational numbers Q.- We adopt the convention N Ø 0, 1, 2, . . .Ù .-

D2 p :

s, t

s p

t2 1,tst

s p¡ 1

.

- If X is a set acted on by a group G, andρ

is the equivalence relation xρ

gx

for all g È G, the set of orbits of G on X is denoted X Ä

ρ .

- If M is a move on X , i.e. it specifies a subset Y of X ¢

X such that x1ρ

x2

iff x1 and x2 are related by a series of M moves iff x1 ¢

x2 È

Y , then theset of equivalence classes of X modulo the equivalence relation ρ is againdenoted X

Ä

ρ .

- The identity Ô

n, nÕ –tangle is denoted I n. The tensor product and composi-

tion of tangle diagrams are given as

T 1

T 2 :

T 1 T 2 T 1T 2 :

T 1

T 2



A KONTSEVICH INVARIANT FOR COLOURED KNOTS 3

Concept Section Example

1 p–Coloured Knot 2.1

0 40

1

21

3

2Untying Linkfor a p–ColouredKnot

2.1

¡ 1

2 01

3

Local Picture foran Untying Linkfor a p–Coloured

Knot

2.3

¡ 1

2

0

4

p–Coloured(Pointed) Non-AssociativeTangle Diagram

4.1

¢

Ô Ô Õ Õ Ô Ô Õ Õ

,Ô Õ Ô Ô Õ Ô Õ Õ

,Ô Ô Õ Õ Ô Ô Õ Õ


¤

¥

1

,

1

, ,

0

,

0¬

-

5p–Coloured Wind-ing Diagram

4.2Æ 1

Æ 2

Æ

2 1 ts

s

s2

sx

y

6p–Coloured JacobiDiagram

4.2

Æ

Æ 1

Æ 2 2 ts s

s

s2

x

y

7 p–ColouredSoft-SkeletonJacobi Diagram

4.2

Æ 1 1

Æ

Æ

t

Æ 1s

8 p–ColouredTrivalent JacobiDiagram

5.1ts1

s2

s

Table 1. Some spaces used in this paper.



4 DANIEL MOSKOVICH

-

k : Ø 1, . . . , k

Ù .- We denote the set of n ¢ n matrices over a ring R by Matn

¢

n Ô R Õ . Theset of n

¢

n diagonal matrices over R is denoted Diagn¢

n Ô

RÕ . The n

¢

n

identity matrix is denoted I n¢

n. For square matrices A and B we setA

B : Ô

A 00 B Õ .

- As in [30], we write multi-indices as k Ø k1, k2, ¤ ¤ ¤ , kµ Ù .

Acknowledgements. The author would like to sincerely thank Tomotada Ohtsukifor his support and encouragement throughout the writing of this paper. Thanksalso to Kazuo Habiro for useful discussions and ideas, to Andrew Kricker for adviceand support, and to Gwenael Massuyeau, Dror Bar-Natan, Jim Davis, and TeruhisaKadokami for useful discussions. Finally, special thanks to my wonderful wife Yona–Chava for always being there for me.

2. The Untying Link

In the following section we express a p–coloured knot as a 1–framed link ina handlebody with some information attached satisfying certain conditions andmodulo certain Kirby-like moves. This section then is concerned with the bottomarrow of the right commutative diagram in Equation 1.1:

Ô

K, ρÕ È

S 3

surg Ô L Õ

S 3¡

K 0 É

L

2.1. Untying p–Coloured Knots by Dehn Surgery. Let G be a fixed discretegroup, and let Ô

K, ρÕ denote a coloured knot. The following definition is useful:

Definition 2.1. Let K0 denote the set of equivalence classes of coloured knotsÔ

K, ρÕ modulo 1–framed surgery by links whose components are unknotted and

which represents a homotopy class in ker ρ. Let Ô

K, ρÕ be a given coloured knot and

let Ô

K 0, ρ0 Õ be a chosen representative of its equivalence class in K0. An untying link L for Ô

K, ρÕ is a 1–framed link in S 3

¡

K 0 whose components are unknottedand which represents a homotopy class in ker ρ such that surgery by L recoversÔ

S 3,Ô

K, ρÕ Õ .

From now on let G

D2 p with pÈ Ø 3, 5 Ù except when otherwise stated. The

coloured knot Ô

K, ρÕ with ρ : π1 Ô

S 3¡

K Õ ։ D2 p is called a p–coloured knot , and K

is said to be p–colourable with p–colouring ρ. This terminology reflects the fact thatÔ

K, ρÕ is uniquely characterized by a knot diagram of K whose arcs are coloured

by elements of Z p (the cyclic group of order p) such that at least two colours areused, and that at each crossing half the sum of the labels of the under-crossingarcs equals the label of the over-crossing arc modulo p. Figure 2.1 presents twocolourings of the 74–knot for different values of p. For more about p–colourabilitywe refer the reader to Fox’s original paper [13].

In the G

D2 p case which we are discussing we call chosen representatives of equivalence classes in K0 are called untied p–coloured knots (the author has resistedthe temptation to call them p–coloured unknots). By the results of [29], when

p

3, 5 (and conjecturally for any odd prime p) they may be chosen to be connectsums of n copies of a Ô

p1, ρ1 Õ knots as given in Figure 2.1, with n 1, 2, . . . , p. The

untied p–coloured knot in the same class in K0 as a p–coloured knot Ô

K, ρÕ is called

the untying of Ô

K, ρÕ and is denoted O

Ô

K,ρÕ

.




01

2

0

12

2

Figure 1. The 3–coloured 74–knot.

04

0

1

21

3

Figure 2. The 5–coloured 74–knot.

01 2

Figure 3. The knot Ô 31, ρ1 Õ .

01 2 3 4

Figure 4. The knot Ô 51, ρ1 Õ .

By [29, Section 6], the number of Ô

p1, ρ1 Õ knots appearing as connect summandsin an untied p–coloured knot which is the untying of a p–coloured knot Ô

K, ρÕ is

determined by the coloured untying invariant of Ô

K, ρÕ which is defined as follows:

cuÔ

K, ρÕ

:

¦ a

aÈ

H 3Ô

C 2;Zß

pZÕ

Zß

pZ

Here C 2 denotes the 2–fold covering of S 3 branched over K ,

¦ denotes theBockstein homomorphism on cohomology, and a denotes the cohomology class inH 1

Ô

C 2;Zß

pZÕ determined by ρ.

2.2. A Kirby–like Theorem. Since in the abelian case any crossing change canbe realized by 1–framed surgery on an unknot representing an element of kerAb,there are (infinitely) many ways of untying any knot by such surgeries. However[16, Theorem 1] tells us that all untying links of K are related by a series of Kirbymoves:

Kirby I.

L

L

L (Stabilization)

Kirby II.

L1 L2

L1 ½ L2(Handle Sliding)



6 DANIEL MOSKOVICH

Here denotes disjoint union.For G D2 p one further move is required:

Kirby III p.

L

L 0 Lnew

(Dihedral Circumcision)

where Lnew is any 1-framed component representing an element in ker ρ. KirbyI, Kirby II, and Kirby III p together are called the p–coloured Kirby moves.

We have the following Kirby-like theorem:

Theorem 1. Two links L and L ½ are both untying links for a p–coloured knot Ô

K, ρÕ

if and only if L is an untying link for Ô

K, ρÕ

and L and L ½ are related by the p–coloured Kirby moves.

This is proved in Section 6.1.

2.3. The Local Picture. Before we move onto the next arrow in Equation 1.1 and

lift to a covering space, we first present a local (easier to lift) picture of the datagiven above. Let LÈ

S 3¡

OÔ

K,ρÕ

be an untying link for a p–coloured knot Ô

K, ρÕ .

