G. Masbaum and P. Vogel- 3-Valent Graphs and the Kauffman Bracket

8/3/2019 G. Masbaum and P. Vogel- 3-Valent Graphs and the Kauffman Bracket

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PACIFIC JOURN AL OF MATHEMATICSVol. 164, No. 2, 1994

3- VALENT G RAPH S AN D THE KAUF FMAN BRACKETG. MASBAUM AND P. VOG EL

We explicitly determine the tetrahedron coefficient for the one-variable Kauffman bracket, using only Wenzl's recursion formulafor the J ones idempotents (or augmentation idempotents) of theTemperley-Lieb algebra.

1. Statement of the main result. In this paper, we consider unori-en ted knot or tangle diagrams in the plane, up to regular isotopy.F u r t h e r m o r e , we impose the Kauffman relations [Kal]

4- 1

u ( ) =-(A+ -')

( th e first relation refers to three diagrams identical except where shown,an d in the second relation, D represents any knot or tangle diagram).Coefficients will always be in Q(A), the field generated by the inde-t e r m i n a t e A over the rational numbers. With these relations, (n, n)-tangles (i.e. tangles in the square with n boundary points on the upperedge and n on the lower edge) generate a finite- dimensional associa-tive algebra T n over Q(A), called the Temperley- Lieb algebra on nstrings (see [Lil] for more details on what follows). Multiplication inT n is induced by placing one diagram above another, and the iden-tity element ln is given by the identity (n, n)- tangle. U p to isotopy,the r e is a finite number of (n, )- tangles without crossings and with-o u t closed loops; they form the standard basis of the algebra T n . Let6" T n Q(A) denote the associated augmentation homomorphism,i.e. (ln) = 1, and is zero on the other basis elements. The algebraT n (with coefficients in Q(A)) is semisimple, and in particular thereis an augmentation idempotent fn eTn with the property

fnx = xfn = e(x)fn361


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362 G . MASBAUM AND P. VOGELfor all x Tn , and f2 = fn.1 These idempotents were first discoveredby Jones [ J l ] . Up to normalisation, they are equal to the "magicweaving elements" of [Ka2]. There is a recursion formula for thesei d e m p o t e n t s due to Wenzl [We], which we will now state graphically.

We make the convention that writing n beneath a component of ak n o t or tangle diagram means that this component has to be replacedby n parallel ones (n may be zero).2 We will often say that thisc o m p o n e n t is colored by n (a color is just a non- negative integer).F o r graphical reasons, we make the further convention that we mayo m i t indicating the color if it can easily be deduced from the rest ofth e diagram. Let us denote the augmentation idempotent fn e Tn bya little box. Then Wenz s formula can be stated graphically as follows:

n - \

[n]

H e r e , we use the notation [n] = (A2n - A- 2n)/ (A2 - A~2). Fora proof of this formula, see [We] or [Lil]. (Notice that in the lastdiagram of the formula above, we did not indicate the color of theline joining the two boxes. As explained above, this does not meant h a t the color is one, but that the color must be calculated form the restof the diagram. Here clearly the color is n - 2, because the little boxesrepresent elements of the Temperely- Lieb algebra on n 1 strings.)

Following Kaufman [Ka2], we define a 3- valent vertex as follows.A triple ( , b, c) of colors is called admissible if a + b + c is evena n d \a- b\


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3- VALENT GRAPH S 36 3The numbers / = (b + c- a)/2, j = (a +c- b)/2, k = (a + b- c)/2

of internal lines of the 3- valent vertex will be called its internal colors.N o t i c e that a triple ( , b, c) is admissible if and only if one can findcorresponding internal colors.

The goal of this paper is to give an explicit determination of thefollowing three coefficients, using only Wenz s formula for the aug-m e n t a t i o n idempotents fn.trihedron coefficient:

= (a,b,c)

tetrahedron coefficient

BIA B E\\D C F/

( O f co u r se , the t r i h e d r o n coefficient is a special case of the t e t r a h e -d r o n coefficient, e.g. (llc) = (a, b,c).)half- twist coefficient:

= (c; a, b)

R E M A R K S . (1) The names trihedron and tetrahedron coefficient aremotivated by the fact that one may imagine the corresponding graphsas a regular trihedron or tetrahedron drawn on a sphere.


