N. Yu. Reshetikhin and V. G. Turaev- Ribbon Graphs and Their Invariants Derived from Quantum Groups

8/3/2019 N. Yu. Reshetikhin and V. G. Turaev- Ribbon Graphs and Their Invariants Derived from Quantum Groups

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Communications nCommun. Math . Phys . 127 , 1 -26 (1990) MathematicalPhysics

Springer-Verlag 1990

Ribbon Graphs and Their Invariants Derivedfrom Quantum GroupsN . Yu. Reshetikhin and V. G.TuraevL.O.M .T., Fontanka 27, SU-191011 Leningrad, USSR

Abstract.The generalization of Jones polynomial of links to the case of graphsinR3is presented. Itis constructed as the functor from the category of graphs tothe category of representations of the quantum group.

1. IntroductionThe present paper isintended to generalize the Jones polynomial oflinks and therelated Jones-Conway andKauffman polynomials to thecase ofgraphs inR3.

Originally the Jones polynomial was defined for links ofcircles inR3via anastonishing use ofvon Neumann algebras (see [Jo]). Later on it was understoodthat this and related polynomials may be constructed using the quantumK-matrices (see, for instance, [TuJ). This approach enables one to constructsimilar invariants for coloured links, i.e. links each of whose components isprovided with a module over a fixed algebra (see [ReJ, where the role of thealgebra is played by the quantized universal enveloping algebra Uq(G) of asemisimple Lie algebraG).

The Jones polynomial has been also generalized in another direction: ingeneralization of links of circles one considers the so-called tangles which are linksof circles and segments in the 3-ball, where it isassumed that ends ofsegments lieon the boundary of the ball. Technically it isconvenient to replace the ball by thestripR2x [0,1], which enables one to distinguish the top and bottom endpoints ofthe tangle. The corresponding "Jones polynomial" of a coloured tanlge is a linearoperator V1...Vk-^V1...V where V,...,Vk (respectively V...V*)a re the modules associated with the segments incident to bot tom (respectively top)endpoin t s of the tangle. H e r e the language ofcategories t u rns out to be veryfruitful. The tangles considered up to isotopy are t r e a t ed as morphisms ofthe"category oftangles." The generalized J o ne s polynomial is a covariant functorfrom this category to the category ofmodules (see [Tu2, Re2])Definitions ofJones- type polynomialsfor embedded graphs in R3 have beengiven byseveral a u t h o r s [KV, Ye] but the subject still remains open. It was clearfrom the very beginning t h a t the graph should beprovided with thickening, i.e.


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2 N. Yu. Reshetikhin and V. G. Turaev

with a surface containing the graph and contracting to it (cf. Sect. 4). Anotherrelated problem is what should be understood by colours of the vertices.The theory of the Jones- type polynomials has proved to be closely related tothe conformal field theories [KT, Wi]. In this connection one should mentionWitten's paper [Wi], where it is shown that the Jones polynomial and itsgeneralizations are related to the topological Chern- Simons action.In the present paper we introduce the so- called coloured ribbon graphs inR 2 x [0,1] and define for them Jones- type isotopy invariants. Ribboness meansthat the role of vertices of graphs is played by small plane squares with twodistinguished opposite bases, the role of edges is played by thin strips (or bands)whose short bases are glued either to distinguished bases of the squares or to

Our approach to colouring is based on the Drinfe d notion of a quasitrian-gular Hopf algebra (see [D r, RS, Re2]). For each quasitriangular Hopf algebra Awe define ^- coloured (ribbon) graphs. The colour of an edge is an yl-module. Thecolour of a vertex is an v4- linear homomorphism intertwining the modules whichcorrespond to edges incident to this vertex. The category of an ^.-coloured ribbongraph happens to be a compact braided strict monoidal category in the sense of[ J S ] .

If A satisfies a minor additional condition, then we construct a canonicalcovariant functor from the category of ,4- coloured ribbon graphs into the categoryofy4-modules. In the case A = Uq(sl2) this functor generalizes the Jones polynomialof links.

We would like to emphasize that coloured ribbon graphs are not merelytopological objects but rather mixed objects of topology and representationtheory. For some algebras a purely combinatorial description of 4-modules and^4- homomorphisms is available (via Young tableaux, Young diagrams, etc.). N o t ealso that for 3- valent graphs the colours of vertices are nothing else but theClebsch- Gordan projectors and their linear combinations (cf. Sect. 7.2).Planof the Paper. In Sect. 2 we recall the notion of a braided monoidal categoryand related notions. In Sect. 3 we discuss quasitriangular and ribbon Hopfalgebras. In Sect. 4 we introduce yl-coloured ribbon graphs. In Sect. 5 we state andprove our main theorem on the functor from the category of graphs in the categoryof modules. In Sect. 6 we collect some applications and variations of the theorem.Section 7 is concerned with examples and further comments on the theorem.

2. Compact Braided Categories2.1.MonoidalCategories(see [Ma, JS]). A monoidal category C= (C o , , /, , r, )consists of a category Co, a covariant functor (the tensor product) : C o x Co- +Co, an object / of Co, and natural isomorphisms

= rv:VI- *V,


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Ribbon Graphs Derived from Their Invariants 3

(where U,V,W are arbitrary objects of Co) so that the following two diagramsc o m m u t e : (UV)(WX)

U(V(WX)) (2.1.1)(U(VW))X

r*i\ / , (2.1.2)UV

Th e diagram (2.1.1) is called the pentagon for associativity and the diagram(2.1.2) is called the triangle for identities. C is called strift if each auvw, rv, (v is anidentity morphism in C.

A braiding for C consists of a natural family of isomorphismsc = cuv:UV- >VU

in C such that the following two diagrams commute:U(VW) U (F W) Uy x

(UV)W V(WU), (2.1.3)c 0 1 \ x 1 ^{VU)Wa- ^V(UW)(UV)W - ^ W(UV)

U(VW) (WU)V. (2.1.4)

U(W V) - ^ (C/PF) FAs noted in [JS] (2.1.4) amounts to (2.1.3) with cuv replaced by e^J. A monoidalcategory together with a braiding is called a braided monoidal category. Below (inSects. 3 and 4) we give two fundamental examples of braided categories; for furtherexamples see [JS].We call a monoidal category C compact if for all objects F there exists an objectF * and arrows ev: F F* -> 7 , nv: 7-> F* F such that the following compositesare identity morphisms:

g) F) -^* (F F*) F - ^U I V e > V,F * - ^- U 7 F * - ^ (V*1^) F* > F * ( F F*) - ^ > F*7 r > F *.

