The AMR -Products and the Turaev-Vassiliev Quantization

72 Chapter 6Quantization of the Loop Algebras

Theorem 6.5. For two partitions P1and P2ofΣ, the composite equivalence TP2T_P₁¹:Z_s,f^p^Σ^qrrh^ss_P₁ ^ÑZ_s,f^p^Σ^qrrh^ss_P₂ is equal to the canonical equivalence fromP₁toP2, cf. Theorem 4.12.

We prove part of Theorem 6.4 in

Lemma 6.6. The map TP: L_h^p^Σ^q ^Ñ Z_s,_f^p^Σ^qrrh^ss_P is aC^rrh^ss-algebra homomorphism and in-duces a morphism of deformation quantizations

As,f^pΣq

T_P

p_s,_f

Z_s,f^p^Σ^qrrh^ss_P

π₀

Zs,f^pΣq

(6.7)

for any partition P ofΣ.

Proof. By construction, TPisC^rrh^ss-linear. To verify that the stack multiplication is taken to_P, letL1,L2be links onΣ. Recalling Theorem 4.12, the definition of the producton C^p^Σ^q(3.10), and the identity (3.11), we obtain

TP^pL1^qPTP^pL2^qRs,fηVP^pL1^qPRs,fηVP^pL2^q

R_s,f^pηVP^pL1^qPηVP^pL2^qq

Rs,fη^pVP^pL1^qVP^pL2^qq

Rs,fηVP^pL1L2^q

TP^pL1L2^q.

The next step is to prove thatTPdescends to the skein algebra, i.e., thatI_s,f^p^Σ^q KerTP. By the first part of the proof, KerTP is an ideal so it suffices to consider a generator of I_s,_f^p^Σ^q. AsTP can be computed locally, we simply consider the skein relation (6.1); we have (cf. (5.25))

exp f 2h

exp f

2hRs,fηV

exp f 2hR_s,f

exp

2 h

exp f

2h exp

1 2

s f

exp

2 h exp

2 h s

2 h

exp

2 h

and analogously TP

exp

f 2h

exp

2 h

6.2 The AMR-Products and the Turaev-Vassiliev Quantization 73

so that TP

exp f

2h exp f

exp

2 h exp

2 h

2 sinh

2 h

2 sinh s 2h

as desired. The induced map TP: A_s,f^p^Σ^q ^Ñ Z_s,_f^p^Σ^qrrh^ss is C^rrh^ss-linear and thereby h-filtered whence it can be completed to a homomorphism of (filtered) C^rrh^ss-algebras TP: A_s,f^p^Σ^q ^Ñ Z_s,_f^p^Σ^qrrh^ss. It is immediate from the definitions that the triangle (6.7)

commutes. ^l

The proof of Theorem 6.4 is complete once we establish thatTP:A_s,f^p^Σ^q ^Ñ Z_s,_f^p^Σ^qrrh^ss is an isomorphism ofC^rrh^ss-algebras. The strategy is to show that theC^rrh^ss-linear homo-morphism (recall (6.4))

FPIdV_P¹:C_h^p^Σ^q^ÑL_h^p^Σ^q^ÑA_s,f^p^Σ^q

descends viaq_s,f: C_h^p^Σ^q ^Ñ Z_s,f^p^Σ^qrrh^ssto yield the inverse of TP. To this end it will be useful to introduce thelargefiltration (larger than the chord filtration) onC_h^p^Σ^q; it is given by

C_h^p^Σ^q_n η¹ hⁿC^p^Σ^qrrh^ssC_h^p^Σ^q, n^PN.

Remark 6.7. By definition,ηand henceqs,f Rs,fηare filtered with respect to the large filtration.