Then we can isotopy L so that it lies inside a ball B

S 3 such that B¡

OÔ K,ρ Õ

is a

genus 1 cuÔ

K, ρÕ solid handlebody Σ in S 3

¡

OÔ K,ρ Õ

which lifts to a single deck of any branched covering space of itself (in other words, the ‘holes’ in the handlebodyhave no crossings). The example p

3 and cu Ô

K, ρÕ 3 is given below, where we

can isotopy any untying link of Ô

K, ρÕ into the dotted box:

We have

Σ Ô

B¡

OÔ K,ρ Õ

Õ Ô Ô

N Õ

BÕ

where N denotes a regular neighbourhood of OÔ K,ρ Õ

in S 3. We denote the ‘holes’—the components of Ô

N Õ

B, by H 1, H 2, . . . , H 1

cuÔ

K,ρÕ

. Each H i has a core whichis an arc in O

Ô K,ρ Õ

. We define the colouring of H i to be the colouring of its core,and require (by correctly choosing Σ) that the colouring of H 1 is g1 È Zß pZ andthe colouring of H j is g2 È

Zß

pZ for all j 2. In addition we choose the numbering

of the ‘holes’ so that the cores of H 1 and of H 2 are in the same prime-summand of O

Ô

K,ρÕ

.Continuing with the example n

2, we now have the following local picture forL, where the coloured line indicates linkage with possibly more than one arc:




(2.1) L

g2g1

This presentation is not unique— to see this, contemplate the following picture:

(2.2) L

g2g1

and slide the line-segment drawn linking to the ‘hole’ drawn in the upper-rightcorner of the picture up and out of the picture and around the knot O

Ô

K,ρÕ

untilit comes back up through the bottom of the picture. We obtain a new link L

½ inΣ which is ambient isotopic in S 3

¡

OÔ

K,ρÕ

but not in Σ. Examining the situationmore closely, we see that because each prime component of O

Ô K,ρ Õ

is a twist-knotwith p twists, L

½ is related to L as follows (for p 3):

(2.3)

L

g2g1

If p 5 then the altered segment goes one more time around the ‘holes’. The move

taking L to L ½ (and its inverse) are called κ –moves.It will be useful in what follows to add slicing information to our picture. We

define a slice to be an incompressible disc in Σ whose boundary is contained in

B

H i for some i 1, 2, . . . , 1 cu Ô

K, ρÕ . The colour of the slice is defined to be

the colour of the ‘hole to which it is adjacent. A maximal non-intersecting familyof slices added to our data gives us what we call the sliced local picture.

Summarizing, corresponding to the first three rows of Table 1 we have at ourdisposal three equivalent presentations for a p–coloured knot:



8 DANIEL MOSKOVICH

(1) As a pair Ô

K, ρÕ .

(2) As an untying link in the complement of an untied p–coloured knot OÔ

K,ρÕ

,modulo the p–coloured Kirby moves.

(3) As the untying link L pushed into a genus 1 cuÔ K, ρ Õ handlebody, modulothe p–coloured Kirby moves, sliced or not sliced.

3. Lifting the Link

Having obtained a 1–framed link in the complement of an untied p–colouredknot in place of our original p–coloured knot Ô

K, ρÕ , we now proceed with our plan

and lift it to S 3 the irregular branched dihedral cover of Ô S 3, Ô OÔ

K,ρÕ

, ρ1 Õ Õ , followingthe right arrow in the right commutative diagram in 1.1:

S 3πρ1

S 3¡

OÔ K,ρ Õ

L L

We then calculate the D2 p-covariant linking matrix of L in terms of the linkingmatrix of L. In this section we take components of L to be oriented.

3.1. The Irregular Branched Dihedral Covering. There are two branched

covering spaces of S 3

naturally associated with a p–coloured knotÔ

K, ρÕ

. The firstis the regular branched dihedral covering M

Ô K,ρ Õ

, defined to be the covering space

of S 3 branched over K associated to ker Ô

ρÕ . The second is the irregular branched

dihedral covering M Ô

K,ρÕ

, which is defined to be the covering space of S 3 branched

over K associated to ρ ¡

1Ô

S 2 Õ where S2 ⊳D2 p. Note that since the trivial group is

an order 2 subgroup of S2, we have that M Ô

K,ρÕ

is a 2–fold cover of M Ô

K,ρÕ

. Thusas in [7], associated to the graph of subgroups of D2 p we have the graph of coveringspaces

(3.1)

Ø

eÙ

Z p S2

D2 p

M Ô

K,ρÕ

M Ô

K,ρÕ

M Ô

K,ρÕ

S 3

f g

h j

where g : M Ô K,ρ Õ

S 3 is a 2–fold branched cyclic covering, j : M Ô K,ρ Õ

M Ô K,ρ Õ

is

a p–fold unbranched covering, h : M Ô

K,ρÕ

M Ô

K,ρÕ

is the quotient by the period 2

covering translation of M Ô

K,ρÕ

, and f : M Ô

K,ρÕ

S 3 is the p–fold irregular brancheddihedral covering.

Irregular branched dihedral covers are an exceptionally rich class of 3–manifolds—in fact any closed orientable 3–manifold is homeomorphic to a 3–fold irregularbranched dihedral covering space of some knot [18, 26, 32].

A cut-and-paste construction of M Ô K,ρ Õ is given in [7]. We present a surgeryconstruction of this space, based on Section 2.

Contemplate the sliced local picture of an untying link L, and let componentsof L be equipped with an orientation. When we cross a slice, the component in




the lift of the link is acted on by a deck transformation in the p–fold irregularbranched dihedral cover of S 3 ¡ O

Ô

K,ρÕ

by conjugation by the colour of the slice.To picture this, drawn p copies of L one above the other representing copies of L inthe irregular branched dihedral cover, and examine how crossing the slices connectson to the other. This is illustrated below for p

3:

L

L

L

We find that the lift of the link in 2.1 is one of the following:

p 3:

L L L

L

p 5:

L L L

L L

L

3.2. Equivariant Linking Matrices. In the present section we calculate the D2 p-

equivariant linking matrix of L in terms of the linking matrix of L. We first assignan orientation and a basepoint to each component of L, a basepoint p for Σ whichis disjoint from L, and paths from p to the basepoints of the components of L such

that the paths and basepoints all lie within a small ball in Σ.

3.2.1. Link Diagrams in Σ. As before, let Σ be a genus n solid handlebody, viewedas D

¢

I for a standard disc with n holes D. Then any link L :

S 1 Σ has a

link diagram in D, unique up to Reidemeister moves.A link diagram of L in D is said to be pointed if the image of each component

of L comes equipped with a basepoint distinct from the crossing points. Such adiagram is unique up to Reidemeister moves away from the basepoints. Pointedlink diagrams are in bijective correspondence with tangles in Σ, whose endpointsare on standard points on D

¢ Ø 0 Ù and on D¢ Ø 1 Ù . Every link in Σ determines a

pointed link diagram by choosing arbitrary basepoints for its components.To incorporate colouring information into link diagrams and into pointed link

diagrams, equip D with n simple non-intersecting curves C 1, C 2, . . . , C n called glu-ing sites from each of its holes to the boundary of the disc which are projectionsof the slices of Σ and have colours induced from the colour of the correspondingslices. Explicitly, for n

2 we can picture D

C 1

C 2 as



10 DANIEL MOSKOVICH

A (pointed or non-pointed) link diagram in D

C 1

C 2 ¤ ¤ ¤

C n is called acoloured link diagram, and is assumed to be in general position with respect to thegluing sites.