Remark that we do not require F ** = F


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4 N. Yu. Reshetikhin and V. G. Turaev2.2. Functorsof Monoidal Categories. Let C, C be two compact braided strictm o n o i d a l categories. Recall that a covariant functor F:C-+C associates with eachobject (respectively morphism) X of C an object (respectively morphism) F{X) of Cso that F(idx) = idF{X) for any XeOb(C) and F(fog) = F(f)F(g) for any twomorphisms g:U-+V,f: V- * W of C. We say that F preserves the tensor product ifF( I c ) = I and for any two objects (respectively morphisms X, Y of C F(XY)= F(X)F(Y). We say that F preserves braiding if F(cuv) =cF{U)F{V) for any twoobjects U, V of C. We say that F commutes with the compact structures if one canassociate with each object V of C an isomorphism Vv: (F(V))*- +F(V*) so that F(nv)= (V v id F{V)) o nF{V) a n deF(V) = F(ev) (id (K )(g> F F ) .2 .3 . Strict Extension of Monoidal Categories.Each monoidal category C isequivalent to a strict monoidal category C D (cf. [Ma]). The objects of C are finitesequences Vl9...9Vk (including the empty sequence 0) of objects of C. Themorphisms from (Vl9..., Vk) into ( W l9..., W)are just C-morphisms

(Here if k= 0 then Fx... W into the same morphism / considered in thecategory C D . The projection p transforms each object (Vu..., Vk) of C into(.. .(F i F2) . . . (x) V

k_

x) F

kand transforms each morphism of C

Din its obviousc o u n t e r p a r t in C. Clearly, p o i= idc. Both i and p are equivalences of categories.(Recall that a co variant functor G: C- *C is called an equivalence of categories C,

C if for any objects V9WofC,G maps Mor/ (K W) bijectively onto Mor f{G{V\G{W))and each object of C is isomorphic to G(V) for someFeO?(C).Speaking non- formally one may say that C has the same morphisms as C butm o r e objects: each decomposition V=(...(VV2).. Vk- )Vk of VeOb(C)gives rise to an object (Vl9 ...,Vk) of C .It is not difficult to show that each braiding in C (respectively a compacts t ruc tu r e in C) extends naturally to a braiding (respectively a compact structure) inC . In particular, (Vl9..., F k)* = (F k *, . . . , Vf).

3 . Quas triangular and Ribbon Hopf Algebras3 . Quasitriangular Hopf Algebras(see [ D r ] ) . Let A be a Hopf algebra with thecomultiplication zl, counit and antipodal mapping s. Let R be an invertiblee l emen t of A2 = AA. Denote the permutation homomorphismab\>ba:AA-^AA by P. The pair (A, R) is called a quasitriangular Hopf


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Ribbon G r a p h s Derived from Thei r Inva r i an t s 5algebra iffor any aeA,

'(a) = R(a)R- \ (3.1.1)( i d A) ( R ) = R 13 R 23 , (3.1.2)( id A A ) ( R ) = R 13 R 12 . (3.1.3)

H e r e A'= ?oA, R12 = R1 eA3, R13 = (idP)(Rl2% R23 = 1R. No te that ifRis a(finite) sum , /?f with ? j8f e4, th en i^ x2 = f ft 1, Kj3 = Y t 1 ft,

The element # iscalled the universal i- mat rix ofA. This element satisfies theYang- Baxter equation#12^13 23=^23^13^12 (3.1.4)

I n d e e d , it follows from (3.1.1) and (3.1.2) that

Let usnote the following importan t equalities:( i d ) ( ) = ( id ( ) ) ( # ) = l, (3.1.5)

( s i d ) (R) = ( i d s~ 1) (R ) =R- 1, (3.1.6)(ss)(R) = R. (3.1.7)

I n d e e d , from the axiom ofcounit wehave ( (x)id)zl= id = (id(x) )A ThereforeR = ( id id) (A id) (R) = ( id id) {Rt R 23 ) = ( id) (R) R,

Since R is invertible, these equalities imply (3.1.5). Toprove (3.1.6) note that ifm: AA-+A isthe algebra multiplication th en the axiom ofantipode states thatm(s id) ( ( a ) ) = m ( i d 5) ( ( a ) ) = (a)A (3.1.8)

for any aeA. Put m 1 2 = mid: A*->A2 and m 2 3 = idm:A3->A2. We havem 12 ((s id id) (A id) ()) = ( id) {R) =1,

( N o t e that s~x and 5 are antiautomorph ism ofA) On the other hand (3.1.2, 3.1.3)imply thatm 12((s id id) (A id) ( R ) ) = (5 id) (R) R, (3.1.9)

1 (3.1.10)Clearly, (3.19), (3.1.10) imply (3.1.6). Also, (3.1.6) implies (3.1.7).

With each quasitriangular Hopf algebra (A9 R) we associate the element u =uA= s{ ^c/Lb where as above R=Yuai i. This element isinvertible and satisfies the


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following identities (see [ D r ] ) :s2(a) = uau~ for all aeA, (3.1.11)

us(u) e (centre of A), (3.1.12)R2 1, (3.1.13)

(3.1.14)Remark. One may more generally define topological quasitriangular Hopfalgebras so that the comultiplication A takes values not in A A but rather in itsce r t a in completion A A. One also has to assume that Re A A. The properties ofquasitriangular Hopf algebras mentioned above may be extended to thetopological case (cf. [ Dr ] ) .3.2. Category Rep A With each algebra A one associates the category Rep>4 of itsfinite dimensional linear representations. The objects of Rep A are left ^4-modulesfinitely generated over the ground field. The morphisms of Rep,4 are ^- linearhomomorphisms. Note that the action of A in ^4-module V induces an algebrahomomorphism A^EndV, denoted by v.Let {A,R) be a quasitriangular Hopf algebra. We shall provide Rep A with astructure of compact braided monoidal category. The comultiplication A in Ainduces the tensor product in Rep A: for objects V, W their tensor product is theordinary tensor product V W of vector spaces equipped with the (left) 4-actionby the formula vw(a)= (QvQw) (^( )) fr a e A. The unit object / ofR e p A is theground field equipped with the action of A by means of the counit of A. Thehomomorphisms a, r, t (see Sect. 2.1) are given respectively by the formulas(xy)z\- ^x(yz\ xj\- +jx, jx\- >jx. Thus Rep A is a monoidal category.