Intuitively, the large filtration measures a ‘degree’ defined in terms of both chords and powers ofh; to make this idea precise we introduceC-linear maps pn: C_h^p^Σ^q^Ñ C_h^p^Σ^q C_h^p^Σ^q,n^PNdetermined by the formula (Dis a chord diagram withmchords)

λ_ihⁱD

0, m^¡n

λnmhⁿ^mD, m^¤n Clearly, these maps are independent projections in the sense that

pnp_n¹δ_n,n¹pn, n,n¹^PN. (6.8) Elements in the subset Impn C_h^p^Σ^qare said to have chord-h degree n. Of course, pn

extends topn:C_h^p^Σ^q^ÑC_h^p^Σ^qC_h^p^Σ^q, and it is a consequence of the definitions that

n£1 i0

Kerp_iC_h^p^Σ^q_n, n^PN. (6.9)

Remark 6.8. Evidently, the concepts of the large filtration and the chord-h degree make sense locally, that is, they can be defined for chord tangles in an embedded squareS Σ. Composition of chord tangles is chord-hgraded.

74 Chapter 6Quantization of the Loop Algebras

Lemma 6.9. The map qs,f:Ch^pΣqÑZs,f^pΣqrrh^ssis surjective, and the induced map qs,f:Ch^pΣq

Kerq_s,_f ^ÑZ_s,_f^p^Σ^qrrh^ss

is an isomorphism of filteredC^rrh^ss-modules when the domain is equipped with the filtration induced from the large filtration.

Proof. Let ˜z ^°_iz˜ihⁱ ^P Zs,f^pΣqrrh^ss. Write ˜zi λi^H zi where zi is a complex linear combination of non-empty diagrams onΣ, and putz^°_izihⁱ. As

qs,f

λihⁱ^H

pλi^Hqhⁱz˜z, it suffices to showz^PImq_s,f. By repeated application of the relation

1 s²f²

s f

mod L_s,_f, we may findD_i^PC^pⁱ^q^p^Σ^qsuch thatR_s,_f^pD_i^qz_i, i.e.,

R_s,f

Dihⁱ

zihⁱz.

ButD^pDi^q^PC^p^Σ^qCh^pΣqsatisfiesη^pD^q^°_iDihⁱ so thatqs,f^pD^qzas desired. The induced isomorphism ofC^rrh^ss-modules

q_s,_f:Ch^pΣq L

Kerq_s,_f ^ÑZ_s,_f^p^Σ^qrrh^ss

is a filtered map. The argument for surjectivity reveals that if z^PhⁿZ_s,_f^p^Σ^qrrh^ss, then the inverse image q_s,¹_f^pz^q contains an element inC_h^p^Σ^q_n so that the inverse map q_s,f¹ is also

filtered. ^l

Lemma 6.10. Kerq_s,f ^tc^PC_h^p^Σ^q^|c^PL^h_s,_f C_h^p^Σ^q_n,n^PN^u.

Proof. Letc^PCh^pΣq. For the inclusionsuppose thatc^PKerqs,f. Lettingπi: Zs,f^pΣqrrh^ss

ÑZs,f^pΣqdenote the projection on theith coefficient, we derive 0πiq_s,f^pc^qπiR_s,_fη^pc^qR_s,_fπiη^pc^q, i^PN

implying thatπiη^pc^q^PLs,f; this means thatpi^pc^q^PL^h_s,f. Givenn^PN, (6.8) and (6.9) yield c

pi^pc^q^PC_h^p^Σ^q_n

so thatc ^PL^h_s,f C_h^p^Σ^q_n. For the other inclusion we assume thatcis in the right hand set.

Since

qs,f L^h_s,f Ch^pΣqn

hⁿC^p^Σ^qrrh^ss, n^PN

by Remarks 6.3 and 6.7, it follows thatq_s,f^pc^q0 as desired. ^l

6.2 The AMR-Products and the Turaev-Vassiliev Quantization 75

Lemma 6.11. The map FP:Ch^pΣqÑAs,f^pΣqis filtered with respect to the large filtration.

Proof. Suppose thatc^pci^q^PCh^pΣqbelongs toCh^pΣqn. SinceFPis filtered with respect to the chord filtration onCh^pΣq, it is enough to show that

FP n1

PA_s,f^p^Σ^q_n. (6.10)

By the characterization of the large filtration (6.9), eachcimay be written as aC^rrh^ss-linear combination of elements of the formhⁿⁱD where Dis a chord diagram with i chords.