3.2.2. Equivariant Linking Matrices. Let x denote a crossing between componentsLi and Lj in L, define lblx Ô

Li, Lj Õ as follows. Beginning at the basepoint of acomponent Li of L, proceed along Li in the direction of its orientation. When wecross a slice we record the colour encountered as an element of D2 p (if the colourof the slice is c then record tsc

È

D2 p). Continue until we arrive at a crossingx, then cross over to the component Lj and proceed along Lj in the direction of its orientation recording colours until we reach the basepoint of Lj. We obtain asequence g1, g2, . . . gk È

D2 p. Define now:

lblx Ô

Li, Lj Õ

:

g1 ¤

g2 ¤ ¤ ¤ ¤ ¤

gk

Note that lblx Ô

Li, Lj Õ lbl ¡

1x Ô

Li, Lj Õ . Using this term we define:

W D2p

ij Ô

LÕ

:

ô

x

Ô ¡ 1 Õ

sgnÔ

xÕ lblx Ô

Li, Lj Õ

where sgn Ô

xÕ denoted the sign of the crossing x. Considering all 1

i, j

ν weobtain an n

¢

n matrix:

W D2pÔ

LÕ

:

¡

W D2p

ij Ô

LÕ

©

È Matµ¢

µ Ô ZÖ

D2 p × Õ

This is a Hermitian matrix (that is, W

D2pÔ

LÕ

is equal to its transpose followed bythe involution g

g ¡ 1). The topological meaning of W D2pÔ

LÕ is as follows:

Consider the diamond of covering spaces 3.1, and let L, L, and L be lifts of L toM

Ô

K,ρÕ

, M Ô

K,ρÕ

, and M Ô

K,ρÕ

correspondingly. By choosing a fundamental domainwith respect to covering translations in each of these coverings, we may choose

K 1, . . . , K µ

µ

,

K 1, . . . , K µ

, and

K 1, . . . , K µ

µ

connected components of the lifts

of Ø

K 1, . . . , K µ Ù . The groups of translations are Zß

pZ, D2 p and S2 correspondingly.Choose actions such that a path which starts at p, crosses the slice coloured tsi,and returns to p (without crossing the slice again) is lifted to a path which starts

at p and ends at tsi ptsi, tsi p, and tp correspondingly. The lifted links L, L, and L

may now be identified with the sets of translates:

K 1, tK 1t,tsK 1t s , . . . , t s p¡

1K 1ts p¡

1, K 2, . . . , t s p¡

1K µts p¡

1µ

K 1, tK 1, . . . , t s p¡

1K 1, K 2, . . . , t s p¡

1K µ

K 1, tK 1, K 2, tK 2, . . . , K µ, tK µ

µ




We define equivariant linking matrices for L, L, and L as follows:

Linki,j Ô

LÕ

:

ô

gÈ

D2p

gLink Ô

K i, gK jgÕ

Linki,j Ô

LÕ :

ô

g È D2p

gLinkÔ

K i, gK j Õ

Þ Linki,j Ô

LÕ : Link Ô

K i, K j Õ

tLink Ô

K i, tK j Õ

Lemma 3.1. W D2pÔ

LÕ

is equal to the equivariant linking matrix for L and for L, and the matrix induced from it by the projection D2 p ։ S2 is the equivariant

linking matrix for L.

Proof.

Note that by projecting D2 p to Ø 1 Ù the linking matrix of L is recovered from

W D2pÔ L Õ . To recover the linking matrices of L, L, and L, make the following

substitutions:

For L:

ψ : Matµ¢

µ Ô ZÖ D2 p × Õ Mat3µ¢

3µ Ô ZÕ

1

I 3¢ 3 t

¡

0 0 10 1 01 0 0

©

s

¡

0 1 00 0 11 0 0

©

For L:

ψ : Matµ¢

µ Ô ZÖ

D2 p × Õ Mat6µ¢

6µ Ô ZÕ

1

I 6¢

6 t Ô

0 11 0 Õ Ô

0 11 0 Õ Ô

0 11 0 Õ

s

¡

0 I 00 0 I I 0 0

©

with I

I 2¢

2

For L:

ψ : Matµ¢ µ Ô

ZÖ

D2 p × Õ Mat2µ¢ 2µ Ô

ZÕ

1

I 2¢ 2 t

Ô

0 11 0

Õ

Lemma 3.2.

(1)

ψÔ

Link Ô

LÕ Õ Link Ô

LÕ

(2)

ψÔ Link Ô

LÕ Õ Link Ô

LÕ

(3)

ψÔ

Þ Link Ô

LÕ Õ Link Ô

LÕ

Proof. This follows from looking at the lifts of these matrices to the universalcover.

Remark 3.3. Note that the linking matrices of ˜L and of

¯L do not change under theκ –move (recall Section 2.3) because the contribution of the arc which has been slid

to these matrices is Ô 1 ¡

tÕ Ô 1

s

s2 ¤ ¤ ¤

s p¡ 1Õ which generates the kernels of

ψ and ψ.



12 DANIEL MOSKOVICH

4. A p–Coloured Diagram Valued Kontsevich Invariant of the Link

In the present section we use the information we have gleaned about liftinguntying links in the complements of untied p-coloured knots in order to define a

p–coloured diagram valued Kontsevich invariant of the link from which we can thencross finally to the LMO invariant of M

Ô K,ρ Õ

. We are continuing then with the arrow

S 3 πρ1

S 3¡

OÔ K,ρ Õ

L

L

In Sections 4.1 and 4.2 we review well-known concepts related to the Kontsevichinvariant in a slightly modified setting. Section 4.1 is a straightforward generaliza-tion of [16, Section 2.2] presented slightly differently. Section 4.2 reviews definitionsand some basic results concerning Jacobi diagrams in our context, in a form some-where in-between [20, Section 3] and [16, Section 3.3].

4.1. p–Coloured (Pointed) Tangle Diagrams. In this section we decompose a(pointed or unpointed) link diagram in D

C 1

C 2 ¤ ¤ ¤

C n into diagrams onsimpler surfaces as in [16]. Consider the following surfaces in R

2:

Σa Σb Σc

t 0

t 1

Σb and Σc contain a distinguished segment of their boundaries called a gluing sitemarked by an arrow. There are well-defined notions of tangle diagrams and of linkdiagrams on such surfaces, with boundary points of the diagram lying on standardpoints of the boundaries of Σa, Σb, and Σc.

Consider an Ô

m1, m2 Õ –tangle diagram in Σi. In the pointed case, the correspond-ing tangle diagrams are also said to be pointed , and images of basepoints of link

components are marked on the diagram by associating to it a subset of “top base-points” S 1

m1 and a subset of “bottom basepoints” S 2

m2. Graphically, werepresent such a diagram as a tangle diagram in Σ i with the kth boundary pointsat t

0 marked if kÈ

S 1, and the lth boundary point at t 1 marked if l

È

S 2.An example of a dihedral based Ô 3, 3 Õ –tangle diagram with S 1 Ø 2, 3 Ù and S 2 À

is given below:

Σb

Composition and tensor products of p–coloured (pointed) tangle diagrams are

defined as with tangles, with S iS j and S i

S j denoting S i composed with S j andS i tensor S j correspondingly. Composed p–coloured (pointed) tangle diagram arerequired to match up on their boundaries— shapes, images of boundary points, andimages of basepoint. For example:




T i

T i

1

Remark 4.1. Our concept of a tangle diagram is more general than in [20, 16]

because we allow tensor products. That is because in this paper we must makeconcrete calculations, and allowing tensor products of tangle diagrams reduces thenumber of associators we must take into account.

A p–coloured link diagram in D

C 1

C 2 ¤ ¤ ¤

C n may be decomposed withrespect to some generic height function on D into a sequence of tangle diagrams

D

E 1, . . . ,E l1,F 1,G1, E l1 1, . . . , E 2

,F 2,G2, . . . ,F n,Gn, E ln ¡

1 1, . . . ,E ln

where 0

l1

l2 ¤ ¤ ¤

ln and E i are tangle diagrams in Σa, F i are tanglediagrams in Σb, and Gi are tangle diagrams in Σc. In the pointed case, we requirein this decomposition that the basepoint of each element be mapped to an pair of endpoints of pointed tangle diagrams (on the top of some T i and on the bottom of

some T i 1) which are marked. The slices C 1, C 2, . . . , C n map to the gluing sites.