I t follows from (3.1.1) that for any ^- modules V, W the mappingcv W = PV>Wo(QvQw){R):VW-+WV,

where P is the permutation homomorphism xy\- ^yx, is an yl- linear homo-morphism, i.e. a morphism of Rep A. The formulas (3.1.2), (3.1.3) imply thecommutativity of the diagrams (2.1.3), (2.1.4) so that Rep^4 becomes a braidedmonoidal category.F o r any - module Vwe provide the dual linear space F v with the (left) action ofA by the formula

(3.2.1)for aG A. Denote by ev the canonical pairing (x, y ) * -*y ( x ) : V F V- > I . Denote by nvth e homomorphism I- +V*V which sends 1 into Y^eeb where {et} is aniarbitrary basis in V and {e1} is the dual basis in F v. It is easy to deduce fromdefinitions that ev, nv are [- linear and that the conditions of the compact categoryare fulfilled.

Therefore, Rep ,4 is a compact braided monoidal category. Its strict extension(Repyl)D will be denoted by Re D(^4).Remark.All objects of R e p A are reflexive up to a canonical isomorphism. Indeed,QVw{a) = v(s2{a)) = uv v(a)uv S where we have identified the vector spaces F w and


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Ribbon Graphs Derived from Their Invariants 7V and where uv = v( u ) . Therefore, the com position of the canonical identificationK

W- F and the homomorphism x^U

1x\ V-*V is an ,4- isomorphism.3.3. Ribbon Hopf Algebras. Let (A, R) be a quasitriangular Hopf algebra with the

comultiplication A, counit and antipode s. P ut u = uAeA. We call the triple (A,R,a central element v e A) a ribbon Hopf algebra ifv2 = us (u ) , s(v) = v, ( v ) = l, (3.3.1)

= (R2lR 2y1(vv), ( 3 . 3 . 2 )where R12 = R and R21 = Perm( ). Since u is invertible, > is also invert ible in A.There exist quasitriangular Hopf algebras which do not contain v as above. Iffexists it may be non- unique. To describe n on- uniqueness of v we define the abeliangroup E(A)CA consisting of central elements EeA such that

2 = 1, s() = , e() = l, J( ) = . (3.3.3)I t is obvious that for any ribbon Hopf algebra (A,R,v) and any EeA the triple{A, R, Ev) is a ribbon Hopf algebra iff E e E(A).

E a c h quasitriangular Hopf algebra (A9 R) canonically extends to a ribbon Hopfalgebra ( ,R, v). Namely, let be the module whose elements are formalexpressions a + bv with a.beA. We provide A with the Hopf algebra structure byt h e formulas

( + bv) {af + V ) = (ad + bb'us(u)) + (ab' + ba')v,

5( + bv) = s(a) + s(fe)t?, (a+ ;) = (aWe identify A with a subset of A by the formula a = a + 0v. Clearly, Re A ACAA.

3.4. Theorem. (A, R, v) is a ribbon Hopf algebra containing A as a Hopf subalgebra.Proof. The equalities (3.1.1-3.1.3,3.3.1,3.3.2) follow directly from definitions. Thuswe have to check only the axioms of Hopf algebra for A. The associativity of Afollows from (3.1.12). Clearly, e (centre of A). Let us show that :A- +AA is analgebra homomorphism. It follows from (3.1.1) that for any aeA,

Since s is a coalgebra antihomomorphism,A(s{u)) = Perm((s(x) s) ( ( u ) ) ) = (s(u)s(u)) (ss) (R2iR2)~ i

= (s(u)s(u))(R21R12)~ i.[The last equality follows from (3.1.7).] Therefore,

(us(u)) = (R21R2)- \us(u)us(u))(R2R12)- ^N ow we can prove that for any a, b, a\ b'eA,

((a+ bv) {af + Vv)) = (a + bv)A{d + b'v).


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8 N. Yu. Reshetikhin and V. G. TuraevI n d e e d

= {aa' + bb'us(u))+ A(ab' + baf) {R2Rl2)~ \v v)= A(aa') + A(bb')(R21R12y2(us(u)us(u))

+ A(abf){R21R12y(v(S) ) + A(b)(R2R2y(vv)A(af)= A(a)A(af) + A{b)(R2R12)- \vv)A(b')(R21R12y1(v )

+ A{ )A{b' )+ A{bv)A{a') = A(a + bv)A(d + b'v).T h e counit axiom for A follows from (3.1.5). It is easy to verify (using 3.1.11) and thefact that the antipode ofA is an an tiau tomo rph ism of the algebra structure) that sis an antiautomorphism of A. Therefore, to check the axiom of antipode (3.1.8) itsuffices to consider the case a = . Let R= ai i. In view of (3.1.6) R^2 5( i) i a n d Ri = js(&j) Therefore,

iim(ids) ( A ( v ) ) = m(ids) ( ( R ^ R ^ 1) (vv) = v2 s( i)i8>s2( J.)s(iSi)

= v2u -

s(u-

) = v

2(s(u)u) ~

= v

2(us(u)) -

x= 1.

H e r e we have used (3.1.11) and (3.1.14). A similar argument shows thatm(sid)(A(v)) = l. Here one should use the equalities s2(( i) i = s( u~ 1) (whichfollows from 3.1.7 and 3.1.14) and 2 sfew" ^ ^ = 1 (which follows from 3.1.11).

i

3.5. Remark. If(A, R, w) is a ribbon Hopf algebra then we A C A, so that v = Ew forsome EsA, E + l. In view of (3.3.3) E generates a two- dimensional Hopfsubalgebra, say, E of A with the basis {1, E}. Clearly, A A E in th e class of Hopfalgebras.

