Francesco Costantino- Integrality of Kauffman Brackets of Trivalent Graphs

8/3/2019 Francesco Costantino- Integrality of Kauffman Brackets of Trivalent Graphs

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arXiv:0908

.0542v1

[math.QA

]4Aug2009

INTEGRALITY OF KAUFFMAN BRACKETS OF TRIVALENT GRAPHS

FRANCESCO COSTANTINO

Abstract. We show that Kauffman brackets of colored framed graphs (also known as quan-tum spin networks) can be renormalized to a Laurent polynomial with integer coefficients by

multiplying it by a coefficient which is a product of quantum factorials depending only on theabstract combinatorial structure of the graph. Then we compare the shadow-state sums andthe state-sums based on R-matrices and Clebsch-Gordan symbols, reprove their equivalence and

comment on the integrality of the weight of the states. We also provide short proofs of most ofthe standard identities satisfied by quantum 6j-symbols of Uq(sl2).

Contents

1. Introduction 11.1. Structure of the paper 21.2. Acnowledgements 32. Representations ofUq(sl2) and graph invariants 32.1. Basic facts on Uq(sl2) 32.2. Elementary operators 42.3. The state-sum computing G,col. 72.4. Invariants of framed trivalent graphs 82.5. Remarks on orientation and framings 103. Integrality 11

3.1. Examples and properties 124. Shadow state sums and integrality 134.1. Shadow state sums 144.2. Simplifying formulas 174.3. Examples and comments on integrality 174.4. Identities on 6j-symbols 185. R-matrices vs. 6j-symbols 20References 22

1. Introduction

The family of Uq(sl2)-quantum invariants of knotted objects in S3

as knots, links and more ingeneral trivalent graphs, can be defined via the recoupling theory ([6]) as well as via the theoryof representations of the quantum group Uq(sl2) ([12],[7]) and the so-called theory of shadows([13], Chapter IX). These invariants are defined for framed objects (framed links or graphs, seeDefinition 2.11) equipped with a coloring i.e. a map from the set of 1-dimensional strata of

the object satisfing certain admissibility conditions (Definition 2.6), and take values in Q(q12 ). As

customary in the literature, we shall call these invariants the Kauffman brackets of the colored,1
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2 COSTANTINO

framed graph G and denote them by G,col. In particular, if G is a framed knot k the set ofadmissible colorings is the set of half-natural numbers (here we use the so-called spin notation)and they coincide with the unreduced colored Jones polynomials of the knot:

k, s

= J

2s

+1(k).

Although the definition of Kauffman brackets via recoupling theory is simple and appealing,the definition based on the theory of representations of Uq(sl2) happens to be more useful forour purposes. In Section 2 we will sketch the proof of the equivalence of the two definitions (therelations have been already studied by S. Piunikhin [11]). Roughly speaking, the invariant ofa graph G equipped with a coloring col (which is a function of q) is the trace of a morphismbetween representations of Uq constructed by composing elementary morphisms associated toa decomposition of a diagram of G into simple tangles. The only difference with respect to thegeneral construction of quantum invariants via representations of quantum groups is that onecan get rid of the orientations of the strands since the very beginning: this can be achieved bymodifying the operators associated to maxima and minima and using the isomorphisms betweenthe representations of Uq(sl2) and their duals.

It is known that, in general, G, col is a rational function of the variable q 12 (there are variousnotations in the literature, for instance our q

1

2 is A in [6]). If L is a framed link, it was shown byT. Le ([9]) that, up to a factor of the form qn4 , L, n is a Laurent polynomial in q 12 (actuallyin [9] a much stronger result is proved which holds for general polynomial invariants issued fromquantum group representations).

On contrast it is well known that G, col is not in general a Laurent polynomial ifG is a trivalentgraph. The main result of the present paper is Theorem 3.3, where we show that multiplyingG,col by an explicit balanced product of quantum factorials which depends only on the abstractcombinatorial structure of G, on col and on the framing on G, one gets a Laurent polynomial inq12 that we will denote G,col. It turns out that G,col = G,col if G is a 0-framed link.

This normalization was proposed and conjectured to be integral by S. Garoufalidis and R. Van derVeen (in [5], where they also proved the integrality in the classical case when q = 1) in orderto define generating function for classical spin networks evaluations. We hope that this result willallow further development in that direction and in the understanding of the categorification of

Uq(sl2)-quantum invariants for general knotted objects.The last sections are almost independent on the preceding ones. In Section 4 we recall the defini-

tion of shadow-state sums to compute G,col, give a new self-contained proof of the equivalence(first proved in [8]) between the shadow-state formulation and the R-matrix formulation of theinvariants, and comment on the non-integrality of the single shadow-state weights. Using shadowstate-sums we also provide short proofs of the most known identities for 6j-symbols (e.g. Racah,Biedenharn-Elliot, orthogonality). In the last section we will quickly comment on the case whenG has non-empty boundary and on the algebraic meaning of the shadow-state sums with respectto the state-sum based on R-matrices and Clebsch-Gordan symbols.

1.1. Structure of the paper. In Section 2 we will recall the basic facts we will need of the repre-sentation theory ofUq(sl2) and provide explicit formulas for the elementary morphisms associatedto each elementary tangle: if you are not interested in explicit formulas, jump to Subsection 2.3.Then, in Section 3 we will define G,col and prove its integrality. Section 4 is almost independenton the first sections (basically it depends only on Lemma 3.10); there we explain how to com-pute G,col via shadow state-sums and provide short proofs of some well known identities for6j-symbols. In Section 5 we comment on the algebraic meaning of shadow-state-sums.


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4 COSTANTINO

2.2. Elementary operators. In this subsection we recall how to associate morphisms betweenUq(sl2) modules to the elementary graphs of Figure 2. Let G be a trivalent graph, E the set ofedges of G, V the set of vertices:

Definition 2.6 (Admissible coloring). An admissible coloring of a KTG G is a map col : E N2(whose values are called colors) such that v V the following conditions are satisfied:

(1) av + bv + cv N(2) av + bv cv, bv + cv av , cv + av bv

where av, bv, cv are the colors of the edges touching v.

Let now D be a diagram of G and let us fix from now on a height function on R2 which allowsus to chop D into elementary graphs as those shown in Figure 2. If G is equipped with a coloringeach subgraph inherits one.

The general construction of quantum invariants via representations of Uq(sl2) ([12]) requires anorientation on every 1-dimensional stratum of G, decomposes D it into elementary subgraphs and,according to the color of each strand and its orientation, considers sutable morphisms between

tensor products of the representations associated to the oriented edges. In our case we want to dothe same but getting rid of the orientations. The rough idea to achieve this, is to orient all theedges towards the bottom. This is possible for all the elementary subgraphs except for the twoextrema: in this case we will modify the standard construction by inserting a coupon representingthe isomorphism between a representation and its dual in order to allow sources on maxima andsinks on minima. All the other operators we will define coincide with the standard ones. Herebelow we define the operators associated to each elementary subgraph:

2.2.1. Half-twists. Let us start by the half twist operator Ha : Va Va which is Ha (1)2aq(a2+a)id. We will later associate it to a vertical a-colored strand whose framing per-

forms a positive half twist with respect to the blackboard framing.