The p–colouring of a gluing site is the p–colouring of the corresponding slice. A p–coloured (pointed) tangle diagram whose gluing sites are p–coloured is said to bea p–coloured (pointed) tangle diagram .

The above decomposition of a p–coloured link diagram into p–coloured tanglediagrams is unique up to the following equivalence relation:

Definition 4.2. Regular isotopy is the equivalence relationship between dihedraltangle diagrams generated by the following moves:

Move I:

T e

T f

T g

T e

T f

T g

T e

T f

T g

where the glued edge T e must have no images of basepoints on it.Move II:

T a

T b

T d

T c

Unit for Tensor Product: If T i is an empty p–coloured tangle diagram onΣx and T j is any p–coloured (based) tangle diagram on Σy then T i

T j

T j

T i

T j , for some x, yÈ Ø

a,b,cÙ .



14 DANIEL MOSKOVICH

Changing Projections: Described below.

To eliminate the basepoints, we must slide them along the link and prove invari-ance under such slides of whatever it is that we are examining. This correspondsto the following moves:

Definition 4.3. Sliding marked endpoints along consecutive tangles is called abasing move. If the gluing site is not crossed this is called a β 1–move, otherwise itis called a β 2–move.

We now attach extra information to the top and bottom endpoints of an Ô

m, nÕ –

tangle, making them what Bar-Natan calls non-associative tangles [3]. The categoryof non-associative tangles is defined as a category whose objects are words in thefree magma on two letters Ø

, Ù (we denote the empty word À ), whose morphisms

are freely generated by the following basic non-associative tangles (A ,B ,C È Ø

, Ù ):

(1) The identity morphism A

A denoted .(2) The associativity morphism Ô

AÔ

BC Õ Õ Ô

ABÕ

C and its inverse, denotedand correspondingly.

(3) The braiding morphism AB BA and its inverse, denoted ¼ and »

correspondingly.(4) Pair creationÞ annihilation À

AB (A

B) and its inverse, denoted

and correspondingly.

For a morphism w1M

w2, we call w1 the top boundary word and w2 the bottom boundary word of M .

An Ô

m, nÕ –tangle diagram in Σa which we call A becomes a non-associative

tangle diagram by assigning to A two words w1, w2 in the free magma on twoletters Ø

, Ù of lengths m and n correspondingly, along with a decomposition of

A into elementary non-associative tangles such that the top boundary word of the decomposition is w1 and the bottom boundary word is w2. For Ô

m, nÕ -tangle

diagrams in Σb we further require that w2 be of the form Ô

v1 Õ Ô

v2 Õ where the number

of letters in v1 corresponds to the number of endpoints of the tangle diagramadjacent to the gluing site and v1 has the standard left bracketing. For Σc wemake the corresponding demand of the top boundary word.

The main definition for this section is:

Definition 4.4. A presentation of a link diagram A in D is a decomposition of A

into 1 cuÔ

K, ρÕ non-associative identity tangles in Σb, 1 cu Ô

K, ρÕ non-associative

identity tangles in Σc, and any number of elementary non-associative tangles in Σa

such that the boundary words match up and the top boundary word and bottomboundary word of the decomposition are the empty words. A presentation is writtenas a vector of boundary words above a vector of tangles where the empty words atthe top and bottom are omitted. Clearly every link-diagram has a presentation.

In the example below (corresponding to the untying link for the 3–coloured 74–knot) we abridge and multiply out some factors so the composite tangles are nolonger elementary, but the meaning is clear:




¢

Ô Ô Õ Õ Ô Ô Õ Õ

,Ô Õ Ô Ô Õ Ô Õ Õ



ª

¤

¥

1

,

1

, ,

0

,

0¬

-

We can now complete Definition 4.2 with the final move:

Changing projections:¢

Ô Ô Õ Ô Õ

,Ô Õ Ô Õ

,Ô Õ Ô Õ

ª

¤

¥

g

g

,

g

g

,

¬

-

is equivalent to¢

Ô Ô Õ Ô Õ

ª

£

g

,

g «

For pointed link-diagrams, we incorporate the basepoint information into theboundary words by considering ordered triples of a non-associative tangles T to-gether with a subset S 1

m and a subset S 2

n. Graphically, we represent sucha tangles by writing the boundary words as elements in the free magma over fourletters Ø

,

,

¢

,

¢

Ù rather than over two Ø

, Ù , where the ith letter of the top (bot-

tom) word w is marked with a cross if iÈ

S 1 (if iÈ

S 2 Õ . For w a word in

,

,

¢

,and

¢ , let w½ denote w with the crosses forgotten. The generalization to the case

of coloured non-associative tangles is straightforward.

4.2. Jacobi Diagrams. In this section we review the Jacobi-diagram terminologywhich is relevant in our context. For general references for the theory of Jacobidiagrams, see e.g. [2, 31]. Some of definitions and conventions we use here differfrom classical usage— in particular our division of a skeleton into a ‘hard’ and‘soft’ part and view of a circle as a trivalent graph with no vertices and no edgesare non-standard.

A uni-trivalent graph is a finite pseudograph (the ‘pseudo’ tells us that we allowmultiple edges and loops) for which each vertex has valence either 1 or 3, and whosetrivalent vertices are oriented, i.e. a cyclic order of the three edges around eachtrivalent vertex is fixed. A uni-trivalent graph which has no legs is called a trivalent graph .

Convention. A circle is considered a trivalent graph with no vertices and no edges.

A skeleton Γ is defined to be a disjoint union of oriented circles ( X) and ori-ented segments ( X), collectively called the hard skeleton ; and two sets ( Æ X and



16 DANIEL MOSKOVICH

x

y

Æ 1

2Æ 1

1

Æ 1

Figure 5. A Jacobi diagram.

X), collectively called the soft skeleton (we refrain from using the standard term“colours” to avoid confusion).

Our main object of study is now defined to be a uni-trivalent graph G whose legsare labeled by points a given skeleton Γ such that no two legs are labeled by the

same point in the hard skeleton (although many different points may be labeled bythe same point in the soft skeleton). This is called a Jacobi Diagram .By definition a circle is a Jacobi diagram with no legs and no arcs. We call this

the null Jacobi diagram and denote it . A Jacobi diagram with two legs (labeledx and y) and no internal vertices is called a chord and is denoted x y.

An example of a Jacobi diagram is given in Figure 4.2. 1–dimensional compo-nents of the skeleton are drawn using thick lines, while the uni-trivalent graphs aredrawn using dotted lines. Colours are denoted Ø Æ 1, . . . ,

Æ n, 1, . . . ,

n Ù .Let D Ô ΓÕ denote the free Q–module of Jacobi diagrams over a skeleton Γ. We

define local moves on elements of D Ô ΓÕ which differ inside a dotted circle as in-dicated below. The equivalence classes of elements of D Ô ΓÕ modulo the relationsØ

AS,IHX,STU Ù and a further global move IB which we define below are denoted

AÔ ΓÕ . Such a space is called an A–space.

The AS relation

¡

The IHX relation

The STU relation

The IB (Infinitesimal Basing) move takes a Jacobi diagram G to the Jacobidiagram G

½ obtained taking the sum of all ways of attaching all legs labeled i to




edges adjacent to legs labeled Æ i with a counterclockwise direction, for all i. Thisis called the x–flavoured link relation in [10]. An example is given below:

Æ

2

Æ 1 Æ 1 Æ 3

Æ

4 1Æ 1

Æ 2

Æ 1 Æ 1 Æ 3

Æ

4

Æ

1

Æ

2

Æ 1 Æ 1 Æ 3

Æ

4

Æ 1

Æ

2

Æ 1 Æ 1 Æ 3

Æ

4

Æ 1

In the same way that the word ‘knot’ is used in Knot Theory for both equivalenceclasses of knots (modulo ambient isotopy) and individual embeddings K : S 1

S 3

which are representatives of those classes, we shall use our terminology looselyand call both elements in A

Ô ΓÕ and elements in DÔ ΓÕ by the same name, Jacobi

diagrams.