4. Coloured Ribbon Graphs4.1. Bands and Annuli. A band is the image of the square [0,1] x [0,1] under its( s m o o t h ) imbedding in R3. The images of th e segments [0, 1] x 0 and [0, 1] x 1u n d e r this imbedding are called bases of the band. The image of the segment (1/2)x [0,1] is called the core of the band.

An annulus is the image of the cylinder S1 x [0,1] under an imbedding in R3.T h e image of the circle S1 x (1/2) under this imbedding is called the core of thea n n u l u s .4.2. Ribbon Graphs. Let fc, be non- negative integers. A ribbon (fc,/ )- graph is anoriented surface S imbedded in R2 x [0,1] and decomposed as the union of finitecollection of bands and annuli, each band being provided with a "type" 1 or 2, sot h a t the following con ditions hold :

(a ) annuli do not meet each other and do not meet bands;(b) bands of the same type never meet each other;(c) ban ds of different types may meet only in the po ints of their bases;


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(d )


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10 N. Yu. Reshetikhin and V. G. TuraevA ribbon graph is called homogeneous if in a neighbor hood of the segments

(4.2.1) thewhite side ofthe graph surface is turned upwards (i.e. to thereader). Forinstance, thegraphs in Figs. 1 and 2 are homogeneous, and those in Fig. 3 are nothomogeneous.4.3. Directed Graphs. Aribbon graph is called directed if the cores ofits bandsanda n n u l i are provided with directions. N o t e that the directed core of a band leadsfrom one base to another one. The former base of the band is called initial, thelatter one is called final.

Two directed ribbon graphs , are called isotopic if there exists a (smooth)isotopy ht:R2x [0,1] - *R2 x [0,1] ofthe identity mapping ho=id so that each htisa self- diffeomorphism of R2 x [0,1] fixing the boundary pointwise, and htransforms onto preserving the decomposition into coupons, ribbons andannul i , and preserving the directions of cores and the orientation of the graphsurface. Isotopy is clearly an equivalence relation.

With each directed ribbon (/c,/)-graph we associate two sequences x( %..., k( ) and \ % ..., \ ) consisting of 1. Here ) = 1 if the segment[i (1/4), i+ (l/4)] x 0 x 0 is the initial base ofthe incident ribbon, and f( ) = + 1otherwise. Similarly, j( )= - 1 if the segment Q/(1/4), j + (l/4)] xOx 1 is thefinal base of the incident ribbon and j( ) = 1 otherwise.

Let Q be a coupon of a directed ribbon graph . Denote by a= a(Q)(respectively by b=b(Q)) the number of ribbons of incident to the initial(respectively final) base ofQ. A small neighborhood ofQ in R2 x (0,1) looks as inFig. 4 where we assue that the white side of Q is turned upwards. Denote theribbons of incident to Q by B\Q\ ...,Ba(Q) and B1(Q),...,Bb(Q) in the ordershown in Fig. 4. The directions of B\ Bj are characterized by numbers (Q), j{Q)e {X ~ 1} Namely, \Q)= 1 if the base ofB* depicted in Fig. 4 is its initialbase; otherwise (Q) = 1. Similarly, fQ)= 1 if the base ofBj depicted in Fig. 4 isits final base; otherwise 7{Q) = l.

Fig. 4

4.4. Colourings of RibbonGraphs. Fix a Hopf algebra A. Denote by = (A) theclass ofv4-modules offinite dimension over the ground field. Denote by N = N(A)th e class of finite sequences {V ^}, ..9(Vk, k) where Vl9...,Vke and 1?..., fc= 1. With such a sequence = ({V1, 1\...,(Fk, k)) we associate the ^- moduleV( ) = VI1... (x) Vk\ where V = V,V~ =V , the tensor product is taken over theground field, and the ^4-module structure in V( ) is determined by the comulti-plication in A.

A colouring (or an ^4- colouring) of a directed ribbon graph is a mapping which associates with each ribbon B of its "colour" (B)e and associates with


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Ribbon Graphs Derived from Their Invariants 11each coupon Q of its "colour" (Q)eomA(V( ), V( % where

= ( ( iB . iQ)) , l ( 0 ) , . . . , ( ( B b(Q)l ' = ( ( (B \Q ) ) , (

There is a natural isotopy relation in the class of coloured directed ribbongraphs, defined as in Sect. 4.3 with the additional condition of preservation ofcolouring. By a CDR- graph we shall mean a coloured directed ribbon graphconsidered up to isotopy. By a HCDR- graph we shall mean a homogeneous CDR-graph (see Sect. 4.2).

Some examples of CDR- graphs and the notation for these graphs are presentedin Figs. 2 and 3. The homogeneous ( k , / )- graph with one coupon of colour /presented in Fig. 2 will be denoted by (f, , f); here = ( ( V , ) , ...,(Vk, k))eN, ' = ((V1, 1),...,(V f, '))eN, where Vb V are the colours of the correspondingribbons and b j are directions of their cores: + 1 means down, 1 means up.4.5. Category H CD R ( ^ l ) . Let A be a Hopf algebra. We define the category ofhomogeneous ^- coloured directed ribbon graphs


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12 N . Yu. Reshetikhin and V. G . Turaev

Fig. 6

Fig. 7

4.6. Theorem. $? = HCDR(^4) is a compact braided strict monoidal category.Proof. Obvious.

We m ay similarly i n t r o d u c e the category C D R ( y l ) of C D R - g r a p h s . In contrastt o H C D R ( ^ ) th e objects of CDR(v4) are sequences (Vl9 l9 vj, ..., (Vk, h9vk), whereV9...9VkE (A) and i9 vf = 1. The numbers {vf} are responsible for the up/downposition of the white side of the border ribbons n e a r the border segments.T h e o r e m 4.6 may be easily extended to the case of C D R ( A ) .