2.2.2. Cup and Cap. Let us now define the a operator: a : C = Va Va Va. Let usfirst define Ba : V

0

Vj

(Va

) by Ba(1) u=a

u=a ea

u Ea

u, and Ta : (Va

) Va

V0

byTa(Eau eav) = u,v1. We define a as a (id D) Ba (where D is the morphism of Proposition2.5). Similarly, we define a : VaVa V0 = C as a Ta(D1 id). An explicit computationin the bases eau of the above operators gives:

(1) a (eau eav) = u,v12uqu

[2a]!e00

(2) a (1) =a

u=a[2a]!

12uqueau eau

2.2.3. R-matrix. We associate to each positive crossing the action of Drinfelds universal R-matrix.

Lemma 2.7. The morphism ab R : Va

Vb

Vb

Va given by the composition of Drinfelds

universal R-matrix with the flip of the coordinates in the basis eau ebv is:(3) R(eau ebv) =

n0

[n]!(q q1)n

a un

a + u + n

n

q2uvn(uv)

n(n+1)2 ebvn eau+n

where the sum is taken over all the n such that |u + n| a and |v n| b. We will denote ab Rh,ku,vthe coefficient of R(eau ebv) with respect to ebh eak.


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INTEGRALITY OF KAUFFMAN BRACKETS OF TRI VALENT GRAPHS 5

Proof of 2.7. We use the formulas provided in [7] (Corollary 2.32: recall that t = q12 ) in the basis

fjm and the diagonal change of basis fjm =

[2j]![jm]! e

jm to compute the coefficients of R in our base

and rewrite them using quantum binomials:

R(eau ebv) =n0

(q q1)n[n]!

[a + u + n]![b v + n]![a + u]![b v]!

[a u]![2a]!

[b v]![2b]!

[2a]![2b]![a u n]![b v + n]! q

2uvn(uv) n(n+1)2 ebvn eau+n =

=n0

[n]!(q q1)n

a un

a + u + n

n

q2uvn(uv)

n(n+1)2 ebvn eau+n

2.7

The morphism associated to a negative crossing whose upper strand is colored by a and whoselower strand is colored by b is the inverse of ab R and can be computed in terms of the one we just

gave and two extrema: (R) (Ida Idb a) (Ida ab R Ida) (a Ida Ida). An explicitformula is then computed out of 3, 2 and 1:

(4) ab R(ebv eau) =

n0

[n]!(q1 q)nq2vu+n(uv)+ n(n+1)2

a un

a + u + n

n

eau+n ebvn

where the sum is taken over all the n such that |u + n| a and |v n| b. Remark thatab R =

ab R

1 = ab R|qq1 because ab R ab R = Ida Idb. We will denote ab (R1)h,ku,v the coefficientof R1(eau ebv) with respect to ebh eak.2.2.4. Clebsch-Gordan theory. To define the operators associated to the Y-shaped elementarygraph (the 7th from the left in Figure 2), we need to fix once and for all morphisms between tensorproducts ofVj s. Recall that, by Clebsch-Gordan decomposition theorem, VaVb is isomorphic toVa+b

Va+b1

. . .

V|ab|. Hence by Schurs lemma, to fix an element Ya,bc

Hom(Vc, Va

Vb)

it it is sufficient to fix Ya,bc (ecc), and this can be done in a unique way (up to a scalar factor). Oncethis is done, the image of ecu will be computable by Y

a,bc (e

cu) = (F

cu)(Ya,bc (ecc)). So suppose for

a moment we fixed Ya,bc (ecc) for each pair (c, (a, b)):

Definition 2.8. The quantum Clebsch-Gordan coefficient Ca,b,cu,v,t is the coefficient in the sum:

Ya,bc (ect ) =

u+v=t

Ca,b,cu,v,t eau ebv

It is clear that Ca,b,cu,v,t is zero if u + v = t because (K)(eau ebv) = qu+veau ebv and the weightof a vector is preserved by a morphism. The co-product formula for F gives

(F)(eau ebv) = qveau1 ebv + queau ebv1and so the equality Ya,bc (F e

ct ) = t C

a,b,cu,v,t(F)(e

au

ebv) implies the following recursion relation:

(5) Ca,b,cu,v,t = Ca,b,cu+1,v,t+1q

v + Ca,b,cu,v+1,t+1qu

which holds for every c > t c (with Ca,b,cu,v,t = 0 if |u| > a or |v| > b). Moreover, the followingrecursion is a consequence of the equality 0 = Ya,bc (E(e

cc)) and holds for u < a:

(6)[a u][b + v]

qv Ca,b,cu,v,c +[b v + 1][a + u + 1]

qu1 Ca,b,cu+1,v1,c = 0


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6 COSTANTINO

=

=

aaa

bb

b

c cc

Pca,b (idc b) Yc,ba(a idc) Ya,cbFigure 1. The first equality is the definition of Pca,b, the second is Lemma 2.10.

Therefore by equations 5 and 6, one can compute all the coefficients Ca,b,cu,v,t out of Ca,b,ca,ca,c; this is

done in the following proposition (where an implicit choice of Ca,b,ca,ca,c is made):

Proposition 2.9.

Ca,b,cu,v,t =1cab(1)(bv)(ct)q (bv)(b+v+1)(au)(a+u+1)2 [a + b c]![b + c a]![c + a b]!

[2c]!

(7)

z,w:z+w=ct(1)zq (zw)(c+t+1)2

a + u + za + c b

b + v + wb + c a

c t

z

where the sum is taken over all z, w N such that z + w = c t and all the arguments of thefactorials are non-negative integers.

Proof of 2.9. In [1], Lemma 3.6.10, using the basis gau = [a + u]!eau for V

a, the following formulawas provided for the Clebsch Gordan coefficients (with the same notation as above, where we are

rewriting the formula via q-binomials and correcting a missing factor of1(tc)):

Ca,b,cu,v,t =1cab(1)(bv)(ct)q (bv)(b+v+1)(au)(a+u+1)2 [c + t]![c t]!

[2c]!

(8) z,w:z+w=ct(1)

z

q

(zw)(c+t+1)

2 a + b

ca u z

a + u + zz

b + v + ww

To get our statement it is then sufficient to multiply by [a+u]![b+v]![c+t]! , to single out of the factorials

the terms [a+bc]![b+ca]![c+ab]![2c]! and to pair the factorials in the denominators of the summands

so that their sums match a + u + z, b + v + w and c t. 2.9In order to get invariants of unoriented graphs, the operators associated to the other elementary

graphs must constructed by moving a leg of the Y-shaped graph up or down (see Figure 1).Hence we define projectors Pca,b : V

a Vb Vc out of Ya,bc as: Pca,b (a Idc) (Ida Ya,cb ).So letting P(eau ebv) =

t P

a,b,cu,v,t e

ct , the coefficients can be explicitly computed as:

(9) Pa,b,cu,v,t = Ca,c,b

u,t,v

12uqu

[2a]!One may also define Pca,b by pulling-up the right leg i.e. by setting P

ca,b (Idc b)(Yc,ba Idb)

(as in the r.h.s. of Figure 1), but as the following lemma shows, the two choices are equivalent:

Lemma 2.10. It holds:

Ca,c,bu,t,v

12uqu[2a]!