Remark 4.5. Our AÔ ΓÕ is usually calledo

AÔ ΓÕ in the literature.

Two maps between spaces of Jacobi diagrams are ubiquitous in the theory of finite-type invariants. The first is the symmetrization map

χX : AÔ Æ X

Y Õ

AÔ X

Y Õ

where Y is any skeleton, which maps a diagram DÈ

AÔ Æ X Õ to the average of all

ways of placing legs coloured x on an oriented interval labeled x, for all xÈ

X , andignored legs labeled by elements of Y . Its inverse is called the formal PBW mapand is denoted by

σX : AÔ X

Y Õ

AÔ Æ X

Y Õ

The space of Jacobi diagrams over a given skeleton AÔ ΓÕ is graded by degree,

where the degree of a Jacobi diagram is defined as half its number of vertices.In the present note, we shall be interested in two classes of Jacobi diagrams which

carry auxiliary structure. An oriented Jacobi diagram whose edges are labeled byfinite series of elements in the set ZÖ

D2 p × and an orientation of such edges is calleda p–coloured winding diagram . Such labels have been drawn in the literature as

boxes with labels inside them and as labeled beads, but we prefer to draw them astriangles on the vertices labeled by the appropriate elements of ZÖ

D2 p × . We think of them as auxiliary bivalent vertices with an distinguished ingoing edge and outgoingedge, and call them coupons following [20]. They are subject to the relations:

The Triv relation

1

The OR relation

g

g ¡

1

The Linearity relation

g

h

g

hgh

hg



18 DANIEL MOSKOVICH

Æ 1

Æ 2

Æ 2

1

ts

s

s2

sx

y

Figure 6. A p–coloured windingdiagram.

Æ

Æ 1

Æ 2

2ts s

s

s2

x

y

Figure 7. A p–coloured Jacobi dia-gram.

The VI relation

g

gg

g g

g

Here OR stands for orientation reversing , and V I for vertex invariance.An (unoriented) Jacobi diagram edge-labeled by elements in Z

Ö

D2 p × shall becalled a p–coloured Jacobi diagram .

We may map p–coloured winding diagrams to p–coloured Jacobi diagrams bythe threading map. This is defined on coupons labeled by non-trivial elements of D2 p by

(4.1) T hrD2p

¤

¦

¥

g

¬

Æ

-

kg

k

kg

g ¤ ¤ ¤

and extended by linearity to edges labeled by elements in the integer group ring

ZÖ D2 p × .Jacobi diagrams (without coupons) and p–coloured winding diagrams are related

by the operation of pushing coupons to legs. For the definition of this operationfor a general (not necessarily dihedral) group see [16, Section 3.6]. In the dihedralcontext, for D

È

AÔ

X

Y Õ , define D

§

§

X Xtsito be the diagram obtained by D by

attaching a coupon labeled tsiÈ

D2 p pointing away from X to each edge adjacentto a leg labeled by an element in the set X . For example:

x

y

1

2Æ 1 D

D§

§

Ø

xÙ ts2 Ø xÙ

x

y

1

2Æ 1

ts2ts2

ts2




4.3. An Invariant of p–Coloured Untying Links. In this subsection we con-struct a non-commutative analogue of Kricker’s winding diagram-valued Kontsevichinvariant of a link [20]. We begin by recalling the combinatorial definition of theKontsevich invariant (see for instance [31, Chapter 6]).

The Kontsevich invariant is a map which eats non-associative tangles and spitsout elements of A–spaces. Consider again the elementary non-associative tangles:

Ô Õ Ô Õ Ô Õ Ô Õ

»

¼

À À

which we assume to be oriented clockwise or down. On these elements we definethe Kontsevich invariant as follows:

Z Ô Õ Φ Z

Ô Õ Φ¡ 1

Z Ô » Õ R Z Ô ¼ Õ R ¡ 1

Z Ô Õ ν 1

2 Z Ô Õ ν 1

2

where

- Φ denotes a rational associator, an element of AÔ x,y,z Õ whose existence isknown but whose exact form is still shrouded in mystery. See [4, 21, 22].

-

ν is the wheels element [8, 12, 35]. It is defined to be

exp

ô

n

1

b2nw2n

where– b2n are the modified Bernoulli numbers defined by

ô

n

1

b2nx2n

12

log

¢

sinh Ô

x2

Õ

x2

ª

– R denotes the Jacobi diagram

exp

¤

¥

1

2

x y¬

-

– w2n are the wheels which are the following Jacobi diagrams

w2

w4

w6

,¤ ¤ ¤

To obtain the Kontsevich invariant of any elementary tangle from the abovedefinition, we make use of the following two operations:



20 DANIEL MOSKOVICH

Comultiplication:

∆

Antipode:

...

S Ô ¡ 1 Õ

n...

where n is the number of legs adjacent to the hard skeleton componentwhose orientation we are reversing.

For an example of how these operations are used in practice to calculate theKontsevich invariant, see Section 6.2.

Given a presentation of a link diagram in D (Definition 4.4), all that remains isto define our analogue of the Kontsevich invariant at gluing sites and at images of basepoints. This is done with the help of the following maps:

(1) At a gluing site labeled n:

GnÔ Õ

tsntsn

tsn

For T i and T i

1 non-associative tangle diagrams on Σb and on Σc corre-spondingly, separated by a gluing site labeled n, and the matching boundaryword is factored as w

w1w2, then:

Z D2pÔ

T Õ

: ¤ ¤ ¤ ¥

Z D2pÔ

T i Õ ¥ Ô

I w1

Gnw2

Õ ¥

Z D2pÔ

T i

1 Õ ¥ ¤ ¤ ¤

(2) Let I ¢

w denote I w ½

with each crossed strand broken. For instance:

I ¢

¢

¢

For T i and T i

1 non-associative tangle diagrams on Σa and w is theboundary word between them:

Z D2pÔ

T Õ

: ¤ ¤ ¤ ¥

Z D2pÔ

T i Õ ¥

I ¢

w ¥

Z D2pÔ

T i

1 Õ ¥ ¤ ¤ ¤

Lemma 4.6. Z D2p is a well-defined invariant of a link diagram in D.

Proof. The proof that Z D2p does not depend on the choice of bracketings and ispreserved by under regular isotopy (Definition 4.2) is the same mutatis mutandisas the proof of [16, Lemma 4.3] and so is omitted.

Lemma 4.7. Z D2p respects basing relations (see Definition 4.3).

Proof. Identical to [16, Proof of Lemma 4.4].

Lemma 4.8.

∆ Ô

Z D2pÔ

LÕ Õ

Z D2pÔ

LÕ

Z D2pÔ

LÕ




Proof. This follows from the corresponding properties for Z , for G, and for allnormalization factors used.

Recall that a Jacobi diagram is said to be X –substantial for a subset X of itsskeleton if it contains no strut components both of whose legs are labeled by pointsin elements of X . It is said to be substantial if X is the entire skeleton.

Proposition 4.9. For any untying link L,

σZ D2pÔ

LÕ exp

¤

¦

¥

1

2

ô

i,j

W D2p

ij Ô

LÕ

xj

xi

¬

Æ

-

R

where R is a series of substantial diagrams.

Proof. Our proof identical to the proof of [20, Theorem 3.5.5] and so is omitted.