5. The Main Theorem5.1. Theorem. Let (A, R, v) be a ribbon Hopf algebra over the field I. There exists aunique covariant functor F = FA:HCDR(A)^RQp A which has the followingproperties:

(i) F preserves the tensor product;(ii) For any A-module V of finite dimension over I the functor F transforms the

object (V, ) into V\ where V = V andV~ 1 =F v ;(iii) F transforms the CDR-graphs r\v, >r\v,XViW (see Fig. 2) respectively inthe canonical pairing

in the pairing

and in the homomorphism

where

(5.1.1)

(5.1.2)

(5.1.3)


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Ribbon G r a p h s Derived from Their Invariants 13(iv) F transforms each C R-graph (f , r) (see Fig. 2) into thehomomor-

phism f:V{ )- +V{ ') (cf Sect. 4A.).Note that according to (i, ii) F transforms each object (Vu 1%...,(Vk, k) ofHC R(A) into the object Vf1,..., Vkkof Rep D A. The action of F on morphisms ofHCDR(^) (i.e. on H CDR- graphs) is more subtle. U sing the identities F( o )

= F ( ) oF ( ) and F{ F V is the multiplication by v~ u).

To prove Theorem 5.1 we need the not ion of generators of HCDR(^4). We saythat certain HCDR- graphs {Xj generate HCDR(^) if each HCD R- graph may beobtained from {Xt} and {Iv, I }ve (A) using composition oand tensor product .This notion is very similar to the notion of generators of a group.5.2. Lemma. The graphs {r\V9 ^> , >r\ v, ^Jv, Xv w, X W\V, WGA(A)} and{ {f^ ')\ , rfeN(A\ feHomA(V( \ V( '))} generate HCDR).

This lemma follows from the obvious fact that each graph (considered up toisotopy) may be drawn so that it lies in a standard position with respect to somesquare lattice in the plane of the picture. The words "standard position" mean thatin each square of the lattice the picture looks as in Fig. 2.

Following the lines of group theory we may consider relations betweengenerators. The following lemma provides us with a "generating set" of relationswhich meansn that all possible relations between our generators follow from thegiven ones. We shall present the relations in a pictorial form.5.3. Lemma. Figure 8 presents a generating set of relations for the generators ofH C D R ( ^ ) given by Lemma 5.2.

Note that all the relations Rel 1- R e l13 of Fig. 8 may be written purelyalgebraically. For example, the first relation Relx means that


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14 N . Yu. Reshet ikhin a nd V. G. Tura ev

Rel7

ReL

ReL


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Ribbon Graphs Derived from Their Invariants

15

Fig. 8

Rel

Th e main point in the proof of Lemma 5.3 is to show that if two isotropicgraphs are expressed via our generators then these two expressions are equivalentmodulo our relations. In the case of graphs with no coupons this is known (see[FY, Tu2]) . To handle coupons note that each graph may be positioned so that allcoupons are parallel to the plane of the page. This condition may be violated in thecourse of isotopy. However we may always assume that in the course of isotopyeach coupon moves as a solid rectangle. Thus isotopy of a coupon gives rise to ap a t h in SO(3) starting in 1 and finishing in SO(2) (since in the end all coupons areagain parallel to the fixed plane). All such paths may be deformed to SO(2). Thisenables one to change isotopy so that in its course all coupons are always parallelto the plane of the page. Now it is easy to see that in a neighbourhood of a couponsuch isotopy changes the picture as in Fig. 8, Relx i- Re^3. This implies Lemma 5.3.5.4. Proof of Theorem 5.1.Uniqueness of F follows from Lemma 5.2. Indeed, thevalues ofF on n*v,


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16 N. Yu. Reshetikhin and V. G. TuraevThe operator counterpart of Rel8 states that ~ o + = idF. We shall prove that is the multiplication by

x. This would imply Rel8. Let { j} be a basis of K A directcom pu t a t i on shows that

and

We have

= v.To prove the equalities corresponding to Rel9, Rel10 fix bases {et}, {f3)respectively inV,W and dual bases {e1}, {fj} respectively in F v and W. The left-h a n d side of Rel9 gives rise to an operator : Fv(x) W ^> F v W which is computedto act as follows:

j )= k S , q

ek(s2(aq)e,)-ei(s(ap)ey( l p)(fj)k,S,p,q

The last equality follows from the fact that s(ocp)s2(aq) q p = (sid) ((sid) ( ) ) = 1.

Thus = id which proves the operator version of Rel9. Similarly, the operator f: V W^> V W corresponding to the right- hand side of Rel10 is the identity:) = f \ s{q)f eY f \ s2{ qpk,,p,q= fJ(s2^ P)s(aq)fM q P)(e)f=eifJ.P,q,t

The operator counterparts of Rel l l 5 Rel12 follow directly from the assumptiont h a t the colour of the coupon is ^4-linear.Let us check the operator equality corresponding to Rel13. Fix a bases {et} inV...Vb and a bases {) } mW 1...W a. The operator

corresponding to the left- hand side of Rel13 acts as follows: ( e j ) = e \ u - e '

jk

H e r e we have used our assumption that the colour / of the coupon is ^- linear.


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Ribbon G r a p h s Derived from Their Invarian ts 176. Applications and Generalizations of Theorem 5.16.1. Isotopy Invariants of Coloured Links. A link in R3 is a finite set of disjointoriented smooth circles imbedded in R3. The circles are called the components ofthe link. Fix a ribbon Hopf algebra A. An ^4-colouring of the link is a functionassociating with each component an 4-module of finite dimension over the groundfield.A link is called framed if each component is provided with an integer. Asusual, one introduces the isotopy relation for coloured framed links [ R e 2 ] .Each framed link L determines a ribbon (0,0)- graph L, consisting of severalannuli. Namely, let L l 5 ...,Lfe be the components of L and let m 1? ...,mk be theassociated integers. Let A{ be theannulus in R3 with the oriented core L t and suchthat the linking number of two circles making dAt is equal to mv Assume that theannuli A1,...,Ak are disjoint. Compressing (J A{ into R2 x (0,1) we get the desiredribbon (0,0)- graph L. Each colouring of L gives rise to a colouring of L in theobvious fashion. The formula L\ - > L defines a bijective correspondence betweenthe isotopy types of coloured framed links and the isotopy types of HCDR- graphswhich consist of annuli.