= Cc,b,at,v,u

12v qv[2b]!


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Proof of 2.10. Since both tensors express maps Va Vb Vc and the space of such maps is onedimensional, it is sufficient to prove the equality for a single vector. It is straightforward to check

them for t = c, recalling that u + v = t. 2.10Similarly, we can define non-smooth minima operators Wa,b,c Hom(C, Va Vb Vc) by

pulling up the only lower leg in Ya,bc , i.e. by setting Wa,b,c

(Ya,bc Idc) c. So, letting:

Wa,b,c(1) a

u=a

bv=b

ct=c

Wa,b,cu,v,t eau ebv ect

the coefficients are:

Wa,b,cu,v,t Ca,b,cu,v,t

12tqt[2c]! = 1abc(1)(bv)2tq (bv)(b+v+1)(au)(a+u+1)2 (10)

[a+bc]![b+ca]![c+ab]!

z,w:z+w=c+t

(1)zqt+ (zw)(ct+1)2

a + u + za + c b

b + v + wb + c a

c + t

z

Finally, an invariant tensor in Ma,b,c Hom(Va Vb Vc,C) can be defined by pulling downthe third leg of Pca,b: M

a,b,c c (Pca,b Idc). Explicitly, its coefficients are:

(11) Ma,b,cu,v,t Ma,b,c(eau ebv ect ) = Pa,b,cu,v,t

12tqt[2c]!

2.3. The state-sum computing G,col.Definition 2.11 (KTG). A Knotted T rivalent Graph (KTG) is a finite trivalent graph G S3equipped with the germ of a smooth surface S S3 such that S retracts on G.Remark 2.12. Not that this is more than a fat graph as S is required to exist around all Gand not only around its vertices. Moreover we are not requiring S to be orientable, even thoughthis assumption can simplify the form of the renormalization constant (see 3.2).

In order to specify a framing S on a graph G we will only specify (via thin lines as in the leftmostdrawing of Figure 2) the edges around which it twists with respect to the blackboard framing in adiagram of G, implicitly assuming that S will be lying horizontaly around G out of these twists.Let now G be a KTG equipped with an admissible coloring col. Decompose G into elementarysubgraphs (i.e. those obtained by possibly adding some parallel vertical strands to those depictedin Figure 2). Associated to each such subgraph there is a map of representations of Uq(sl2) asexplained in the preceding subsection. The composition of all the morphisms associated to theelementary subgraphs provides a morphisms G, col : V0 V0 = C therefore a rational functionin q

14 (the 14 is only due to the Hj operators: all the other operators provide terms in C(q

12 )). This

section is devoted to explain in detail how to compute this morphism via a state sum based on theformulas provided in the preceding sections.

Let G be a closed KTG, E be the set of its edges, V the set of its vertices and col : E N2an admissible coloring. Let also D be a diagram of G and for every e E, let ge

N

2 be thedifference between the framing of e in G and the blackboard framing on it (it is half integerbecause the two framings may differ of an odd number of half twists). Let C, M, Nbe respectivelythe set of crossings, maxima and minima in D (recall that we are fixing a height function on R2

to decompose D into elementary subgraphs). Then let f1, . . . f n be the connected components ofD \ (V M N C). Remark that each fj is a substrand of an edge of G therefore it inherits acolor which we will denote cj . To express G,col as a state-sum, let us first define a state:


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8 COSTANTINO

Definition 2.13 (States). A state is a map s : {f1, . . . f n} Z2 such that |s(fj )| cj , j =1, . . . n and s(fj ) is integer iff cj is. Given a state s, we will call the value s(fj ) the state of fj .(Equivalently, a state is a choice of one vector e

cj

s(fj)of the base of Vcj for each component of

D \ (V M N C).)The weight w(s) of a state s is the product of a factor ws(x) per each x crossing, vertex,

maximum and minimum of D. To define these factors, in the following table use the letters a,b,cfor the colors of the strands and u,t,v,w for their states.

ws(a

u v ) = u,v12uqu

[2a]!ws( a

u v) = v,u[2a]!

12uqu

ws(w

a bu v

t

) = ab Rt,wu,v (see Formula 3) ws(

w

a bu v

t

) = ab (R1)t,wu,v (see Formula 4)

ws( a bc

u v

t

) = Ca,b,cu,v,t (see Formula 2.9) ws( a bc

u v

t

) = Pa,b,cu,v,t (see Formula 9)

ws(a b cu v t

) = Wa,b,cu,v,t (see Formula 2.2.4) ws(a b c

u v t) = Ma,b,cu,v,t (see Formula 11)

Finally, to take into account the action of the half-twist operators Hj on the edges of G, let

F(G,col)

eedges14gecol(e)q2ge(col(e)2+col(e)) be the framing factor (note that it does not

depend on any state). The weight of a state s of G,col is then defined as follows:(12) w(s) = F(G, col)

Mmaxima

ws(M)

mminima

ws(m)

ccrossings

ws(c)

vvertices

ws(v)

The value of G,col is then given by :(13) G,col =

sstates

w(s)

Remark 2.14. Note that, up to isotopy, one can always suppose that D contains only positivecrossing, maxima minima and vertices with 3 top legs.

2.4. Invariants of framed trivalent graphs. The state-sum defined in the preceding subsectionis just a reformulation of the standard recoupling theoretical invariants (and, in particular, is aninvariant of colored KTGs):

Proposition 2.15. The function G,col C(q 14 ) is an invariant up to isotopy of (G, col) andcoincides with the Kauffman bracket G,col of [6] after replacing A = q 12 .

Note that a way to prove that G,col is an invariant consists to check Relations 1-13 of [12] (seeFigure 8) for the operators we provided. This is easy because the only difference with respect tothe standard case of oriented graphs consists in the movements involving the creation/deletion ofa maximum or minimum. Therefore one only needs to check relations 1 4 (which are straightfor-ward), 8 (which is easily computed on a highest weight vector), 9 , 10 and 13 (which is a consequenceof Lemma 2.10). We will instead sketch a different approach which justifies also the second state-ment of the proposition.

Example 2.16. IfL = L1, . . . Ln is an un link with (possibly half-integral) framings g1, . . . gn andcolored by colors c1, . . . cn then:

L, col =n

i=1

(1)2ci [2ci + 1](1)4giciq2gi(c2i+ci)


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a

a

a

a

a

a

a

aa

bb

b

b

bb

c

c

c

c

aa ab R ab R Ya,bcHa Wa,b,c Ma,b,cPca,bFigure 2. The elementary graphs and the associated morphisms. In all thedrawings except the leftmost, the framing is the blackboard framing.

aa

ab

b

b

c

cc = Ma,b,c Wa,b,c

Figure 3. The operator associated to a colored theta graph.