5. Diagrammatic Integration

In the following section we construct a D2 p–equivariant invariant of the irreg-ular dihedral covering space M

Ô

K,ρÕ

from the p–coloured winding diagram valued

Kontsevich invariant of its surgery presentation L. In this way we realize the final

top arrow of the right commutative diagram in Equation 1.1:

M Ô

K,ρÕ

surg Ô LÕ

S 3É

L

An invariant of M Ô K,ρ Õ

which gives a non-commutative analogue of the rationalKontsevich integral should be analogous to the LMO invariant of M

Ô

K,ρÕ

[36]. TheLMO invariant of a 3–manifold is defined by performing an operation called LMO integration in [11] on the Kontsevich integral of its surgery presentation. The basicidea of LMO integration is to kill the skeletons of Jacobi diagrams appearing in theKontsevich integral by gluing legs in a certain way, until we obtain a sum over Qof Jacobi diagrams whose underlying graphs are trivalent. It is straightforward toperform an analogous gluing of legs in the p–coloured setting, and this is what wedo in Section 5.1. Proving that the formal sum obtained is indeed an invariant of

M Ô

K,ρÕ

(and therefore of Ô

K, ρÕ

) is less simple, and this is put off until Section 6.2.If the linking matrix of L is invertible over Z (this occurs if M

Ô K,ρ Õ

happens

to be an integral homology 3–sphere) then LMO integration simplifies to Arhusintegration (also called formal Gaussian integration )— see [9, 10, 11]. This is thesubject of Section 5.2.

5.1. LMO Integration. Let Z D2pÔ

LÕ be the p–coloured winding diagram valued

Kontsevich invariant of a p–coloured untying link L for Ô

K, ρÕ which has µ com-

ponents, and let Z D2pÔ

LÕ denote Z D2p

Ô

LÕ

ν

µ. In this section we do not needbasepoints for L, so we are free to assume that Z D2p

Ô L Õ has only oriented circles asits skeleton. When we change L by p–coloured Kirby moves, or when we performa κ –move, Z D2p

Ô L Õ changes. LMO integration may be naıvely thought of as amethod of obtaining invariance under these moves by gluing legs and introducing

relations tailor-made to cancel the changes. There does exist a more conceptualinterpretation of LMO integration as outlined in [11], but it is still hand-wavy.

We begin by introducing the relations we would like to mod out by (assume thatn

2):



22 DANIEL MOSKOVICH

The On relation g

¡ 2ng

The P n relation

All ways of

completingwith dottedarcs

¤ ¤ ¤

2n legs

0

The ZM relation

Ô

t¡ 1 Õ Ô 1

s

s2 ¤ ¤ ¤

s p¡

1Õ

0

A conceptual explanation of the On and P n relations (as the dimension of somephantom vector space and its symmetric algebra) is given in [11, Section 3.2]. Thefirst few relations P n are:

The P 2 relation

0

The P 3 relation

0

.

.

.

For the ZM relation, recall Remark 3.3 which implies that a κ –move corresponds

to introducing the elementÔ

1¡

tÕ Ô

1

s

s2

¤ ¤ ¤

s p

¡

1Õ

which ought to be degenerate.The next step is to replace solid circle components of the skeleton with dashed

graphs. Following [36] we define a series of elements T m È

AÔ Æ 1,

Æ 2, . . . ,Æ m Õ as

follows. Set T 0

T 1 0 and set T 2

Æ 1

Æ 2. For m

3, define




T m :

ô

τ È Sm

¡

2

Ô ¡ 1 Õ

rÔ

τ Õ

Ô

m¡ 1 Õ

m¡

2r Ô τ Õ

T τ

where rτ is the number of kÈ Ø 1, 2, . . . , m

¡ 2 Ù satisfying τ k

τ k

1 and T τ is theJacobi diagram corresponding to the permutation τ as follows:

1 2 3 m ¡ 2

1 2 3 m¡ 2

¤ ¤ ¤

0 m ¡ 1

1 2 3

¤ ¤ ¤

m¡ 2

Figure 8. The Jacobi diagram T τ .

To eliminate the skeleton, define the map ı1 : AÔ 1,

2, . . . . l Õ

AÔ À Õ to

be the linear map which replaces a solid circle component of the skeleton with m

legs attached to it by T m with legs glued up in pairs correspondingly:

¤¤¤

T m

¤ ¤ ¤

where if orientations are reversed then we multiply T m by Ô ¡ 1 Õ

m. The result is takenmodulo On. The maps ın : A

Ô 1, 2, . . . .

ln Õ

AÔ À Õ are defined similarly,

except that first we act n¡ 1 times on each solid circle by the comultiplication map

∆, and multiply the result by the normalization factor 1n!

.The first few values of T m for m

2 are:

T 3

1

2

T 4

1

6

1

6

.

.

.

Let σ

and σ¡

denote the number of positive and negative eigenvalues of Link Ô

LÕ

respectively.



24 DANIEL MOSKOVICH

Definition 5.1. Let L be a p–coloured untying link for a p–coloured knot Ô

K, ρÕ

with irregular dihedral cover M Ô

K,ρÕ

. Then we define the p–coloured LMO series :

Z D2p,gpn Ô

M Ô K,ρ Õ

Õ

:

ın Ô

Z Ô

U

Õ Õ

¡ σ

ın Ô

Z Ô

U ¡

Õ Õ

¡ σ¡

ın Ô

Z D2pÔ

LÕ Õ

where U

and U ¡

denote the 1 and ¡ 1–framed unknots respectively.

The sum of the p–coloured LMO series is the p–coloured rational Kontsevich invariant :

Z D2p,gpÔ

M Ô

K,ρÕ

Õ

: 1

Z D2p,gp1 Ô

M Ô

K,ρÕ

Õ

Ô 1Õ

Z D2p,gp2 Ô

M Ô

K,ρÕ

Õ

Ô 2Õ

¤ ¤ ¤

where Z D2p,gpn Ô

M Ô

K,ρÕ

Õ

Ô

iÕ denotes the degree i part of Z

D2p,gpn Ô

M Ô

K,ρÕ

Õ .

Theorem 2.

Z D2p,gp : Ø

p–coloured knotsÙ

Agp,0ß Ü

ZM Ý

is an invariant of p–coloured knots.

This theorem is proven in Section 6.2.

5.2. Arhus Integral. For the present section we assume that M Ô K,ρ Õ

is an integral

homology 3–sphere, and that therefore LinkÔ

LÕ is invertible over Z. In this case

the definition of Z D2p,gp simplifies allowing us to view it as a p–coloured formalGaussian integral. The simplified version of the LM O integral is called the ArhusIntegral .

We briefly recall the definition of Gaussian integration. Let V be a vector spacewith Lebesgue measure dv and consider I T

V eT dv where T is a polynomial

which may be written as a sum of the form T

12

Q

P where Q is a non-

degenerate quadratic and P is some perturbation. The Fourier transform meansthat up to homothety we have I T

eP , e¡

Q ¡

1ß

2

where Ü ¤

,¤ Ý denotes the usual

pairing S ¦

Ô

V ¦

Õ

S ¦

Ô

V ¦

Õ

C.The diagrammatic analogue of this [9, 10, 11] is based on the formula

σZ D2pÔ

LÕ exp

¤

¦

¥

1

2

ô

i,j

W D2p

ij Ô

LÕ

xj

xi

¬

Æ

-

R

of Proposition 4.9 where the equivariant linking matrix plays the role of the qua-dratic and R is the perturbation. The analogy to the pairing is given by Ü ¤

,¤ Ý : A

Ô Γ

X Õ

AÔ

Γ

X Õ

AÔ

ΓÕ

which sends a pair of Jacobi diagrams to the sum of allways of pairing up all X –labeled legs in the first argument to all X –labeled legs inthe second argument, and to zero if the number of X –labeled legs in each argumentis different. For example:




y

x

x ,

x

z

x

y

z

Ø

xÙ

z

y

zy

zy

zy

To define an analogue to Gaussian integration we must invert the equivariantlinking matrix. For W D2p

Ô

LÕ to be invertible if and only if Link Ô

LÕ is invertible

over Z, we must consider it over the Cohn localization Rloc of the quotient ringR :

ZÖ

D2 p × ß Ô 1 ¡

tÕ Ô 1

s ¤ ¤ ¤

s p¡

1Õ ([16]). It seems an interesting problem to

identify Rloc explicitly.One key lemma in the construction is:

Lemma 5.2 (Translation Invariance). For any matrix power series M È Matν ¢ ν Ô

RÕ

set

F Ô

x

M x ½

Õ

:

F

¤

¥ x1

ô

i1

M 1i1x ½

1i1

ô

i2

M 1i2x ½

1i2 ¤ ¤ ¤

ô

iµ

M 1iµx½

1iµ

¬

-

Then:÷

Ô

nÕ

F Ô

xÕ

dX

÷

Ô

nÕ

F Ô

x

M x½

Õ

dX

Proof. Our proof is identical to [11, Proof of Proposition 3.1] (see also [20, Proof of Theorem 7.2.1]), and is omitted.