ChooseLj^PL^V_h^p^Σ^q_isuch that^prLj^sj^qV_P¹^pD^q^PL_h^p^Σ^q_i. Since FP^phⁿⁱD^qhⁿⁱ^prLj^sj^q^prhⁿⁱLj^sj^q^PA_s,f^p^Σ^q with (cf. (6.3))

hⁿⁱLj ^PhⁿⁱL^V_h^p^Σ^q_ihⁿⁱ^pI_s,f^p^Σ^q hⁱL_h^p^Σ^qqI_s,f^p^Σ^q hⁿL_h^p^Σ^qL_h^p^Σ^qn, we infer thatFP^phⁿⁱD^q^PA_s,f^p^Σ^q_n. This implies (6.10) and thereby the lemma. ^l Lemma 6.12. L^h_s,_f KerFP.

Proof. Any generator ofL^h_s,f is the composition of the element g sh f h ^PCh^pS;^ÒÒ,^ÒÒq

located in some squareS Σ, with a suitable chord tangle inΣS. By the compatibility of the universal Vassiliev invariant with this decomposition, we need only consider the squareSand show thatFP^pg^q0. Define

exp f

2h exp f

2h 2 sinh s 2h

PC_h^pS;^ÒÒ,^ÒÒq. By definition ofIs,f^pΣq,

FP^pX^q0. (6.11)

Refining the computation in the proof of Lemma 6.6 a little, one establishes X2 sinh

1 2g s

2 sinh s 2h Sincegis homogeneous of chord-hdegree 1, it follows that

pn^pX^q

0, n0

g, n1

gxn,xn ^PCh^pS;^ÒÒ,^ÒÒqhas chord-hdegreen1; n^¥2

(6.12)

Applying induction we construct a sequenceYi^PCh^pS;^ÒÒ,^ÒÒq,i1, 2, . . . of the form YiXyi, yi^PC_h^pS;^ÒÒ,^ÒÒqhas chord-hdegreei1

76 Chapter 6Quantization of the Loop Algebras

and satisfying

Yig^PCh^pS;^ÒÒ,^ÒÒq_n ₁. (6.13)

To initiate the process we set y1 ^ÒÒ so thatY1 X; by (6.12) and (6.9) this is sound.

Assume thatY1, . . . ,Ynhave already been defined. Then pn 1

pn 1^pXyi^q n

pn 2i^pX^qyi n

gxn 2iyi.

Thus we are lead to putyn 1

°n

i1xn 2iyi, which by induction has chord-hdegree n, and thereby obtain

pn 1 n 1

Y_ipn 1 n

Y_i p1^pX^qyn 10.

Taken together with the hypothesis (6.13), this implies pj

n 1

Yig0, j0, . . . ,n 1

completing the induction step by (6.9). Now, (6.13) means by definition that

°n

i1Yi ^Ñ

g,n^Ñ⁸in the large filtration so by (6.11) and Lemma 6.11 0

FP^pX^qyiFP n

ÑFP^pg^q, n^Ñ⁸.

SinceA_s,f^pS;^ÒÒ,^ÒÒqis Hausdorff, this impliesFP^pg^q0 as desired. ^l Lemma 6.13. The map TP: A_s,_f^p^Σ^q^ÑZ_s,f^p^Σ^qrrh^ssis aC^rrh^ss-algebra isomorphism.

Proof. From Lemmas 6.10, 6.11 and 6.12 follow that Kerqs,f KerFP. Consequently, Lemma 6.9 yields an induced homomorphism

FP:Zs,f^pΣqrrh^ss^Ý^ÑC_h^p^Σ^q^L_Ker_q_s,f ÝÑAs,f^pΣq

of filteredC^rrh^ss-modules. We verify thatFPis the inverse ofTP. From the diagram L_h^p^Σ^q ^V^P

Ch^pΣq

q_s,f

Ch^pΣq

V_P¹

Lh^pΣq

A_s,_f^p^Σ^q ^T^P Z_s,_f^p^Σ^qrrh^ss C_h^p^Σ^q^L_Ker_q_s,f ^F^P A_s,_f^p^Σ^q

follows thatFPTPιι: L_h^p^Σ^q^Ñ A_s,f^p^Σ^q. SinceFPTPis a filtered endomorphism of A_s,_f^p^Σ^q, this means thatFPTPId. ThatTPFPis the identity onZ_s,f^p^Σ^qrrh^ssneed only