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364 G. MASBAUM AND P. VOGEL(2 ) Knowing the t e t r a h e d r o n coefficient is equivalent to knowing

th e (more popular) recoupling coefficients or 6-7- symbols (see 2).(3 ) Reversing the sign of the crossing in the half- twist coefficientreplaces (c \a,b) by its conjugate (c a,b), where the conjugation

on Q(A) is defined by A = A~ .Recall [n] = (A2n - A- 2n ) / (A 2 - A~2). This n o t a t i o n is motivated

by the fact t h a t [n]\ A=x = n. N o t i c e [0] = 0. We also define [n]\ =[1] '[n] for n> 1, and [0]! = 1.THEOREM 1 (Trihedron coefficient). Let (a, b, c) be admissible and

let / , j , k be the internal colors of a 3- valent vertex (a, b, c). Then(a b c)- ( t

T H E O R E M 2 (Tetrahedron coefficient). Let A, B,C, D,E, F becolorssuch that the triples (A,B,E), {B,D,F), [E, D, C) and(A,C,F) are admissible.Set = A + B + C + D + E + F and

b = ( - A- D)/2 ,b2 = ( - E- F)/ 2,b3 = ( - B- C)/ 2.

ThenA B E\_ t i )= i fa - ajV. (ax a2 a3 a4D C F/ [A]\[B]\[C]\[D]\[E]\[F]\ \ bx b2 b3

wherea 2 a3 a ^ ( - 1 ) C E C + 1 ] !

b\ b2 b3 J )


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3-VALENT GRAPH S 365interval colors of a 3- valent vertex (a, b, c). Then

{c\ a,b) = (- \)kAij- k^J+k+2\REMARKS. (1) Similar (but not identical) formulas were deduced

by Reshetikhin and Kirillov [KR] from the representation theory ofth e quantum group UgS\ 2 3 They are in fact #- analogues of formulast h a t have been used by physicists for a long time in the quantumtheory of angular momentum (see for example [BL]). Piunikhin [P]has worked out the precise relationship between Kauffman's definitionof a 3- valent vertex, which is the one we use here (up to normalisation),an d the definition coming form the representation theory of UgS\ 2(we will state his formula in an appendix to 2). Using the resultof [KR], he thus obtains a formula for the tetrahedron coefficient.Nevertheless, it seems fair to say that the approach we take here isa much simpler way to get our formulas, because the representationtheory approach involves much more complicated calculations thano u r approach through the Kauffman bracket.

(2) The results of this paper are used in [BHMV3], where it is shownhow the invariants p of [BHMV1, BHMV2] lead to an elementaryc ons t r uc t ion of a "Topological Quantum Field Theory".

T h e remainder of this paper is organized as follows. In 2, we willshow how the knowledge of the three coefficients above allows one toevaluate any colored link diagram or any colored 3- valent graph. (Thisis a fact well known to specialists.) The bulk of this paper is 3, wherewe will give the proof of Theorems 1- 3.

T h e authors wish to acknowledge helpful conversations with O.Viro, who pointed out a great simplification in our original proof ofT h e o r e m 3. The proof of Theorem 3 given here is due to him.

2 . Evaluating colored link diagrams and colored 3-valent graphs.This section presents a graphical calculus which allows one to evalu-a te any colored link diagram or any colored 3- valent graph, using thecoefficient determined in Theorems 1-3.

We first need to recall some concepts. Let &t be the Q(^4)-modulegenerated by link diagrams in an annulus, modulo regular isotopy andth e Kauffman relations.4 Given a ^- component link diagram L, and

In order to make contact with #- formulas and 17_S12 , the correct substitution seems to beA = - q~1^4 . N o t e that the Kauffman bracket is related to Jones' original F- polynomial [J2]through a similar substitution (see [Kal]).

4 & was called s/ in [Li2]. It is also the same as the Jones- Kauffman module of a solidto r u s as defined in [BHMV1], because of the usual correspondence between planar link diagramsu p to regular isotopy, and banded links in 3-space.