We continue by repackaging LMO integration as in [11, Proposition 2.1]:

Lemma 5.3.

Ô mÕ

¥

σ

ım where:

÷

Ô

mÕ

G dX :

õ

xÈ

X

1

m!

¡

x x

2

© m

, G

X

Ç

Ô

Om, P m

1, ZM Õ

Proof. Compare coefficients.

Continuing, for a cycle σ È Sm consider the product

Ü

x1, x2, . . . , xm Ý σ :

1

m

mô

i 1

xσiÔ

1Õ

xσiÔ

2Õ

¤ ¤ ¤

xσiÔ

mÕ

With respect to this product we have the function

detÔ M Õ :

ô

πÈ

Sµ

Ô ¡ 1 Õ µsgnÔ π Õ

ô

σ cycle of π

µô

i

1

M i,σ Ô iÕ , M i,σ2 Ô i Õ , . . . M i,i

σ

Lemma 5.4. detÔ

W D2pÔ

LÕ Õ

is an invariant of Ô

K, ρÕ

.



26 DANIEL MOSKOVICH

Proof. Our proof consists of two stages. First we show that det is a Whitehead determinant . Then we prove that any Whitehead determinant of Ô W D2p

Ô L Õ Õ is aninvariant of Ô

K, ρÕ .

Recall that a Whitehead determinant is defined as a function Mat Ô R Õ R

satisfying the following properties [24]:

i) det Ô

ABÕ detÔ

AÕ det Ô

BÕ .

ii) det Ô

eij

Ô

rÕ Õ 1 for each elementary transvection e

ijÔ

rÕ (an elementary transvec-

tion is the identity matrix plus a matrix which is zero everywhere except in itsÔ i, j Õ th coordinate where it is r).

iii) det Ô

A

I m

m Õ detÔ

AÕ for all m.

Properties 2 and 3 are clear from the definition. It remains to prove Property 1.

6. Proofs

6.1. Proof of Theorem 1. Our theorem may be proved in much the same waythat the ‘relative version of the Kirby Theorem’ is proved by Garoufalidis andKricker [15]. Our notation closely follows that of Roberts [33].

Let L1 and L2 be two untying links for a p–coloured knot Ô

K, ρÕ , and let M

denote the ambient manifold S 3¡

OÔ K,ρ Õ

. Pick a base-point for M which avoidsthe L1 and L2. Let M L1

and M L2be diffeomorphic manifolds relative to their

boundaries, obtained from M by surgery on the framed links L1 and L2, with base-points induced by the base-point of M . We can assume that the diffeomorphismbetween them preserves base-points. Our goal is to prove that L1 and L2 areequivalent modulo our expanded set of Kirby moves.

Let W 0 and W 1 be 4–manifolds obtained by taking M ¢

I and attaching 2-handles to the top surface along L0 and L1. Put in another way, W 0 and W 1are cobordisms between M and M L0

and M L1correspondingly. The boundaries

Ô

W 0 Õ and Ô

W 1 Õ are homeomorphic (we write this boundary as M Ô

M ¢

I Õ

N

where N is homeomorphic to M Li , i 0, 1, following Roberts), and so we can cross

the boundaries with I and glue them together using the homeomorphism, to get aclosed smooth 4–manifold W . See Figure 6.1.

W 0W 1

M L0M L1

M M

Let π :

π1 Ô

W 0 Õ

π1 Ô

W 1 Õ be the fundamental group of W , and let K Ô

π, 1 Õ

be the (pointed) Eilenberg–MacLane space associated to this group. We have a




well-defined map η : W

K Ô

π, 1 Õ from W into K Ô

π, 1 Õ which maps the ‘baseline’(the basepoint of M times an interval ending at the basepoints of W L0

or W L1)

to the base-point of K Ô

π, 1 Õ . By [15, Theorem 4] we know that L1 and L2 areequivalent by stabilization and handle-slides if and only if η Ô W Õ 0 È H 4 Ô π Õ .

If H 4 Ô

πÕ would vanish as it does in the abelian case, we would be finished and

p–coloured circumcision would not be necessary. But already for the simplest caseO

Ô

K,ρÕ

Ô 31, ρ1 Õ and L1

L2 À this is not the case. But fortunately we maykill this homology group by p–coloured circumcision as follows:

For L an untying link for a p–coloured knot, let κ3 Ô

C, LÕ denote the union of L

with a component C in the kernel of the p–colouring which ringed by a 0–framedunknot (we call this p–coloured circumcision of L by C ). The manifolds M L andM κ3 Ô

C,LÕ

have the same homology because all we have done is to add a cancelingpair of handles, but π is divided by the relation induced by C .

In the degenerate case where L1

L2 À are empty links, we rename W byW .

For simplicity, we assume that L1, L2 È ker ρ both consist of a single component(since cobordisms may be ‘stacked’, proving the theorem for this case proves it in

general). Here W differs from W by a single 2-handle attached along the curve L1

(or L2) [17]. Take C a component parallel to L1 but with opposite framing and

orientation, and apply p–coloured circumcision by C to W (C may be chosen to bedisjoint from L2 in W ). This will kill the contribution of L1 (and so also of L2) to

the fundamental group of W , reducing it to the fundamental group of W .It remains to prove the theorem for W . We assume that the coloured untying

invariant of Ô

K, ρÕ is one (the proof easily generalizes to the general case). Laying

out the Ô

p, 2 Õ –torus knot symmetrically on the boundary of an imaginary unknottedsolid torus and looking at it from ‘above’ (consider Figure 2.1 or the embeddinggiving explicitly at the beginning of [27]), we obtain a knot diagram with p arcsØ

a0, . . . , a p¡

1 Ù . Denoting the meridians of these arcs Ø

m0, . . . , m p¡

1 Ù , by the p–colouring rule we may assume without loss of generality that ρ

Ô

mi Õ

tsi for i

0, . . . , p¡ 1.

The Wirtinger presentation of the knot group of t Ô p, 2 Õ (see [6]) is

G

:

x0, . . . , x p¡

1

xixi

1x

¡

1

i

2x

¡

1

i

1

u, v

u

2

v

¡

p

with iÈ

Zß

pZ and

u Ô

x0x1 Õ

1¡

p2 x ¡

11

v

x0x1

Choose C 1 to be a curve that goes p times around a0 and a1. The homotopy classcorresponding to C 1 is in ker ρ because ρ

Ô

x0x1 Õ È

Z p

D2 p, and so p–coloured cir-

cumcision by C 1 is a Kirby III p–move, which takes W to a homeomorphic manifold

W 1 with fundamental group

G1 :

u, v

u2v ¡ p, v p Z p Z2

To turn W 1 into a homeomorphic manifold W 2 with fundamental group D2 p weperform p–coloured circumcision to introduce the relation uvu ¡

1v 1, or equiva-

lently x1x20x1 1 or x2

0x21 1. We achieve this by taking C 2 to be a curve which



28 DANIEL MOSKOVICH

links to a0 twice and to a1 twice. Again this is in kerρ as its image under ρ ist2

Ô tsÕ

2 1.

We conclude by recalling the homology of D2 p.

Lemma 6.1. For n odd, H 4 Ô

D2n Õ 0.