6.2 The AMR-Products and the Turaev-Vassiliev Quantization 77

be verified onZs,f^pΣq Zs,f^pΣqrrh^ssby theC^rrh^ss-linearity, cf. (3.1). LetDbe a diagram onΣ. We may considerDas an element ofCh^pΣq, and clearlyqs,f^pD^qD. Therefore

FP^pD^qId^pV_P¹^pD^qq (6.14) so that

TP^pFP^pD^qqTPIdV_P¹^pD^qqs,fVPV_P¹^pD^qD

as desired. ^l

Proof (Theorem 6.4). Lemmas 6.6 and 6.13. ^l

Proof (Theorem 6.5). For a diagramDonΣthe canonical equivalence fromP₁ toP₂ on Zs,f^pΣqis given by

D^ÞÑRs,fτ^pD^qRs,f

pVP₂V_P₁¹^pD^qq^p^r^qh^r Rs,fηVP₂V_P₁¹^pD^q, cf. Theorems 4.6 and 4.12. But (cf. (6.14))

TP₂T_P¹

pD^qTP₂IdV_P¹

pD^qq_s,fVP₂V_P¹

pD^qR_s,fηVP₂V_P¹

pD^q.

This completes the proof since by (3.1) it suffices to consider the restrictions of the two

maps toZs,f^pΣq. ^l

Chapter 7

The Case SL ₂

C

Revisited

We present in this chapter a canonicalΓ pΣq-invariant-product onO^pM^p^Σ; SL2^pC^qqqdue to Bullock, Frohman and Kania-Bartoszy ´nska [BFK]. This-product is defined on a model forO^pM^p^Σ; SL₂^pC^qqqespecially suited for the purpose; we develop the model in the first section. Subsequently we prove, in the case^BΣH, that the BFK-product is canonically equivalent to each of the AMR-products onO^pM^p^Σ; SL₂^pC^qqq. We end the dissertation with an investigation of the differentiability of the BFK-product.

7.1 A Good Model for the Moduli Space

Using the abbreviationsZ^p^Σ^qZ_1,

1 2

pΣqandI2^pΣqI^p^Σ; SL2^pC^qq, Theorem 5.34 implies that we have an algebra homomorphism

Ψ:Z^p^Σ^q^LI₂^p^Σ^q^ÑO^pM^p^Σ; SL₂^pC^qqq.

Remark 7.1. We also learn from Theorem 5.34 thatΨ is a surjection with KerΨ

Shortly we shall see that

0 0 so that, by the same theorem, Ψ is actually a Poisson isomorphism (we know this already in the case^BΣHby Corollary 5.35).

Ignoring the orientation of the loops in diagrams on Σinduces an equivalence relation;

the equivalence classes are called unoriented diagrams onΣ and we denote byZ^p^Σ^q the free complex vector space generated by them. Of course,Z^p^Σ^qis a commutative algebra under union of unoriented diagrams, and the orientation-forgetting map

u:Z^p^Σ^q^ÑZ^p^Σ^q, u^pD^qD

is an algebra homomorphism. DefineK0^pΣqZ^p^Σ^qto be the subspace generated by the local relations

(7.1a)

2 (7.1b)

As usual,K0^pΣqis an ideal. Consider the linear mapu^r:Z^p^Σ^q^ÑZ^p^Σ^qgiven by

ur^pD^q^p1^qⁿD ^p1^qⁿu^pD^q, Da diagram with n loops. (7.2) It is a homomorphism of complex algebras.

7.1 A Good Model for the Moduli Space 79

Remark 7.2. Unlike many other maps defined on (chord) diagrams,u^rcannot be computed locally because of the sign. In expressions below whereu^r is seemingly evaluated on a tangle, it is understood that this tangle is part of a particular diagram onΣ.

Proposition 7.3. The mapu^r:Z^p^Σ^q^ÑZ^p^Σ^qinduces an isomorphism ur:Z^p^Σ^q^LI2^pΣq

ÑZ^p^Σ^q^LK0^pΣq (7.3)

of complex algebras.