N o t e that for an arbitrary HCDR- graph which has zero number of borderribbons the operator F( ) is a linear endomorphism of the ground field, i.e. themultiplication by an element of thisfield.This element is an isotopy invariant of .In particular, with each framed coloured link L we may associate its isotopyinvariant F( L) which is an element of the ground field. For instance, ifA = Uh(sln((E)\ (cf. Sect. 7), if all integers which determine the framing of L arechosen to be zero, and if the colours of all components of L are chosen to be thevector representation of A in C" then the invariant F( L) depends on q = eh as apolynomial. Up to a reparametrization this polynomial equals PL(qn,q q~ ),where PL is the 2- variable Jones- Conway polynomial of L. In the case n = 2we getthe Jones polynomial of L. For A=UhG, G = so(n\ sp(2k)a similar constructiongives an infinite set of 1-variable reductions of the 2- variable Kauffmanpolynomial of L (see [T uJ).The invariants corresponding to the exceptional Liealgebras and to the spinor representation of Uhso(n) seem to be new (see6.2. EvenGraphs. A HCDR- graph is called even if it may be obtained from graphsv v, r\v, Xyt w, (f9 , ') and the graphs Zyt w shown in Fig. 9 by applications ofthe operations (x) and .

There is a simple necessary and sufficient criteria which enables one to decide ifa given HCDR- graph is even. Draw a picture of this graph so that all coupons arerepresented by rectangles with horizontal bases (as in Figs. 1,2 and 4). The bases ofribbons will be automatically horizontal. Assume also that white sides of thecoupons are turned up and that all ribbons lie "parallel" to the plane of the picture,i.e. have no twists (cf. Fig. 1). Now, moving along the directed core of a ribbon or anannulus one meets a certain number of local extremums rv, *^? v>, KJ of the heightfunction. Compute the number of left oriented extremums \J, * lying on the

Y Y YFig. 9 V V \


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ribbon (respectively the annulus). It is a simple topological exercise to show that aH C D R - g r a p h is even if and only if these numbers are even for all ribbons andannu l i of the graph. For example, the 1-ribbon graphs * \ \j do not satisfy thiscriteria and therefore they are not even. The HCDR- graph L corresponding to aframed coloured link L is even if and only if all framing numbers of L are odd.

Let ( ,R,v) and ( ,R,v') be two ribbon Hopf algebras with the sameunderlying quasitriangular Hopf algebras. Let F, F' be the corresponding functors

-^RepQ] A It follows directly from definitions that the values ofF , F' on> Xv,w> {f, , ') are the same. Using the assumption that v, ' lie in thecen t r e ofA one easily shows that F(Z w) = F'(Zyt w). This implies that on the set ofeven graphs we have F= F. In other words, for even graphs the functor F does not

depend on the choice of v.Moreover, for even ^- coloured HCDR- graphs we can define operators F

without any use of v, i.e. for such graphs we may use an arbitrary quasitriangularHopf algebra (A,R).Here is a sketch of the construction. First, extent the groundfield to an algebraically closed field. Note that each representation of A may beextended to a representation of A (see Sect. 3.3); we use that for any finite-dimensional operator over an algebraically closed field there exists an operatord acting in the same vector space such that d1 ^) and d commutes with alloperators commuting with 21 ) . Thus ^- colouring giver rise (non- uniquely) to an^[- colouring. Apply now the functor F = F and the reduction homomorphismRepDy4- >Rep A The composition restricted to even graphs does not depend onth e choice of square roots.6.3. Theorem. (An extension of Theorem 5.1). Let (A,R,v) be a ribbonHopfalgebra. Assume that for each A-module V we have fixed two nvertible A-linearoperators y, y'.V- ^V. There exists a unique covarant functor F :C R(A)- > R e p D ;4 which has the following properties: (i) on the subcategory HCDR(^4) ofC R(A) F equals F ; (ii) F preserves the tensorproduct; (iii) for any A-module V

F (Jv) = v and F (Jy) = y .We give here a construction ofF . Note that an arbitrary CDR- graph with kbot tom ends and t top ends may be uniquely represented in the form

(/ . . / 0 (J l h) > where is a homogeneous CDR- graph and each is either v or J v, = + 1 , each Jf is either v or Ky, e = + 1 (see Figs. 2 and 3). PutF (Jy) = y,F (KV) = ( y) - \ F (Iy) =idyE, Where V+ = V, V~ = V\ Put

F( ) =(Foil1)... F ( I ' ) ) o F( ) o (FeilJ... F ( I k ) ) .All the properties of F easily follows.6.4. Remarks. 1. It is obvious that the operator invariant F( , ) of a directedribbon graph with a colouring is additive with respect to the colours of thecoupons of . This invariant is also additive with respect to the colours of thoseribbons which connect 2 distinct coupons. More exactly, if the colour of such aribbon B is a direct sum of ^- modules, V, W then we split the colours of thecoupons incident to B and obtain thus two yl- colourings, say, X and " of .Here, ' and " coincide outside B and the incident coupons and f(B)= V9 "{B)= W.


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Th e n F( , ) = F( , ') + F( , "). This follows from the isotopy invariance ofF( , \ since B is isotopic to a standard plane vertical ribbon for which theassertion is obvious. Similarly, the invariant F( , ) is additive with respect to thecolours of annuli and of those ribbons which connect a coupon with one of theborder segments (4.2.1). Note also that each annulus of may be replaced by twocoupons and two ribbons without changing F( , ) (see Fig. 10).

2. Let be a HCDR- graph and let the colour of a ribbon B of is the tensorp r o d u c t V W of two ^4-modules, V, W . Cutting out B along its core we get two newribbons which make together with \B another DR- graph, say, F'. The colouringof obviously induces a colouring of f. Note that looking on B from its white sidewe see one of the new ribbons to lie to the left of the directed core ofB, the other onelying to the right; we colour them respectively by Vand W . It turns out that F( r)= F( ). This follows from 3.1.2, 3.1.3; these equalities directly imply equalitiespresented in Fig. 11. A similar assertion holds for each annulus coloured by atensor product VW\ We may replace such an annulus by two subannuli,obtained by cutting along the core, and coloured V and W.Such a replacementdoes not change F( ).