Proof. First we calculate the cohomology of D2n as in [1, Lemma IV.6.3]. We have

H ¦

Ô

D2n;Z2Õ

Z2Ö

e×

(the polynomial algebra on one generator with coefficientsin Z2), and H ¦

Ô

D2n;Zn Õ

Zn Ö

b2× ¢

E Ö

eb× where E is the exterior algebra, e is

as before, and b is the order n Bockstein operation. Combining to get integralcohomology, we obtain:

H iÔ

D2n Õ

°

³

³

³

²

³

³

³

±

Z for i 0,

Z2 for i 2 mod 4,

0 for i odd,

Z2n for i 0 mod 4.

Since D2n is finite, H i Ô

D2n Õ is finite abelian for all i 0. The Universal Coef-

ficient Theorem now tells us that H i 1Ô

D2n Õ is isomorphic to H i Ô

D2n Õ , which inparticular implies the vanishing of H 4 Ô

D2n Õ .

Summarizing, p–coloured circumcision has allowed us to simplify the fundamen-tal group of W until it becomes D2 p, whose fourth homology vanishes. This provesour p–coloured version by Garoufalidis and Kricker’s relative version of the Fenn-Rourke Theorem.

Remark 6.2. The above proof is deeply unsatisfying as it artificially works on thebasis of a difficult proof which has since been substantially streamlined and im-proved. As pointed out by Andrew Kricker, a good proof of our theorem (and themain theorem of [15]) would use the simpler “mapping class group” methods of [23]and [25].

Remark 6.3. Many thanks to Kazuo Habiro for pointing out that Fenn and Rourke’s

condition that η map the ‘base-line’ of W to the base-point of K Ô

π, 1Õ

is necessaryfor η to be well-defined. Consider for instance a closed 4-manifold with fundamentalgroup Z4, in which case K

Ô

π, 1 Õ is the 4-torus. Without the condition on the base-points, η

Ô

W Õ would have Z indeterminacy because it could ‘wrap’ W around the

4-torus any number of times.

6.2. Proof of Theorem 2.

6.2.1. Invariance under orientation change and Kirby moves. The arguments of [36] for proving invariance under change of orientation of a single link componentand invariance under Kirby I and Kirby II carry over to our setting, where theslight differences in conventions between this paper and [36] are handled by thecomments in [16, Proof of Theorem 4].

We prove invariance under Kirby III p. Let L be an untying link for a p–colouredknot Ô

K, ρÕ , and let L ½ denote the union of L with a 1–framed component x

È ker ρ

ringed by a 0–framed unknot y. By the magic formula for the Kontsevich invariantfor the long Hopf link [12, Theorem 4] (see also [5, 8]) we have:




(6.1) Z D2p

£

0C

«

C R ν

Since ın Ô

L ½

Õ vanishes on Jacobi diagrams which do not have 2n legs on eachcomponent, in particular it vanishes for Jacobi diagrams which do not have 2n legson y.

For D a homogenous summand of the RHS of Equation 6.1, we have by [31,Lemma 10.23]:

ın Ô

DÕ

¤

¤

¤

¤

¤

¤

P n ν ιn Ô

D½ Õ

where D ½ is D minus y and all legs labeled by points of y, and n1 the number of dotted arcs connecting ın Ô

D½ Õ to P n and n2 the number of dotted connecting P n to

ν satisfy n1

n2

n.If n1 0 then ın Ô

DÕ 0 by [36, Lemma 3.3] since each connected diagram in ν

has a trivalent vertex. Thus the only contributions to Z D2pÔ

L ½

Õ with legs labeledx or y come from exp

Ô x y Õ . These are the same contributions as we would havefrom L ¾ the disjoint union of L and a 0–framed Hopf link, and since L and L ¾ arerelated by Kirby I and Kirby II, we have:

Z D2pÔ

L ½

Õ

Z D2pÔ

L ¾

Õ

Z D2pÔ

LÕ

6.2.2. Invariance under the κ –Move. Invariance under the κ –move results in a non-trivial way from the ZM relation. Since any coupon marked Ô 1 ¡

tÕ Ô 1

s ¤ ¤ ¤

s p ¡

1Õ

will kill the Jacobi diagram it is on when we LMO integrate, we take the liberty of redefining our coupons to be over

R : ZÖ

D2 p × ß Ô 1 ¡

tÕ Ô 1

s ¤ ¤ ¤

s p¡

1Õ

To further simplify notation, when there is a Ô 1 ¡ tÕ label in the diagram we write1

s ¤ ¤ ¤

s p¡ 1Õ 0 as such a diagram will also vanish when LMO integrated. Sim-

ilarly when a Jacobi diagram D is in the kernel of LMO integration or is equivalentto a Jacobi diagram with fewer chords, we write D

0.By Section 4.1 without the limitation of generality concerning the possible colour-

ings of slices, we may write the κ –move as



30 DANIEL MOSKOVICH

t t t

ts ts ts

L

¤ ¤ ¤

¤ ¤ ¤

ts

t t

ts

t t

ts

ts ts ts

L

¤ ¤ ¤

¤ ¤ ¤

¤ ¤ ¤

where there are no associators around the coupons. To see how the p–coloured wind-ing diagram valued Kontsevich invariant changes when the κ –move is performed,we begin by recalling that the associator Φ satisfies the following two relations (see

for instance [31]):The pentagon relation

∆3Φ

∆1Φ

Φ

∆2Φ

Φ

The hexagon relation

∆1R

Φ

R

Φ¡ 1

R

Φ

We now make the following calculation:




(6.2)

Z D2p

£

)(

g«

Z D2p

¤

¦

¦

¦

¦

¦

¦

¥

)

)

)

)

)

)(

(

(

(

(

(

g¬

Æ

Æ

Æ

Æ

Æ

Æ

-

gν1ß 2

S 2Φ

S 1R

S 1Φ¡

1

R

S 1Φ

ν 1

ß

2 g

S 1∆1R

ν 1

ß

2

g

S 1∆1gR

ν 1

ß

2

g

S 1∆1gR

with

gR : exp

¤

¥

1

2

x

g

y¬

-

and the third equality follows from the hexagon relation. Taking the degree 1 termof the final diagram gives:

g

g¡ 1

ν 1

ß

2

Based on this we have the following lemma (where we are taking degree 1 termsfor ease of notation, but replacing these with exponents does not change the resultor the proof):

Lemma 6.4.t t

Ô

t¡ 1 Õ

g

Ô

t¡ 1 Õ

g

0

for any gÈ

R such that ghÈ Z

for some h È R.



32 DANIEL MOSKOVICH

Proof.

tt

t

g g ¡ 1t

tg g ¡

1t

t-1

t-1

We also know that

ts

t

ts

t

ts¡ 1

1 ¡

ts

Remembering that we have s

¡

1

–valued coupons on both ends of the left skeletoncomponent, this last term equals:

A :

t t

Ô 1 ¡

tÕ

s

Ô t ¡ 1 Õ s ¡

1

Define

B :

t t

Ô

t¡ 1 Õ

s ¡

1

Ô 1 ¡

tÕ

s

Because for p 3 we have:

(6.3)

Ô

s

s2Õ Ô

s

s2Õ 2

s

s2 1

Ô

s¡

s2Õ Ô

s¡

s2Õ ¡ 2

s

s2 ¡ 3.

while for p 5:




(6.4)

Ô

s

s4Õ Ô

s2

s3Õ

s

s2

s3

s4 ¡ 1

Ô

s¡

s4Õ Ô 2s

¡

s2

s3¡ 2s4

Õ ¡ 4

s

s2

s3

s4 ¡ 5.

the conditions of Lemma 6.4 are fulfilled for s, for s¡ 1, for s

s ¡ 1, and for s¡

s ¡ 1,and therefore both A B 0 and A ¡ B 0. Thus A 0, finishing the proof.

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Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502

JAPAN

E-mail address: [email protected]

URL: http://www.sumamathematica.com/

Daniel Moskovich- A Kontsevich invariant for coloured knots

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