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366 G. MASBAUM AND P. VOGELb\ , . . . , bk G 38, let (b\, ... 9 b^)L denote the Kauffman bracket ofth e linear combination of link diagrams obtained from L by replacingth e v'th component by bj. Notice that ( , . . . , )L is a fc- linear formo n 38.5 There is a "trace map" Tn 38 given by mapping a tanglein the square to the diagram in the annulus obtained by identifyingth e upper and lower edges of the square. The image of an elementx eTn under this map is called the closure of x, and will be denotedby Jc. Set en = f ne3B.I t turns out that the en (n > 0) form a basis of 38, and hence themeta- bracket of L is determined by the values

( e i 9 . . . 9 e i k ) L .We call this the colored bracket polynomial of L. This is up to nor-malisation the same as the colored Jones polynomial appearing in thec ons t r uc t ion of 3- manifold invariants in [RT] and [KM]6 (see also[ M S ] ) .

T h e goal of this section is to evaluate fa , . . . , e^L - With ourgraphical conventions, this is the value of the diagram obtained formL by writing ij beneath the jth component, and inserting one littlebox into each component. Such a diagram will be called a coloredlink diagram. (We may place the little box wherever we want on thec o m p o n e n t . Also, we may insert several boxes since a box representsa n idempotent.) Let C denote the diagram of a circle, and set

(This was denoted by (ek) in [BHMVl], and is equal to ( - l)*[fc+ l] =( - l ) k(A 2 k+ 2 - A- 2k~2) / (A2 - A~2).) Gluing two annuli together so asto get a third endows 38 with a multiplication for which eo = 1 andexe\ = et+i + e^\ [BHMVl]. It follows easily that in 38, one has theequa t ion

k( th e sum being over all k such that the triple (/ , j , k) is admissible).

5Th is >linear form was first considered in [Li2]. We call it the meta- bracket of L [BHMVl].6See also a remark in the proof of Prop. 2.2 in [BHMV2].


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3-VALENT GRAPHSThis implies the following graphical equation in Ti+J :

367

We now show how to use this and the coefficients determined inTheorems 1-3 to evaluate colored link diagrams. (The rangle of sum-m atio n is not indica ted explicitly in the formulas below. It is de-termined precisely by the requirement that the colors meeting at a3-valent vertex should form an admissible triple.) First of all, we canget rid of crossings using the above form ula an d the half-twist coeffi-cient, as follows:

i V jk

i

Thus we are left with a colored 3-valent graph. Notice that an edgecolored w ith zero may jus t as well be removed, since fo is representedby the empty (0, 0)-tangle. Since

(*) oany diangular face (th at is a face with precisely two vertices) of a graphcan be elim inated. Similarly, any triangular face can be simplifiedusing the tetrahedron coefficient as follows:

A B ED C F

C


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368 G. MASBAUM AND P. VOGELM o r e generally, any triangle, square, pentagon, etc. can be reduced

to a diangular face (which can then be eliminated by formula (*)above) using

b\ cb ]

c d j]

where {ac bd is the recoupling coefficient or 6- j;- symbol defined byi b c

a b i\ {i)\j d a)c d jj (i,a,d)(i9b9c)(see [Ka2] for a graphical proof of th is formula). We now are left

with a connected graph each of whose faces has at most one vertex.As explained above, we may assume no edge is colored by zero. Thenif the graph has a vertex, the value is zero, because for k > 1 one has

as follows immediately from the fact that fa is the augmentationi d e m p o t e n t . If however the graph has no vertex, then it is simply adisjoint union of circles colored by k\ 's, and its value is the productof the (ki) 's. (By the very definition of the Kaufman bracket, thevalue of a disjoint union of diagrams is equal to the product of thevalues of the individual components.)

This completes the evaluation of colored link diagrams.REMARK. The reader will find two more useful graphical identities

in the proof of Theorem 3.REMARK. In this paper, we have made the convention that writing

n beneath a component of a knot or tangle diagram just means thatthis component has to be replaced by n parallel ones. It is sometimesuseful to make the additional convention that writing n beneath a


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3-VALENT GRAPH S 369c o m p o n e n t also includes insertion of the idempotent fn . For example,this would be convenient when using thegraphical calculus to evaluateth e colored bracket polynomial of a link diagram. Indeed, in thiscontext it is sufficient to consider only diagrams where each componentc o n t a i n s an fn , and it would then no longer be necessary to drawthelittle boxes explicitly. However, for the purpose of this paper, whichis to give an elementary proof of Theorems 1-3, it is useful not tomake this additional convention.