In particular Proposition 2.15 holds in this case. Indeed it is sufficient to prove it for the case ofan unknot colored by cj and with framing gj . In that case the value is the trace of the operator

cj (Id H2gjcj ) cj which equals:

tr(cj (Id H2gjcj ) cj ) = (1)4giciq2gi(c2i+ci)

cju=cj

14uq2u =

(

1)4giciq2gi(c

2i+ci)(

1)2cj

cj

u=cj

q2u = (

1)2cj [2cj + 1](

1)4giciq2gi(c

2i+ci)

Remark 2.17. We allow half integer framings i.e. some component of L may be framed by aMoebius band instead of an annulus. The standard case corresponds to gi N.Proof of 2.15. We bound ourselves to sketch the idea of the proof. First remark that the operatorPca,b we defined coincides with the operator defined in [1] (Definition 3.6.7), up to operating the

diagonal change of base gjm = [j + m]!ejm. To prove this claim it is sufficient to show that the

operators are equal on vectors of the form eau ebv with u + v = c. This is proved by comparingthe explicit formulas given in [1] (Lemma 3.6.8) and our definition of Pca,b.

Then the following results of [1] hold true in our case without changes: Theorem 3.7.1, Lemma3.10.9, Theorem 3.12. These results are all the basic tools of skein theory which allow one to reducethe computation of G,col to that of a union of unlinks and in Example 2.16 we already saw that

in that case the invariants coincide. 2.15

Example 2.18 (The theta graph). Consider a theta graph as in Figure 3. Equip it with theblackboard framing and color the edges by Va, Vb and Vc. Its invariant is then:

(14) (a,b,c) a

m1=a

bm2=b

cm=c

Wa,b,cm1,m2,mMa,b,cm1,m2,m


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10 COSTANTINO

By Proposition 2.15 the above formula equals the standard skein theoretical value:

(15) (a,b,c) = (

1)a+b+c[a + b + c + 1]![a + b c]![a + c b]![b + c a]!

[2a]![2b]![2c]!Remark 2.19. The above example shows that in general G,col is not a Laurent polynomial:consider for instance the case a = b = c = 1 in Formula 15.

2.5. Remarks on orientation and framings. Before concluding we stress that the machinerywe recalled applies to any framed, colored trivalent graph G and in particular to those whoseframing S is not an orientable surface. Of course, up to twisting around some edges it is alwayspossible to change the framing S ofG in order to make it an orientable surface. Moreover, ifD is adiagram ofG there is a framing SD induced on G simply by considering a surface containing G andlying almost parallel to the projection plane. Pulling back the orientation ofR2 shows that SD isorientable. The following is a converse: every orientable framing is (up to isotopy) the blackboardframing of some diagram of G.

Lemma 2.20. If G is a framed graph and S is and orientable framing on G then there exists adiagram D of G such that S coincides with the blackboard framing on G.

Proof of 2.20. The idea of the proof is to fix a diagram D and count the number of half twistsof difference on each edge of G between SD and S. The reduction mod 2 of these numbers formsan explicit cochain in H1(G;Z2) which is null cohomologous because S and SD are orientable.The coboundary reducing it to the 0 cochain corresponds to a finite number of moves as those inLemma 2.21 which change D and isotope G into a position such that the number of half twists ofdifference between S and SD is even on every edge. Then up to adding a suitable number of kinks

to each edge of G this difference can be reduced to 0 everywhere. 2.20

Lemma 2.21 (Half-twist around a vertex). For every admissible 3-uple (a,b,c) it holds:

ab

c=

a

b R Ya,b

c = (H1

b H1

a ) Yb,a

c (Hc

) =

ab

c

Proof of 2.21. It is sufficient to prove the equality for the highest weight vector ecc Vc; to dothis it is sufficient to check that ab R Ya,bc (ecc) and (H1b H1a ) Yb,ac (Hc)(ecc) have the samecoefficient with respect to the element ebb eacb of the basis ebj eai , |i| a, |j| b of Vb Va.By formulas 3 and 2.9 this coefficient is for the left hand side:

u+v=c

ab R

b,cbu,v Ca,b,cu,v,c = ab Rb,cbcb,b Ca,b,ccb,b,c =

= q2(cb)b 1cab(1)0q 12 (a(cb))(a+(cb)+1) [2b]![a + c b]![2c]!

The coefficient on the r.h.s. is (Hbb )1(Haa )

1Cb,a,cb,cb,c(Hcc ) which equals:

12c2a2bqc2+c(a2+a)(b2+b)1cba(1)a(cb)q 12 (a(cb))(a+(cb)+1) [2b]![a + c b]![2c]!

=

=1cbaqc2+c(a2+a)(b2+b)q 12 (a(cb))(a+(cb)+1) [2b]![a + c b]!

[2c]!

A straightforward computation shows that the two coefficients are indeed equal. 2.21


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3. Integrality

Let G be a closed KTG, E be the set of its edges, V the set of its vertices and col : E

N

2 an

admissible coloring. We will define the Euler characteristic (e) of an edge e E as 1 ife touchesa vertex and 0 otherwise (some edges of G may be knots). Let also D be a diagram of G and forevery e E, let ge N2 be the difference between the framing ofe in G and the blackboard framingon it (it is half integer because the two framings may differ of an odd number of half twists). Wedefine a renormalization for G,col as follows:

G,col G, colF(G,col)1

eE,(e)=1[2col(e)]!

vV [av + bv cv]![bv + cv av]![cv + av bv]!where by av, bv, cv we denote the colors of the three edges surrounding v.

Remark 3.1. Except for the term F(G,col)1 =1

PeE

4gecol(e)eE q

ge(col(e)2+col(e)), thenormalization factor depends only on the combinatorial structure of the abstract graph G and noton its embedding in S3. The factor F(G,col) encodes only how much the framing of G is twisted

with respect to the blackboard framing and does not influence the integrality of the invariant, sowe could omit it from our normalization. The reason why we keep it is that it may shift powers ofq of the form q n4 , n N present in G,col so that only to half integral powers of q are presentin G,col.Remark 3.2. Of course if the framing on G is the blackboard framing of its diagram then thefactor F(G,col) is 1, and, by Lemma 2.20 this situation can be achieved up to isotopy iff theframing is an orientable surface. Hence, if one restricts to graphs whose framings are orientablesurfaces then one may drop the factor F(G,col)1 from the above definition and the followingTheorem still holds with no changes in its wording.

Theorem 3.3 (Integrality of the renormalized Kauffman brackets).

G,col Z[q 12 ]Moreover, > is divisible inZ[q 12 ] by [2n + 1] for each n color of an edge of G.

Proof of 3.3. Up to isotopy, we can suppose that the diagram D of G is the closure of a (1, 1)-tangle G whose boundary strands are colored by n and also (by small isotopies around vertices andcrossings) that D contains only maxima, minima, positive crossings and vertices with 3 top legs.The factor F(G, col)1 in our normalization simplifies with the factor F(G,col) in the weight ofeach state (see formula 12), so that we may suppose from now on that all the framings ge are 0 (i.e.that G is equipped with the blackboard framing). By Schurs lemma the morphism represented by

G is IdVn . We claim that Q

eE,(e)=1[2col(e)]!QvV

[av+bvcv]![bv+cvav]![cv+avbv]! belongs to Z[q 12 ]; this will

conclude because G,col = (1)2n[2n + 1].To prove our claim let us define renormalized operators associated to each maximum, mini-

mum, crossing and vertex of D equipped with a state as follows:

N Hj Hj (Na)u,v u,v 1[2a]!

u v

a (Na)u,v u,v[2a]!vu

a

(ab N R)t,wu,v :=

ab R

t,wu,v (N W)

a,b,cu,v,t

Wa,b,cu,v,t[a + b c]![b + c a]![c + a b]!