Proof. The first step of the proof is to show thatI2^pΣqmaps to 0 under the composition Z^p^Σ^q^Ý^u^Ñ^r Z^p^Σ^q^Ý^ÑZ^p^Σ^q^LK0^pΣq.

It is sufficient to consider the generators (cf. (5.22)) of the idealI2^pΣq. We have

taking care of (5.22a). For the other two relations we need the following intermediate result

σ^PS₂

ǫ^pσ^q σ

ǫ (7.4)

whereǫis equal to1 raised to the number of loops in the diagram corresponding to^ÒÒ. This formula implies (5.22b) since

σ,τ^PS₂

ǫ^pσ^qǫ^pτ^q σ τ

ǫL

and

σ,τ^PS₂

ǫ^pσ^qǫ^pτ^q σ τ

ǫR ǫR

with appropriate signsǫ_L andǫ_R easily seen to be equal. Applying (7.4) to the left hand side of (5.22c) yields

σ^PS₂

ǫ^pσ^q σ γ γ

ǫγ γ⁻¹

Here the decoration of the middle diagram means that traversing the upper strand from right to left amounts to first traversingγ¹and thenγ. Comparing this formula with (7.4) verifies relation (5.22c). Consequentlyu^rdescends to an algebra homomorphism

ur:Z^p^Σ^q^LI2^pΣq

ÑZ^p^Σ^q^LK0^pΣq

In order to invertu, we record a simple observation.^r

80 Chapter 7 The CaseSL2^pC^qRevisited Claim. For anyγ^Pπ1^pΣqwe have

σ^PS2

ǫ^pσ^q σ 1 γ γ

σ^PS2

ǫ^pσ^q σ γ 1 mod I2^pΣq.

The two identities are analogous; we prove the former one:

σ^PS₂

ǫ^pσ^q σ 1 γ 1 γ 1 γ γ

Now letv:Z^p^Σ^q^ÑZ^p^Σ^q{I2^pΣqbe theC-algebra homomorphism determined by

v^pγ^q~^γ ^(7.5)

where γ denotes some unoriented loop on Σand~^γ is one of the two possible oriented versions of it. Thatvis well-defined follows from the claim and an application of relation (5.22c):

γ⁻¹

σ^PS₂

ǫ^pσ^q σ 1 γ⁻¹

σ^PS₂

ǫ^pσ^q σ γ γ

1 γ⁻¹

σ^PS₂

ǫ^pσ^q σ γ 1 γ

The next step is to verify thatK0^pΣqKerv. Relation (7.1b) is trivial:

20.

Given an instance of relation (7.1a) we assume without loss of generality that the two ver-tical strands belong to separate loops. Lettingαandβdenote oriented versions of them we

7.1 A Good Model for the Moduli Space 81

deduce

α β α β

σ^PS₂

ǫ^pσ^q σ α β

σ^PS₂

ǫ^pσ^q σ α⁻¹

α⁻¹

α β

σ^PS₂

ǫ^pσ^q σ 1 α⁻¹β

α⁻¹β α⁻¹ β

as desired. It is obvious from (7.2) and (7.5) that the induced map v:Z^p^Σ^q^LK0^pΣq

ÑZ^p^Σ^q^LI2^pΣq

is the inverse ofu. ^l

The usefulness of this proposition stems from the fact that the relations (7.1) are similar to the skein relations defining the Kauffman bracket; recall that this is a polynomial invariant

xL^y ^P Z^rA¹^sof the framed, unoriented linkL R³, satisfying (and determined by) the conditions

A A¹ (7.6a)

A²A² (7.6b)

Substituting A 1, over- and undercrossings cannot be distinguished, and the skein relations reduce to (7.1).

Remark 7.4. In light of what we have just said, it is easy to see that Z^p^Σ^q{K₀^p^Σ^qis iso-morphic to the complex algebraS2,⁸^pΣI;C,1^qstudied in [PS]. The latter algebra has no zero-divisors, in particular no nilpotent elements, by Theorem 4.7 of that paper. It thus follows from Proposition 7.3 (cf. Remark 7.1) thatΨ:Z^p^Σ^{q {}I2^pΣqÑO^pM^p^Σ; SL2^pC^qqqis a Poisson isomorphism. We transfer the Poisson structure onZ^p^Σ^{q {}I2^pΣqtoZ^p^Σ^q{K0^pΣq

by requiring that the algebra isomorphismu^ris a Poisson isomorphism.