Appendix. Comparison with the representation theoretical approach.H e r e is th e relationship of ourgraphical calculus with the one of [KR](see [P] for more details). Let Vn denote the quantum deformationof the standard irreducible representation of SI2 of spin n/ 2 (i.e.of dimension n + 1). The 3- valent vertices of [KR] represent certainintertwining homomorphisms Kcab:VaVb- >Vc, K%b: Vc- *VaVhas follows:

As explained in [P], the trivalent vertex as defined in 1 may alsobe interpreted as an intertwiner, and there are coefficients *b, ca bsuch that

P i u n i k h i n has shown that

(see formula 4.12 of [ P ] ) . 7 ' 8Actually, formula 4.12 of [P] containsa certain 6-7-symbol which can however be evaluated.

8 Themeaning of the square root is no t specified in [P].


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370 G. MASBAUM AND P. VOGELREMARK. In view of formula (*) above, the equation

is equivalent to

(assuming (a, b, c) is an admissible triple).3 . Proof of Theorems 1-3. The properties of the fn that we willneed are Wenz s recursion formula (see 1) and the fact that they are

au gment a t i o n idempotents, which implies the following two equations:n- 2

(i) = 0, ( )

(Recall that writing n beneath a component of a tangle diagrammeans that this component has to be replaced by n parallel ones.)

LEMMA 1.

[ f t + 2]

Proof, Let us denote by gn the element of Tn represented by thediagram on the L.H.S. Using (ii), one sees that gn = fn for somec ons tan t . The closure ^ G %of gn satisfies


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3- VALENT GRAPHS(these notations are defined in 2). Hence

, (en+) c [n + 2](en)c [n + l]

as asserted.L E M M A 2. For j > 1 :

371

W / - IU+j- U

k+ l

Proof. The proof is by induction on j . For 7 = 1, this followsfrom (ii). For j > 2, we apply Wenz s formula on the L.H .S. to thebox representing fi+j- i. The first term of that formula yields zero by( i ) : hence we get

U+j- 2]

Using (ii), we can draw this as follows:

U+j- 2]

Applying the induction hypothesis to the upper part, we get


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372 G. MASBAUM AND P. VOGEL

[i+y-l] [i+y-2]

Using (ii) again, the lemma follows.REMARK. Lem m a 2 rem ains valid for / = 0 if we ma ke the con-

vention that a diagram is zero if it has a negative number of stringssomewhere.LEMMA 3. [n + r][m + r] = [n][m] + [n + m + r][r] .Proof, This can be verified by a direct calculation. Alternatively, italso follows easily from the equation

(the sum being over all k such that the triple (i, j 9 k) is admissible)(see 2). Details are left to the reader.Proof of Theorem 1. It is convenient to set

( a , b , c) = [ i , j , k]where the / , j , k are the inter nal colors of a 3-valent vertex (a, b , c).We have


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3- VALENT GRAPH ST h e proof is by induction on j . If j = 0, then clearly

373

an d the theorem is true in this case. If j > 1, we apply Wenz sformula to get

V+j]

We can simplify this, using Lemmas 1 and 2. Thus we get thefollowing recursion formula:

[j + [i] 2

Applying the induction hypothesis to the R.H.S., we get after simpli-fying

[ i + JV U + k ] \ [ i + k ]\But [/ + j]\ j +k+l] -[ i] [k+l] = [j][i + j3 , whence the result.

L E M M A 4. For y > 1

because of Lemma


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374 G. MASBAUM AND P. VOGELProof. Wenz s recursion formula applied to the box representing

f x+y yields

[x + y]

T h e first diagram on the R.H.S. can be simplified using Lemma1. The second diagram can be simplified using Lemma 2 twice, asindicated below:

H e n c e

T h e result now follows from Lemma 3.Proofof Theorem 2. We start by reformulating Theorem 2 as follows.

N o t i c e that the numbers aj and ftz defined in Theorem 2 satisfy dj = bi = and max( / ) < min(Z?z). Conversely, it is nothard to see that whenever a\ , ^2 >^3 ? ^4?

^i ? 2 >^3 a r e nonnegativeintegers satisfying these two conditions, then they uniquely determineA9 B, C etc. such that there is a well- defined tetrahedron coefficient.With th is in mind, we define

#3 0

whenever

A BE) if max( ) < min(^)D c Fl0 if max( 7 ) > min(6/ )

i (and 0/ > 0, ft/ > 0 ).