3. The functor F:HCDR(,4)- >RepD,4 is surjective since each morphismf: - + ' of Rep G A is the image of the graph (f, , ') . Moreover, / may be alwayspresented by a 3-valent HCDR- graph, i.e. a graph whose coupons are all incidentto ^ 3 ribbons. This follows from the fact that the local operation on the H C D R -graphs depicted in Fig. 12 preserves the image of the graph in Rep ^4 anddecreases the valence of the coupon by 1 by virtue of introducing an additionalcoupon of valence 3. Note that in Fig. 12d is the identical endomorphism ofVk^1Vk and the colours / , / ' are equal.

Fig. 10

Fig. 11V V W V V W V W V V W V

Fig. 12


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4. To colour a directed ribbon graph is a problem in itself. It is natural tocolour such a graph in two steps: first ribbons and t h en coupons. The secondstep is somewhat more complicated as it is not easy to describe theyl-homomorphisms in a combinatorial language. The following construction givesa useful method of colouring coupons.

Assume that the ribbons of are coloured. Assume that with each coupon Qof we have associated a DR- graph Q with coloured ribbons so that the borderribbons of Q have the same directions, colours, and orientations (white/black) ast h e ribbons of incident to Q. Gluing Q instead of Qwe get a "simpler" directedribbon graph with coloured ribbons. This in principle enables us to reduce theproblem of colouring coupons of to the same problem for simpler graphs { Q}.F o r instance, in many cases we may take Q without any coupons at all. (Ribbongraphs without coupons are called ribbon tangles.) This construction suggests aspecific point of view on coupons: one may think that coupons are black boxeshiding inside some blocks { Q}.7. Examples7.1. Algebra Uh&. The theory of quantum groups developed in [Drl, J] givesnontrivial examples of (topological) ribbon Hopf algebras. We mean the quantizeduniversal enveloping algebras of simple Lie algebras, called also briefly quantizedLie algebras.

Let be a simple Lie algebra. The quantized Lie algebra Uh( is the associativealgebra over the ring of formal power series C[ ] generated (as an algebracomplete in the ft- adic topology) by elements Hb Xf+, Xf, = l , . . . , r = ranksubject to the following relations:

[Hb Hj] = 0, [HbX]

J o ( " 1 ) l [ l ] ( ' ^ ( X i where { f}JL1 are the simple roots of; ( f, o C j ) is the value of the canonical bilinearform in the root space and

r h n \

H -P u t U = l/ h(5. The algebra U is provided with the structure of (topological)ribbon Hopf algebra as follows. The antipodal mapping S:U-^U and the counit

: 1/->


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D e n o t e by UU the completion of UQ jU in the /z-adic topology. Thecomultiplication A :U-^U U is given on generators by the formulas

?)= Xt exp fc H) + exp

The universal JR-matrix ReU = uexp( - h iil- (7.1.1)I t is straightforward to show that i?satisfies relations 3.3.1, 3.3.2 so that the triple(U,R,v) is a (topological) ribbon Hopf algebra.7.2. Representations of U and U- Colourngs of Graphs. There exists an isomor-phism of C[/ i]- algebras U= Uh( -+(UQ ) [ ] identical an the Cartan subalgebra /generated by Hl9 ...,Hr (see [ D r l] ) . This implies that i.h.w. (irreducible highestweight finite dimensional) / - modules have the same structure as the correspond-ing - modules, i.e. they are identical as linear spaces and have the same weightdecompositions with respect to / (see [L, Ro]). Finite dimensional representationsof U are completely reducible. In particular for any i.h.w. ^/- modules V, V wehave

(7.2.1)

where W v is a vector space over (C, Fv( ) W v is the primary component of F v inV V with multiplicity \mW^ .F o r the algebra U = Uh& we have a more or less satisfactory combinatorialdescription of [/ - colourings of ribbon graphs. The i.h.w. (/ - modules are para-metrized by the highest weights of (5. Thus we can colour ribbons by associatingwith them some highest weights , ,v,... of. Remark 6.4.3 show that the processof colouring coupons may be reduced to the case of 3-valent coupons. Consider the

Fig. 13


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3-valent coupons with coloured ribbons depicted in Fig. 13. The colours of suchcoupons are linear mappings respectively.

/ : F v - > F F , g:VV-+V\Such homomorphisms bijectively correspond to elements of W v (respectively( W v ) * ) . There are rather explicit parametrizations of the elements of W v . In thecase (5 = sln the simplest parametrization is given by the Young tableaux.7.3. Homomorphsm w. Let V be the finite dimensional i.h.w. representation ofU=Uh& with the highest weight. Let (Vf be the dual representation (3.2.1). Put * = wo(/), where w0 is the element of the Weil group of maximal length withrespect to the Bruhat order. There is an 17- linear isomorphism

Thus for any aeU,

Since WQ = 1 we have (A*)* = 1.F o r representations V with multiplicity free weights acts in the weight basis byth e formula [Rel],_hc h {Q'W 2 2 / 2\ e Wo9 ( $ 2 = 1.

Th e action of w in arbitrary V may be defined using imbeddings of V in thetensor powers of the vector representation of (5. (For the case = sl2 see [KR]).Assume that we have a HCDR- graph without coupons and that is thesame graph in which the directions of several ribbons and annuli have beenreversed and their colours have been changed by the rule i *= wo( ).Thenon e may use w to compute from F( ). In particular, for generators of thesubcategory of H C D R ( t / ) consisting of graphs without coupons we have

I t is easy to extend these formulas to all generators of H C D R ( l / ) if one agrees tochange not only the colours of ribbons and annuli but also conjugate by w thecolours of coupons.Remark.The homomorphisms {xv} suggest the extension UhQ of t / h(5, generatedby a certain element w subject to relations

= eh,


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Ribbon G r a p h s Derived from Thei r Invariants 23where isthe non- trivial involution ofthe Dynkin diagram. The formulas

A(w) = R~1(w


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24 N. Yu. Reshetikhin and V. G. Turaevs i o n a l for \ j jj\ t^j ji +j2 a n d a r e equal t o zero for o t h e r j . In o t h e r words