Since the morphism represented by G is diagonal, the only non-zero weight states are thosewhere the states of the top and bottom strand ofG are equal. Therefore, if in formula 13 one fixes


12/23

12 COSTANTINO

the same state u on the top and bottom strand of G and replaces each weight by its normalizedversion defined above, the result will be:

G,col = G,col

eE([2col(e)]!)(cap(e)

cup(e))

vV[av + bv cv]![bv + cv av]![cv + av bv]!where for each edge, cup(e) (resp. cap(e)) are the number of minima (resp. maxima) on e. Onesees that the above formula gives the same normalization factor as in the claim by remarking that,since by our hypothesis on D all the vertices have 3 top legs, for each edge e different from the topstrand it holds (e) = cap(e) cup(e), and that for the top strand cap(e) cup(e) = 0.

To prove the claim remark now that that each normalized operator belongs to Z[1][q 12 ].

This is straightforward because of Lemma 3.5 and Formulas 3 and 2.2.4. We are then left to showthat actually all the coefficients are non-imaginary. Let us remark that in the state-sum for G,col,

since each edge with (e) = 1 has two endpoints and each N Wa,b,c belongs to1abcZ[q 12 ],

the product of the factors1col(e) coming from these vertices is 12col(e). Similarly the

product of the factors

1

2uicoming from the cups and caps on e is

1

2col(e)(cap(e)cup(e))=

12col(e) (because each ui is half integer iff col(ei) is) and this cancels with the previousimaginary phase. 3.3

Corollary 3.4. If G,col is a colored KTG and a1, . . . ak are pairwise co-prime colors of someedges of G then G,col is divisible byj [2aj + 1] inZ[q 12 ].Lemma 3.5. Leta1, . . . as be integers and let the q-multinomial be defined as

a1 + + asa1 a2 . . . as1 as

[a1 + + as]![a1]! [as]!

Then the q-multinomial is a Laurent polynomial with positive, integer coefficients.

Proof of 3.5. If s = 2 the statement is a direct consequence of the fact that, if yx = q2xy are twoskew-commuting variables, then:

(x + y)n =n

j=0

qn(n+1)

2 j(j+1)

2 (nj)(nj+1)

2

nj

xj ynj

which is easily proved by induction. The general case follows by induction on s by remarking that:a1 + + as

a1 a2 . . . as1 as

=

a1 + + as

(a1 + a2) . . . as1 as

a1 + a2a1 a2

To prove the last statement, remak that we just proved that each state in the state-sum 3.5

3.1. Examples and properties. The following examples can be proved by re-normalizing theformulas provided in [10] for the standard skein invariants of tetrahedra and -graphs.

Example 3.6 (Unknot).

(16) a = (1)2a[2a + 1]

Example 3.7 (Theta graph).

(17) a

b

c = (1)a+b+c

a + b + c + 1

a + b c, b + c a, c + a b, 1


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Example 3.8 (The tetrahedron or symmetric 6j-symbol).

(18) a

bc d

e f

=z=MinQj

z=MaxTi

(1)z z + 1

z T1, z T2, z T3, z T4, Q1 z, Q2 z, Q3 z, 1 where T1 = a + b + c, T2 = a + e + f, T3 = d + b + f, T4 = d + e + c, Q1 = a + b + d + e, Q2 =a + c + d + f, Q3 = b + c + e + f.

From now on we will denote respectivelya

,a

b

cand

abc

d

e fthe values provided respectively

in Formulas 16, 17 and 18 and often drop the : all the invariants we will be dealing with willbe the normalized version unless explicitly stated the contrary.

Example 3.9 (The crossed tetrahedron).

(19)

a

bc

d ef =

a

b

c

d e

f = ab

c de

f =

1

2(f+ceb)qf

2+f+c2+cb2be2e efa

b

c d

The following lemma can be easily proved by starting from the analogous statements for thestandard skein theoretical normalization of the invariants:

Lemma 3.10 (Properties of the renormalized invariant). The following are some of the propertiesof G,col:

(1) (Erasing 0-colored strand) If G is obtained from (G,col) by deleting a0-colored edge, thenG,col = G,col.

(2) (Connected sum) If G = G1#G2 along an edge colored by a, then

aa

G1 G2 = 1(1)2a[2a + 1] aaG1 G2

(3) (Whitehead move) If G and G differ by a Whitehead move then:

a

b c

d

j =

i

i abi

c

d j

ai

d

bi

c

a

b c

d

i

where i ranges over the all the admissible values.(4) (Fusion rule) In particular, applying the preceding formula to the case j = 0 one has:

a b =

i

i

ai

c

a

a

b

b

i

4. Shadow state sums and integrality

In this section we will first provide a so-called shadow-state sum formula for the invariantsG, col and give some examples. Then we will discuss some integrality issues and in Subsection?? will provide short proofs of known identities on 6j-symbols.


14/23

14 COSTANTINO

4.1. Shadow state sums. Let G,col be a fixed colored graph, D R2 be a diagram of it andV, E be the sets of vertices and edges ofG, and C, F the sets of crossings and edges ofD (each edgeof D is a sub-arc of one of G therefore it inherits the coloring from col). Let the regions r

0, . . . rm

of D be the connected components ofR2 \ D with r0 the unbounded one; we will denote by R theset of regions and with a slight imprecision we will say that a region contains an edge of D or acrossing if its closure does.

Definition 4.1 (Shadow-state). A shadow-state s is a map s : R F N2 such that s(f) equalsthe color of the edge of G containing f, s(r0) = 0 and whenever two regions ri and rj contain anedge f of D then s(ri), s(rj ), s(f) form an admissible three-uple.

Given a shadow-state s, we can define its weight as a product of factors coming from the localbuilding blocks ofD i.e. the regions, the edges of D, the vertices of G and the crossings. To definethese factors explicitly, in the following we will denote by a,b,c the colors of the edges of G (orof D) and by u,v,t,w the shadow-states of the regions and will use the examples given in Section3.1.

(1) If r is a region whose shadow-state is u then ws(r) u

(r)

(where (r) is the Euler

characteristic of r).(2) Iff is an edge ofD colored by a and u, v are the shadow-states of the regions containing it

then ws(f) u

a

v

(f)

where (f) = 0 if f is a closed component of D and 1 otherwise.

(3) If v is a vertex of G colored by a,b,c and u,v,t are the shadow-states of the regions

containing it then ws(v) a

bcu

v t.

(4) Ifc is a crossing between two edges of G colored by a, b and u,v,t,w are the shadow-states

of the regions surrounding c then ws(c)

a

uv

b tw .