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3- VALENT GRAPH ST h e o r e m 2 can now be stated as follows:

375

a4] _ fax

where the R.H.S. is defined by

ax a2b\ b2a2 a3 ab2 b3

a i a2 #3 #4

T o obtain a recursion formula for [ b 2 h 3b 4 ] , it is convenientto introduce some more notat ion. Let / , j , k denote the internalcolors of the vertex labeled a\ , and similarly , , those of a2,an d x,y,z those of a$, as in the following figure:

Assume that j > 1 and > 1. Applying Wenz s formula to theedge labeled B = i + j = a + ,we obtain the above

[ I + 7 - 1 ][i+ 7

Observe that if y = 0, then the first term is zero, and if / = 0 ora = 0, then the second term is zero (cf. the remark following Lemma2 ) . We apply Lemmas 4 and 2 and obtain the above


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k+ l

F o r this recursion formula to be valid it is sufficient to supposet h a t j > 1 and > 1, but we may allow some of the other sevennumbers / , k, a, , x, y, z to be zero. (We make the conventiont h a t a diagram is zero if it has a negative number of strings somewhere.This is consistent with the convention that

d\ CL2b\ b2

a4= 0

if max( / ) > min(Z>/) , because the 12 numbers bi aj are the internalcolors of the four vertices of the tetrahedron .) If j = 0 or =0, however, the R.H.S. is zero whereas the L.H .S. may be non- zero.But if we multiply both sides of the recursion formula by [j][ ] =[b\ - d^\[b - a^], the result will be true without restriction. Afterrewriting things in terms of the aj and the b\ , we thus obtain:

- 1 a2 - 1 min(bi), this is true by definition.Next, we consider the case where max( 7) = min(6/ ). Then (atleast) one of the 12 numbers b\ - aj is zero, and without loss of gen-

erality, we may suppose y = b - a$ = 0. We proceed by inductiono n min(7 , ) = min(&i -


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3-VALENT GRAPHS 377follows easily from Th eorem 1. N ow suppose m in( y, ) > 1. Thefirst t e r m on the R.H .S. of our recursion formula being zero, we have

a i a2b\ b2

- a2 + 1] \a - 1 a2 - 1 a3 a4[b \ - a4][b2 - a4] [ bx - 1 b2 - 1 b3I t is easy to verify that the same relation holds for ( *') in place of[. ] (tk e s u m reduces to the term for = b3 = a4). Th is yields thei n d u c t i o n step. Thus we have shown Theorem 2 in the case max( 7) =

T o prove the theorem in the general case, we wish to simplify therecursion formula obtained above. At th is point, it is useful to lookat symmetries. Observe that the coefficient ( \ \ \ 4) has symmetryJ?4 x J?3 that is, it is unchanged under any permutation of the


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378 G. MASBAUM AND P. VOGELr 3 + 1, we find

[03 + ll fol + 2 ~ *3] - [ 2 + I p l + 03 ~H e n c e , the equation above has [63 - a\ + 1] (which is equal to[01 3 1]) a s a common factor. If 3 a\ - 1, we may di-vide by this common factor, and obtain

This is the "simpler" recursion formula we sought. Using Lemma3 as above, it is not hard to verify that the same formula is true withObW"*) in place of [%VV 4 ] . We leave this to the reader. (Itis true "term by term", that is, even before summing over .) Noticet h a t it follows that our formula rem ains true if 63 = a\ 1, becausein that case we know already that ( '") = [ "' ] for the three termsappearing in that formula.

Setting

Ia \ 0 2 0 3 0 4 ] 0 1 0 2 03 0 4 ] _ ( CL \ a 2 0 3 a6 1 b 2 b 3 j ~ [ b x b 2 b 3 \ V b x b 2 b 3

we thus obtain after a slight change of variables the following finalformula:

( 1 02 - 1 03 04

0 1 02 03 04* h b,

We now conclude the proof of Theorem 2 as follows. We must showI b 2b 3b 4 } = 0. We use double induct ion on = aj: = bi ando n d = min(6z) - max( / ). For = 0 there is nothing to show, and


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3-VALENT GRAPHS 379if d = 0, then we know already the result. If > 1 and d > 1,we look at our final formula. By the hypothesis of the induction on , the L.H.S. is zero. Since { \ \ \

4} is unchanged under anyp e r m u t a t i o n of the a.j, we may assume that max( / ) = a?>hence by

th e hypothesis of the induction on d, the second term of the R.H .S.is zero. Hence

a i # 3 # 4

But we assumed d > 1 hence [63 - a^] is non- zero, and the resultfollows.