Pursuing the similarity of (7.1) and (7.6), we define aBFK-diagram onΣto be the isotopy class of a finite collection of unoriented circles embedded intoΣsuch that no loop bounds a disk inΣ; informally speaking, a BFK-diagram is an unoriented diagram onΣwith no crossings and no homotopically trivial components. LetB^p^Σ^qbe the complex vector space freely generated by all BFK-diagrams onΣ. Define a linear map κ: Z^p^Σ^q ^Ñ B^p^Σ^q by

82 Chapter 7 The CaseSL2^pC^qRevisited the Kauffman bracket procedure, i.e., given a generic unoriented diagram Dreplace all crossings by the right hand side of (7.1a) and remove all arising trivial loops at the cost of a factor2 to obtain a linear combinationκ^pD^q^PB^p^Σ^q.

Proposition 7.5. The mapκ:Z^p^Σ^q^ÑB^p^Σ^qis well-defined and descends to an isomorphism κ: Z^p^Σ^q^LK0^pΣq

ÑB^p^Σ^q (7.7)

of complex vector spaces.

Proof. Two generic unoriented diagrams onΣare homotopic if and only if they are related by isotopy and the three Reidemeister moves. Of course,κ^pD^qis invariant under isotopies ofD. Regarding the first Reidemeister move, we follow the steps in the computation ofκ to obtain:

as required. For the second and third Reidemeister moves one can simply substituteA

1 (and ignore over-/undercrossing information) in Kauffman’s proof of the invariance of his bracket under these moves, cf. [K]. Henceκ is well-defined, and by construction it descends to a quotient map as in (7.7).

Any BFK-diagram onΣcan be considered as an unoriented diagram onΣ; this defines a linear mapι:B^p^Σ^q^ÑZ^p^Σ^q. Evidently, the composition

B^p^Σ^q^Ý^Ñ^ι Z^p^Σ^q^Ý^ÑZ^p^Σ^q^LK0^pΣq

is the inverse ofκ. ^l

We equipB^p^Σ^qwith the Poisson algebra structure induced byκand have thus constructed the diagram

C^p^Σ^q

R_1,

Z^p^Σ^q Z^p^Σ^q^LI₂^p^Σ^q ^u^r

Z^p^Σ^q^LK₀^p^Σ^q ^κ B^p^Σ^q

O^pM^p^Σ; SL₂^pC^qqq

(7.8)

of Poisson homomorphisms. Hereν: B^p^Σ^q ^Ñ O^pM^p^Σ; SL₂^pC^qqqis the unique map (iso-morphism) making the diagram commutative. From (7.5) follows that

ν^pγ^qΨp~^γ^q

whereγ is a non-trivial loop onΣ. Thereforeν is equivariant with respect to the natural action ofΓ pΣqonB^p^Σ^q, cf. Theorem 2.22.

It will facilitate computations later on to adapt diagram (7.8) slightly. Consider the map Ψr:C^p^Σ^q^ÑO^pM^p^Σ; SL₂^pC^qqqgiven by

ΨrpD^q^p1^qⁿΨpD^q, Da chord diagram withncore components

7.1 A Good Model for the Moduli Space 83

SinceΨ:C^p^Σ^q^ÑO^pM^p^Σ; SL₂^pC^qqqis a Poisson homomorphism and the Poisson structure onC^p^Σ^qpreserves the skeletons of chord diagrams,Ψ^r is also a Poisson homomorphism.

We know thatΨmaps the loop relation

1 2

to 0. Since smoothing a chord in-/decreases the number of core components by 1, it follows thatΨ^r satisfies a different loop relation:

1 2 Thus we obtain a triangle of Poisson homomorphisms

C^p^Σ^q

1,1 2

Ψr

1,₂¹^pΣq

Ψr

O^pM^p^Σ; SL2^pC^qqq For a diagramDwithnloops we have by (7.2)

νκu^pD^q^p1^qⁿνκu^r^pD^q^p1^qⁿΨpD^qΨ^rpD^q so that (7.8) transforms into the commutative diagram

C^p^Σ^q

1,1 2

Ψr

1,¹₂^pΣq

Ψr

Z^p^Σ^q^LK0^pΣq

κ B^p^Σ^q

O^pM^p^Σ; SL2^pC^qqq

(7.9)

As a composition of Poisson homomorphismsuκ¹ν¹Ψ^r is a Poisson homomorphism.