From now on, to avoid a cumbersum notation, given a shadow-state s we will not explicit thecolors of the edges of each graph providing the weight of the local building blocks of D as they arecompletely specified by the states of the regions and the colors of the edges of G surrounding theblock. Then we may define the weight of the shadow-state s as:

(20) w(s) =

rR

(r) fF

(f)vV

cC

Then, since the set of shadow-states ofD is easily seen to be finite, we may define the shadow-statesum of (G,col) as

shs(G, col)

sshadow statesw(s)

As the following theorem says, the shadow state-sums provide a different approach to the compu-tation ofG,col:Theorem 4.2. It holds: G,col = F(G,col)shs(G,col), where (as in the preceding sections)F(G,col)

eedges

14gecol(e)q2ge(col(e)2+col(e)).The original definition of the shadow state sums and proof of the above result (but for the

standard normalization of the invariants) is due to Kirillov and Reshetikhin ( [8]) and was later


15/23


generalized to general shadows by Turaev ([13]). We used this formulation to extend the definitionof colored Jones polynomials to the case of graphs and links in connected sums of copies of S2 S1([3]) and to prove a version of the generalized volume conjecture for an infinite family of hyperboliclinks called fundamental hyperbolic links ([2]). These links were already studied in [4] for theirremarkable topological and geometrical properties.

Proof of 4.2. Multiplying by the factor F(G,col)1 we can reduce to the case when the framingof G is the blackboard framing. Let D be a diagram of G; we can add to G some 0-colorededges cutting the regions of D (except r0) into discs (this changes G and D but not the valueof the resulting invariant); for each region we will need (r) 1 such arcs. Fix also a maximalconnected sub-tree T in D and let o R2 be a 0-colored unknot bounding a disc containing D; itis clear that G,col = (G,col)

0

. Let also A be the trivalent graph defined as follows:A (N(T) D) N(T) (where N(T) is the regular neighborhood of T in R2). The idea ofthe proof is to apply a sequence of fusion rules and inverse connected sums in order to express

G, col

as a

colic(coli)

A, coli

for some colorings coli of A and coefficients c(coli); then to

show that each summand c(coli)A,coli is the weight of exactly one shadow-state. The unknot ois isotopic to N(T) and, while following the isotopy, at isolated moments it will cross some edgesof D \ (N(T) D) (but no vertices or crossings because they are all contained in N(T)). Let uschoose the isotopy so that every edge of D \ (N(T) D) is crossed exactly once (this can be donesince each region is a disc and T is a maximal connected sub-tree of D). We say that u enters aregion r if during the isotopy a subarc of u not contained in r crosses an edge contained in r. Weclaim that, since T, is connected each region ri, i = 1, . . . n will be entered by o exactly onceduring the isotopy. Indeed if u enters twice a region ri, let , the subarcs of o in ri, connectingthem by an arc we may produce two unknots whose connected sum is u. Since T is connected andcontains all the vertices and crossings of D, one of the two discs bounded by these unknots cannotcontain not contain vertices and so and cross the same edge of ri, against our hypothesis onthe isotopy.

We interpret each crossing moment as a fusion ruleso that the isotopy of u progressively erases eacharc ofD \ (N(T)D) exactly when entering a regioncontaining that arc, and the sum is taken over alladmissible u:

a b

=

u

u

au

b

a bu

At the end of this isotopy, since for each i n u entered ri only once, all the components ofN(T) ri (which are arcs) will be colored by the same colors ui (see Figure 4 for an example ofthis construction in the case of a planar graph). Therefore each summand in the final expressionwill be associated to a shadow-state s given by s(r0) = 0, s(ri) = ui. Moreover, the other edges ofA (i.e. those of N(T) D) are included in those of G and therefore inherit the coloring col. Thenwe proved the following equality:

G, col

=

u1,...un

rR

ui

fF\T

1

A, col

{u1, . . . un

}where the colors of the edges of the graphs are specified by the uis and col. Remark that thesummation range is exactly the set of shadow-states because the colors of the arcs of N(T) riare all ui and the admissibility conditions for a three-uple of colors around an edge are satisfiedat every moment we apply the fusion rule. Moreover, in the above formula we already got part ofthe weights of each shadow-state (i.e. those of the regions and of all the edges out of T). We are


16/23

16 COSTANTINO

Figure 4. On the left a planar graph G to which we apply the construction ofthe proof of Theorem 4.2. In the middle the graph A constructed by shrinkingon a maximal subtree of G an unknot bounding a disk containing G (the dottedparts are left just for reference). On the right the final union of planar tetrahedra(in this case there are no crossings).

left to prove that what is missing equals

A,col {

u1, . . . un}

, i.e. we claim the following:

A,col(u1, . . . un) =

fT

1 vV

cC

where in each factor the colors are specified by the combinatorics of D and by the state u1, . . . un col on R F. To prove this, remark that:

abu

=

i

i

ab

i

iabu

=a

b

c

1 aa bb

u u

where the first equality is a fusion rule and the second is the inverse of a connected sum. Applyingthis identity on all the edges of T we split A,col {u1, . . . , un} into the product of the graphs

remaining in the neighborhoods of the crossings and vertices (which are respectively and

) divided by the product of the s corresponding to the edges of T. This proves the

claim and completes the proof when all the regions are discs. If in the beginning we added some0-colored edge to G to cut D into a diagram D whose regions are discs, then it is clear that everyshadow state s of D can be lifted to a unique shadow state s of D: indeed the compatibilityconditions around a 0-colored edge force the states of the neighboring regions to be the same.Moreover, since the only difference between D and D is given by the presence of the 0-colorededge, it holds:

w(s) = w(s)

fF0

uu

0

1= w(s)

fF0

u 1= w(s)

rR

u (r)1

This concludes the proof. 4.2

The formula given by Theorem 4.2 is often re-written by means of the so-called gleams:


17/23


Definition 4.3 (Gleam). The gleam of a diagram D ofG is a mapg : R Z2 which on a region ri equals the sum over all the sectorsof crossings contained in ri of

12 times the local contributions of

the crossing determined according to the pattern on the right.

+ +

Remark 4.4. Do not confuse a shadow state with the gleam. In general for a given diagram Dthere is a unique gleam but many different shadow-states.