This completes the proof of Theorem 2.Proof of Theorem 3. We reproduce here an argument of O. Viro [V].

O n e has the following two equations:

= A~ik

where , = {- \)A + 2 / . Indeed, the first equation is an immediateconsequence of the equation t i = zez of [BHMV1] (see also [Lil,Li2]), and the second equation follows from applying the KaufFmanrelations to the ik crossings of the diagram, together with the factt h a t the box represents the augmentation idempotent f^+i. Hence


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b

whence (c;a,b) = - Aij~ ik- kj = ^

as asserted.R E M A R K . One has

(c;a,by =_This follows immediately from Theorem 3.

R E F E R E N C E S

[BHM Vl] C. Blanchet, N . Habegger, G . Masbaum and P. Vogel, Three- manifold in-variants derived from the Kauffman bracket, Topology, 31, N o. 4, (1992),685- 699.

[BHMV2] , Remarks on the Three- manifold Invariants p , N ATO AdvancedResearch Workshop and Conference on Operator Algebras, MathematicalPhysics, and Low- Dimensional Topology, I stanbul, July 1991 (to appear).

[BHMV3] , Topologicalquantum field theoriesderivedfrom the Kauffman brac-ket, preprint May 1992 (in revised from January 1993).

[BL] L. C. Biedenharn and J. D . Louck, Angular momentum in quantum phys-ics, Encyclopedia of Mathematics and its Applications, vol. 8, Addison-Wesley 1981.

[J l] V. F . R. Jones, Index of subfactors, Invent. M a t h . , 72 (1983), 1-25.[J2] , A polynomial invariant for links via von Neumann algebras, Bull.

Amer. M a t h . Soc , 12 (1985), 103- 111.[Kal] L. H . Kauffman, State models and the Jones polynomial, Topology, 26

(1987), 395- 401.[Ka2] , Knots, spin networks, and 3- manifold invariants, (preprint 1990).[KM] R. C. Kirby and P. Melvin, The 3- manifold invariants of Witten and

Reshetikhin- Turaev for sl(2, C ) , Invent. M a t h . , 105 (1991), 473- 545.[KR] A. N . Kirillov and N . Yu. Reshetikhin, Representations of the algebra

Uq{ ), q- orthogonalpolynomials, and invariants of links, LOMI preprint1988.


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3-VALENT GRAPHS 381[ L i l ] W. B. R.Lickorish, Three- manifold invariants and the Temperly-Lieb al-

gebra, M a t h . Ann., 290 (1991), 657- 670.[Li2] , Calculations with theTemperley-Lieb algebra, C o m m e n t . M a t h .

Helv., 68(1992), 571- 591.[MS] H. R. M o r t o n and P.Strickland, Jones polynomial invariants for knotsand satellites, M a t h . P r o c . Cambridge Philos. Soc , 109 (1991), 83- 103.

[P] S.Piunikhin, Turaev-Viro and Kaufman invariantsfor - manifolds coin-cide, J.K n o t Theory and its Ramifications, vol. I no. 2, (1992), 105- 135.

[RT] N . Yu.Reshetikhin and V.G. Turaev, Invariants of ^- manifolds via linkpolynomials and quantum groups, Invent. Math . , 103 (1991), 547- 597.

[TV] V. G. Turaev and O. Ya.Viro, State sum invariants of ^- manifolds andquantum 6- j- symbols, Topology, 31 (1992), 865- 902.

[V] O. Ya.Viro, private communication.[We] H . Wenzl, On sequences of projections, C. R. M a t h . Rep. Acad. Sci. C a n a -

da, IX(1987), 5-9.Received M a r c h 15, 1992.U N I V E R S I T Y DE N A N T E SU R A n758 D U C N R S2 RUE DE LAHoUSSINlfeRE44072 N A N T E S C E D E X 03, F R A N C EANDU N I V E R S I T Y PARIS VIIU R A w212 D U C N R SCASE POSTALE 7012, 2 PLACE J U SSI E U75251 PARIS C E D E X 05, F R A N C E


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G. Masbaum and P. Vogel- 3-Valent Graphs and the Kauffman Bracket