We are now set to derive formulas for the product and the Poisson bracket onB^p^Σ^q. Let DandEbe BFK-diagrams in general position. RegardingDandEas unoriented diagrams onΣ, their product is simply the union D^YE. Hence relation (7.1a) leads us to define a state for^pD,E^qto be any mapS: D#E^Ñ^t0,^8uand its corresponding diagramD^pS^qto be the one obtained fromD^YEby resolving all crossingsD#Eas follows

D E

ifS^pp^q0

ifS^pp^q⁸

(7.10)

Notice thatD^pS^qis not necessarily a BFK-diagram since it may contain trivial loops. Obvi-ously we have

DE^p1^q^|D#E|

D^pS^q^PZ^p^Σ^q^LK₀^p^Σ^q^. ^(7.11)

84 Chapter 7 The CaseSL2^pC^qRevisited To calculate^tD,E^u, liftDandEto diagrams_D~ and~_EonΣso thatu^p~_D^q_Dandu^p~_E^q_E.

Since

t~_D,~_E^u ^¸

p^P~D#~E

ǫ^pp;~_D,~_E^q_D~ ^Y_p~_E^PC^p^Σ^q,

diagram (7.9) implies

tD,E^u^tuR

1,¹₂^p~_D^q_,_uR

1,¹₂^p~_E^qu

u R

1,¹₂^pt~_D,~_E^uq

p^PD#E

ǫ^pp;~_D,~_E^q_{u R}

1,¹₂^p~_D^Y_p~_E^q^PZ^p^Σ^q^LK0^pΣq. Focusing on the chord inD~ ^Yp~E, we get

1,¹₂

1 2

modK0^pΣq

(7.12)

so that

ǫ^pp;~_D,~_E^q_{u R}

1,₂¹^p~_D^Y_p~_E^q¹

2^rD^Yp,⁸ED^Yp,0E^s

whereD^Yp,sE,s0,⁸is obtained fromD^YEby resolving only the crossing atp accord-ing to the rule (7.10). Hence we have

tD,E^u1 2

p^PD#E

rD^Yp,⁸ED^Yp,0E^s^PZ^p^Σ^q^LK₀^p^Σ^q^.

The diagram D^Yp,sE contains the crossings D#E^tp^uwhich may also be resolved via (7.1a); doing so leads to the various state diagrams for^pD,E^q. Putting 0^pS^q|S¹^p0^q|and

8pS^q|S¹^p8q|for a stateS, we therefore derive

tD,E^u^p1^q^|D#E|1 2

p0^pS^q^8pS^qqD^pS^q^PZ^p^Σ^q^LK0^pΣq. (7.13) This formula (up to a sign) was obtained in [BFK].

Remark 7.6. In the case^BΣHwe have the-product_P,P^PP^p^Σ^qonO^pM^p^Σ; SL₂^pC^qqq induced via the-equivalenceΨ:Z^p^Σ^q{I2^pΣqÑO^pM^p^Σ; SL₂^pC^qqq, cf. Theorem 5.27 and Corollary 5.35. We may transfer_PtoZ^p^Σ^q{K0^pΣqandB^p^Σ^qby requiringu,^r κand, thus, νto be-equivalences. The adapted mapΨ^r: C^p^Σ^q^Ñ O^pM^p^Σ; SL2^pC^qqqis a morphism of

Psince this-product preserves skeletons of chord diagrams. From diagram (7.9) follows thatu:Z

1,¹₂^pΣqÑZ^p^Σ^q{K0^pΣqis also a morphism of_P.

In document Afhandling (Sider 77-91)

The AMR -Products and the Turaev-Vassiliev Quantization

Chapter 7

The Case SL 2

C

Revisited

7.1 A Good Model for the Moduli Space

The Case SL ₂