2j=2j1 +2j2{mod2)Therefore, the ^s/ 2- colouring of3-valent ribbon graphs is extremely similar. Itsuffices to associate with each ribbon and integer or half- integer j and with eachc oup on a complex number. Ofcourse, the invariant F ofthe graph maybe non-zero only iffor every coupon t h e 'colours j1J 2J ofthe adjacent ribbons satisfy theinequalities \j- j2\ j jjtj2, 2j = 2j1+ 2/ 2(2).Th e modules Vj are self- dual: (V*)v VK This implies that the correspondinginvariants ofcoloured ribbon tangles do not depend on the choice ofdirections ofribbon cores and annuli cores. This generalizes the fact that the Jones polynomialof a framed link L does not depend on the choice ofo r i e n t a t i on . Indeed, ifall annuliof the graph L (see Sect. 6.1) are coloured by V/ 2 we get exactly the Jonespolynomial. Also, ifone colours allthe annuli of L by V1 one gets a one- variabler educ t ion of the Kauffman polynomial.

Since the modules {Vj} have the preferred bases one may study thetransformation matrix of (7.4.1). F orany j l 9 2 9j 3 and m l5 m 2, m 3 with 1^

JdJ 1lm 1m 2m - t

x l Ygrim+j+1)

These numbers are called q 3/- symbols (see [KR,K]). They constitute thetransformation matrix for (7.4.1):for \j1j2\


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Ribbon Graphs Derived from Their Invarian ts 25

choose in Rep A a "multiplicative bases" consisting of finite dimensional irredu-cible A - modules {V} . The commutativity morphism VV-+VV and theassociativity morphism (VV)Vv ^>V(VVv) are given by a calculusgeneralizing the Racah- Wigner calculus for groups. The compositiono opoF:C R(A)- +C {A) is a representation of HCDI l) in C (A) realizedvia Racah- Wigner ^-coefficients (the case A = Uhsl2was treated in [KR]).A functor similar to io pF was constructed in [MS] using fusion ruleconstants in the conformal field theory. This similarity once more suggests a closerelationship between the conformal field theory and the theory of quantumgroups.

References1. [Br] Bremmer, M.R., Moody, R.V., P atera, J.: Tables of dom inant weight multiplicities for

representation of simple Lie groups. New York: Dekker 19842. [D ] Deling, P., Milne, J.S.: Tannakian Categories in Hodge deles, motives and simura

varieties. Lecture Notes in Mathematics, vol. 900. Berlin, Heidelberg, New York: Springer1982

3. [D r l] Drinfeld, V.G.: Proceedings of the International Congress of Mathematicians, vol. 1,p . 798-820. Berkeley, California: New York: Academic Press 1986

4. [Dr2] Drinfeld, V.G.: Quasicommutative Hopf algebras. Algebra and Anal. 1, (1989)5. [F RT] Faddeev, L., Reshetikhin, N., Takhtajan, L.: Quantized Lie groups and Lie algebras.L O M I - p r e p r i n t , E- 14- 87, 1987, Leningrad

6. [F Y] Freyd, P.J., Yetter, D .N.: Braided compact closed categories with applicat ions to lowdimensional topology. Preprint (1987)

7. [JS] Joyal, A., Street, R.: Braided Manoidal categories. Macquare Math. Reports. ReportN o . 860081 (1986)

8. [JO] Jones, V.F.R.: A polynomial invariant of Kn ots via von Neumann algebras. Bull. Am.M a t h . Soc. 12, 103-111 (1985)

9. [KT] Tsuchiya, A., Kanlie, Y.: Vertex operators in the conformal field theory on P I andmonodromy representations of the braid group. In: Conformal field theory and solvablelattice models. Lett. Math. Phys. 13, 303 (1987)

10. [KV] Kauffman, L.H., Vogel, P.: Link polynomials and a graphical calculus. Preprint (1987)11. [K] Kirillov, A.N.: ^- analog of Clebsch- Gordon coefficients for slz. Zap. Nauch. Semin.

L O M I , 170, 198812. [K R] Kirillov, A.N., Reshetikhin, N.Yu.: Representation of the algebra Uq(sl2), q-orthogonal polynomials and invariants of links. LOMI- preprints, E-9- 88, Leningrad 198813. [L] Luztig, G.: Quantum deformations of certain simple modules over enveloping algebras.

M.I.T . Preprint , December 198714. [M a] MacLane, S.: Natural associativity and commutativity. Rice Univ. Studies 49, 28- 46

(1963)15. [MS] Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Commun.M a t h . Phys. 123, 177- 254 (1989)16. [R el] Reshetikhin, N.Yu.: Quantized universal enveloping algebras, the Yang-Baxter

equation and invariants of links. I. LOMI- preprin t E-4-87 (1988), II. LOMI- preprin t E-17-87(1988)

17. [Re2] Reshetikhin, N.Yu.: Quasitriangular Hopf algebras and invarian ts of Tangles. Algebraand Anal. 1, (1988) (in Russian)

18. [RS] Reshetikhin, N.Yu., Semenov Tian- Shansky, M.A.: Quantum .R- matrices and factori-zation problem in quantum groups. J. Geom. Phys. v. V (1988)


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26 N. Yu. Reshetikhin and V. G. Turaev19. [R O] Rosso, M.: Representations irreductibels de dimension flnie du ^- analogue de algebre

enveloppanted'une algebre de Lie simple. C.R. Acad. Sci. Paris, t. 305, Se'rie 1,587-590 (1987)20. [ Tul] Turaev, V.G.: The Yang-Baxter equation and invariants of links. Invent. Math. 92,

527- 553 (1988)21 . [Tu2] Turaev, V.G.: The Conway and Kauffman modules of the solid torus with an appendixon the operator invariants of tangles. LOMI Preprint E- 6- 88. Leningrad (1988)

Turaev, V.G.: Operator invariants of tangles and R- matrices. To appear in Izv. An SSSR22. [Ye] Yetter, D.N.: Category theoretic representations of Knotted graphs in S3, P reprint23. [Wi] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. (in

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C o m m u n i c a t e d by A. JaffeReceived March 21, 1989

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N. Yu. Reshetikhin and V. G. Turaev- Ribbon Graphs and Their Invariants Derived from Quantum Groups