Then one can re-state Theorem 4.2 as follows:

Corollary 4.5. Under the same hypotheses as Theorem 4.2, it holds:

(21) G,col = F(G, col)

s

rR

14g(r)uq2g(r)(u2+u)u

(r) fF

(f) VC

Proof of 4.5. Using Example 3.9 one can rewrite the factors coming from crossings in term of

tetrahedra multiplied by extra factors of the form 12u

q(u2

+u) for each of the 4 sectors arounda crossing. To conclude, collect these factors according to the region containing the corresponding

sectors and compare with Definition 4.3. 4.5

4.2. Simplifying formulas. Both formulas 21 and 20 are far from being optimal: indeed most ofthe factors in the state-sum can be discarded from the very beginning. Instead of giving a generaltheorem for doing this let us show why this happens through some examples. We will say that anedge, vertex or crossing is external if it is contained in r0 and a region is external if its closurecontains an external edge. Then the following simplifications can be operated:

(1) If an external region contains two distinct external edges whose colors are different, thenG,col = 0

(2) If an external region r contains external edges f1, . . . , f k whose colors are all c, then, for ev-

ery shadow-state s on D, the total contribution coming from rf1, . . . f k isc

(r)Pi (fi)

(3) If an external vertex v is the endpoint of a non-external edge f then their contribution

simplify becausea

bcc

0 a=

ab

c(beware: if f is composed of two external vertices only

one of them can be simplified with f)

(4) For the same reason, if D contains a sequencea b r0

of n Z \ {0} half-twistsseparating r0 from a region r, then, in each summand of formula 21 the total contribution

of the crossings and internal edges of the twist (excepted the initial and final ones) isa

b

u

where u is the state of r and a, b are the colors of the strands (beware: the power of qcoming from the gleams do not simplify).

4.3. Examples and comments on integrality. According to Theorem 3.3, G,col is a Lau-rent polynomial, and this is proved is by showing that the weight of each state in the state-sumexpressing it via R-matrices and Clebsch-Gordan symbols is a Laurent polynomial. Surprisinglyenough, this is not true for shadow-state sums: the weight of a single shadow-state may be a rationalfunction, but the poles of these functions will cancel out when summing on all the shadow-states.We will now clarify this by explict examples.


18/23

18 COSTANTINO

4.3.1. Complicated formulas for unlinks. Consider the n-colored unnormalized Jones polynomialsof the following unlink:

Jn(12

12

-1

) =12nq4(n2+n)

2nu,v=0

12(u+v)q(u2+u)+(v2+v)u v n

nun

n v

nn

u

nn

v

In the picture we included the gleams of the regions to help the reader recovering formula 21. Of

course this is a very complicated way of re-writing (1)2n[2n + 1]2q2(n2+n), but what is interestingis that the single states are again not Laurent polynomials: for instance when n = u = v = 1 the

weight is [3]([5]1)[2][4] .

4.3.2. A more complicated link example. Fix a, b N and consider the n-colored unnormalizedJones polynomials of the following link:

Jn(

a

b ) =2n

u,v=0

12(a+1)u+2(b+1)v4n(a+b+1)qg(u,v)u v n

nun

n v

nn

u

nn

v

where a box with a Z stands for a sequence of a half twists and g(u, v) (a +1)(u2 +u) + (b + 1)(v2 + v) 2(a + b + 1)(n2 + n). To get the above formula, before applying 21, remarkthat up to isotopy of the diagram, the region r0 can choosed freely. Therefore, it is better to pickr0 as the region touching the two boxes contemporaneously. Again, the summands are not Laurentpolynomials but the sum is (take for instance a = b = n = u = v = 1). More surprisingly, by

Theorem 3.3, for every a, b 0 the resulting invariant will be divisible by [2n + 1] in Z[q, q1].

4.3.3. Some planar graphs. If G is a planar graph equipped with the blackboard framing, thenthe gleam of its regions are 0 and so by Corollary 4.5 G,col has a simple expression. In theexample of Figure 4, if all the edges of G are colored by n N then

(22) G, n =2n

u,v=0,|uv|


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4.4.1. Normalizations of 6-symbols. It holds:

(23) b,0a c

=

u

u a

ab c

c u

ac

u

where u ranges between |a c| and a + c. This is proved by applying Formula 21 to: ab

c

and recalling that the invariant of a union of two unlinked graphs connected by a single arc is zerounless the color of the arc is 0, in which case the invariant is just the product of the invariants ofthe graphs. Similarly, it holds:

(24)

ab

c

1

4b

q

2(b2+b)

=

u

1

2(u+a2c)

q

u2+u+a2+a

2(c2+c))

u abc

u

b c

bc

u

where u ranges between |b c| and b + c. This is proved by applying Formula 21 toa b c

and

to the same framed graph after undoing the kink on the b-colored edge.

4.4.2. Orthogonality relation. A direct corollary of Formula 21 is the well-known orthogonalityrelation:

b,d

ab

c

ef

b

b=

u

12(db)qd2+db2bu a

bcf

u e acd

f

e u

a

eu

c

fu

where u ranges over all the admissible colorings of the union of the two tetrahedral graphs on theright. To prove it, just apply formula 21 to the following two isotopic graphs and simplify thecommon factors:

a

b

c

d

ef a bc

d

e

f

4.4.3. Racah identity. It holds:(25)

12(a+bc)qa2+a+b2+bc2c a bcd

e f=

u

12(u+fed)qu2+u+f2+fd2de2eu

a efb

d u a cbe

u d

ab

u

bd

u

where u ranges over all the admissible colorings of the union of tetrahedra on the right. Indeedit is sufficient to apply Formula 21 to the following two isotopic graphs and simplify the common


20/23

20 COSTANTINO

factors:

a

b c

d

ef

a

b c

d

ef

4.4.4. Biedenharn-Elliot identity. Another direct corollary of Formula 21 is the well-known Biedenharn-Elliot identity:

(26)

abc

d

e f ghe

c

d i

ce

d

=

u

u aef

g

u h dbf

u

g i abc

i

h u

ah

u

bi

u

fg

u

where u ranges over all the admissible colorings of the union of tetrahedra on the right. Indeedit is sufficient to apply Formula 21 to the following two isotopic graphs and simplify the common

factors:

a

b

c

d

e

f

g h

i

a

b

c

d

e

f

g h

i

5. R-matrices vs. 6j-symbols

In the preceding sections we showed to compute invariants of colored graphs by means of twodifferent state-sums. Although for practical computation shadow state-sums turn out to be easierto deal with, state-sums based on R-matrices and Clebsch-Gordan symbols allow one to proveintegrality results. In this section we will compare the state-sums when applied to graphs withboundary. So let a (n, m)-KTG be a framed graph embedded in a square box which contains only

3-valent vertices (inside the box) and n (resp. m) 1-valent vertices on the bottom (resp. top) edgeof the box. A typical example is a framed (n, m)-tangle. Let as before E, V be the set of edgesand 3-valent vertices of G, G G box and, once chosen a diagram D of G, let F, C bethe set of edges, and crossings of D. Let also b1, . . . bn be the bottom (univalent) vertices of Gand t1, . . . tm the top vertices. The definition of admissible coloring of a ( n, m) is the same as thestandard one, but in this case, a second coloring is needed to get a numerical invariant out of G,namely a coloring on G.

Definition 5.1 (-colorings for (n, m)-KTGs). Let G,col be a colored (n, m)-KTG; a -coloringfor G is a map col : G Z2 such that if ik (resp. jk) is the color of the edge containing bk (resp.tk) then |col(bk)| ik and col(bk) ik Z.

Equivalently a -coloring is a choice of a vector in Vi1 Vin of the form ei1col(b1)

e

in

col(bn) and a vector in V

j1

Vjm

of the form e

j1

col(t1) ejm

col(tm). Given a (n, m)-KTGequipped with a coloring col and a -coloring col, one can compute G, col col exactly as inFormula 13: it is sufficient to restrict the set of admissible states to those such that the state of theboundary edges coincide with col. Then G represents a morphism Z(G,col) : V

i1 Vin Vj1 Vjm and G,col col is an entry in the matrix expressing Z(G, col) in the basesformed by tensor products of basis elements.

The integrality result 3.3 still holds true (the idea of the proof is exactly the same):


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Theorem 5.2 (Integrality: case with boundary). The following belongs to Z[q12 ]:

G,col col G,col colF(G,col)eE

[2col(e)]!vV[av + bv cv]![av + cv bv]![cv + bv av]!

mk=1

1

jk

nk=1

1ik

where E is the set of all the edges ofG which do not intersectG+ and F(G, col) is defined as inthe preceding sections.

What is interesting is that one can re-compute the invariant of (G,col) also via shadow-statesums and Clebsch-Gordan symbols. To explain this, we will use the following:

Definition 5.3. Given a finite sequence j1, . . . , jm a Bratteli sequence associated to it is a sequences0, s1, . . . sm such that s0 = 0 and for each 0 k m 1 the three-uple sk, jk, sk + 1 is admissible.

It is not difficult to realize that the set of Bratteli sequences associated to j1, . . . , jm is in bijectionwith the set of irreducible submodules ofVj1Vjm and that the submodule V(s) correspondingto a Bratteli sequence s = (s0, . . . sm) is isomorphic to Vsm . Moreover, using the Clebsch-Gordan

symbols defined in Subsection 2.2.4, we may fix explicit maps (s) : Vj1 V

jm

Vsm

and

i(s) : Vsm Vj1 Vjm . Graphically, (s) =s2

sm

j1 j2 j3 jm

and i(s) =sn

s2

j1 j2 j3 jm

.

More explicitly, to construct (s), apply the Clebsch-Gordan morphism Ps1j1,j2 to the first two

factors of Vj1 Vjm (the isomorphism is fixed by the choice of Clebsch-Gordan projectorsmade in Subsection 2.2.4). The result is in Vs2 Vj3 Vjm and composing iteratively withwith P

sk+1sk,jk+1

one gets the seeked map (s) : Vj1 Vjm Vsm . Similarly, i(s) : Vsm Vj1 Vjm is defined by composing recursively from right to left the Clebsch-Gordan morphismsof Subsection 2.2.4 Vsm Vsm1 Vjm Vsm2 Vjm2 Vjm . . . Vj1 Vjm .

Let G G box viewed as a framed graph in R2 by embedding the box in R2 with theblackboard framing around its boundary. Let s+1 , . . . s

+m and s

1 , . . . s

m be Brattelli sequences

associated to j1, . . . , jm and i1, . . . , in and suppose that s+m = s

n x. We can extend col to a

coloring col

s

s+ ofG: the color of the edge in the top (resp. bottom) edge of the box boundedby jk and jk+1 (resp. ik and ik+1) is s+k (resp. s

k ), the color of the left edge of the box is 0 and

that of the right edge x (see Figure 5).

Theorem 5.4 (Shadow state-sums vs. R-matrices and Clebsch-Gordan symbols). Let (s, s+)be the coefficient such that the morphisms (s+) Z(G,col) i(s) : Vx Vx and (s, s+)IdVxcoincide. Then:

(s, s+)x

= G, col s s+and

Z(G,col) = ss+

n1

t=1

st

itst

st+1

m1

t=1

st

jtst

st+1

G, col s s+sn

s+2

s2

s

n

s+m

i1 i2 i3 in

j1 j2 j3 jm

.

Proof of 5.4. The value on the right hand side is the invariant of the colored graph ( G, cols+s)depicted on the right of Figure 5. But, as shown in the picture, G is the closure of the graphrepresenting (s+) Z(G,col) i(s).


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22 COSTANTINO

s+1s+1

s1s1

s+m1s+m1

sn1 sn1

xx 00 G GG

Figure 5. Using a pair of Bratteli sequences it is possible to close (G,col) to acolored closed KT G (in the middle).

The second statement follows by applying n 1 times the fusion rule on the bottom legs of Gand m 1 times on the top legs to get:

G

=

s

s+

n1t=1

st

itst

st+1

m1t=1

st

jtst

st+1

s+2

s2

sn

s+m

i1 i2 i3 in

j1 j2 j3 jm

G

where s and s+ range over all Bratteli sequences associated to i1, . . . , in and j1, . . . , jn respectivelyand G

is the (1, 1) colored KTG obtained by opening G, col s+ s along the right edge of the

box. 5.4

Example 5.5 (R-matrices vs 6j-symbols). It holds:

ab R

t,wu,v =

a+bc=|ab|

Cb,a,ct,w,v+u

12(cab)qc2+ca2ab2bc

ab

c

Pa,b,cu,v,u+v

Indeed it is sufficient to apply the preceding theorem to (G,col) being a crossing.

References

1. J. S. Carter, D. E. Flath, M. Saito, The classical and quantum 6j-symbols, Math. Notes, PrincetonUniversity Press (1995).

2. F. Costantino, 6j-symbols hyperbolic structures and the Volume Conjecture, Geom. Topol. 11, (2007),

1831-1853.3. F. Costantino, Colored Jones invariants of links in #S2S1 and the Volume Conjecture, J. London Math.

Soc. (2) 76, (2007), no 1, 1-15.4. F. Costantino, D. P. Thurston, 3-manifolds efficiently b ound 4-manifolds, J. Topology (1), 3, (2008).

5. S. Garoufalidis, R. Van der Veen, Asymptotics of classical spin networks, www.arXiv.org/0902.3113(2009).

6. L. Kauffman, S. Lins, Temperley-Lieb recoupling theory and invariants of 3-manifolds., Annals of Mathe-matics Studies, 134. Princeton University Press, Princeton, NJ, 1994.


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7. R. Kirby, P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,C), Invent. Math.105, 473-545 (1991).

8. A.N. Kirillov, N.Yu Reshetikhin, Representations of the algebra Uq(sl2), q-orthogonal polynomials and

invariants of links, Infinite dimensional Lie Algebras and Groups (V.G. Kac, ed.), Advanced Ser. In Math.Phys., Vol. 7, 1988, 285-339.

9. T.Q. Le, Integrality and symmetry of quantum link invariants, Duke Math. J. 102 (2000), no. 2, 273306.10. G. Masbaum , P. Vogel, Three-valent graphs and the Kauffman bracket, Pac. J. Math. Math. 164, N. 2

(1994).11. S. Piunikhin, Turaev-Viro and Kauffman-Lins invariants for 3-manifolds coincide. J. Knot Theory Ramifica-

tions 1 (1992), no. 2, 105135.12. N.Yu. Reshetikhin, V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups.

Comm. Math. Phys. 127 (1990), no. 1, 126.

13. V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, Vol. 18,Walter de Gruyter & Co., Berlin, 1994.

Institut de Recherche Mathematique Avancee, Rue Rene Descartes 7, 67084 Strasbourg, France

E-mail address: [email protected]

Francesco Costantino- Integrality of Kauffman Brackets of Trivalent Graphs

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