Kauffman bracket versus Jones polynomial skein modules
aa r X i v : . [ m a t h . G T ] F e b KAUFFMAN BRACKET VERSUS JONES POLYNOMIALSKEIN MODULES
SHAMON ALMEIDA AND R ˘AZVAN GELCA
Abstract.
This paper resolves the problem of comparing the skeinmodules defined using the skein relations discovered by P. Melvin andR. Kirby that underlie the Reshetikhin-Turaev model for SU (2) Chern-Simons theory to the Kauffman bracket skein modules. Several applica-tions and examples are presented. Motivation
In 1984 V.F.R. Jones introduced a polynomial invariant of knots andlinks [14]. Immediately after, L. Kauffman defined a similar polynomialknot and link invariant, the Kauffman bracket, in fact this polynomial isan invariant of framed knots and links [15]. Kauffman has shown how theJones polynomial of a knot can be computed from the Kauffman bracket.In 1989 E. Witten in [21] has explained the Jones polynomial by meansof a quantum field theory based on the Chern-Simons functional. The Jonespolynomial corresponds to the particular case of the Chern-Simons theorywith gauge group SU (2). By making use of physical intuition, Witten pre-dicted the Jones polynomial to be part of a more general family of knot,link, and manifold invariants. Motivated by Witten’s ideas, Reshetikhinand Turaev constructed the knot, link, and manifold invariants of the SU (2)Chern-Simons theory using a quantum group associated to sl (2 , C ) [19]. Thistheory fulfills Witten’s predictions. An analogous theory was developed forthe Kauffman bracket by Blanchet, Habegger, Masbaum, and Vogel in [1],and this theory parallels that of Reshetikhin and Turaev.Within the Reshetikhin-Turaev theory, and already present in previousworks by Reshetikhin himself, lies the Jones polynomial of framed knots andlinks, but with a slightly different normalization. This polynomial fits ex-actly the quantum field theoretical model from Witten’s paper, it is the poly-nomial that Chern-Simons theory would associate to a link whose compo-nents are colored by the 2-dimensional irreducible representation of SU (2).We will refer to this polynomial as the Jones polynomial in the Reshetikhin-Turaev normalization (or simply the Jones polynomial, when there is nopossibility of confusion). The coloring of a knot by the n -dimensional ir-reducible representation of SU (2) yields a polynomial invariant of knots Mathematics Subject Classification.
Key words and phrases.
Kauffman bracket, skein modules, Chern-Simons theory. called the colored Jones polynomial, of which the Jones polynomial in theReshetikhin Turaev normalization itself corresponds to n = 2. The con-vention is that the n th colored Jones polynomial of a knot K , denoted by J ( K, n ), corresponds to the coloring of K by the n + 1st irreducible repre-sentation. L H V
Figure 1.
A framed link in the 3-dimensional sphere S , or more generally in acompact, orientable, 3-dimensional manifold M , is an embedding of finitelymany annuli. Both the Kauffman bracket and the Jones polynomial in theReshetikhin-Turaev normalization of a knot or link in S can be computedusing skein relations, and these skein relations are quite similar. We denotethe Kauffman bracket of a link L by h L i and this version of the Jonespolynomial by J L , both in the variable t . To write down the skein relations,let L, H, V be three framed links that coincide except in a ball where theyare as shown in Figure 1. The Kauffman bracket has the skein relations h L i = t h H i + t − h V i , h O i = − t − t − . Here and below O is the unknot. The skein relations of the Jones polynomialin the Reshetikhin-Turaev normalization have been computed by R. Kirbyand P. Melvin in [16]; they are J L = tJ H + t − J V or J L = ǫ ( tJ H − t − J V ) , J O = t + t − . Note that there are two skein relations from resolving a crossing, the oneon the left is used when different link components cross, and the one on theright is used when a link component crosses itself, with ǫ being the sign ofthe crossing.A great amount of Chern-Simons theory is dedicated to the study ofthe combinatorial properties of knots and links decorated by irreduciblerepresentations of quantum groups (the so called quantized Wilson lines),and the algebraic topological concept that lies at the heart of this study isthat of a skein module. Following J. Przytycki [18], we construct the skeinmodule of a compact, orientable, 3-dimensional manifold M by consideringthe free C [ t, t − ]-module with basis the isotopy classes of framed links in M and then factoring it by the skein relations. In the case of the Kauffmanbracket we obtain the Kauffman bracket skein module K t ( M ), obtained byfactoring the above mentioned free module by the submodule spanned bythe elements of the form L − tH − t − V , where L, H, V are framed links thatcoincide except in an embedded ball where they are as described in Figure 1,and also by the relation that states that every link that contains a trivial
AUFFMAN BRACKET VERSUS JONES POLYNOMIAL 3 link component is equivalent to the same link with that component erased,multiplied by − t − t − . For the Jones polynomial, the skein module of M was defined in [13]; it is denoted by RT t ( M ) to point out that it comes fromthe Reshetihin-Turaev theory. It is defined like for the Kauffman bracket,but with the Kirby-Melvin skein relations instead. In [13] it was explainedhow several constructs of SU (2) Chern-Simons theory can be reduced tothese skein modules.For a better understanding of the need to introduce the skein modules ofthe Reshetikhin-Turaev theory, let us contrast the two skein relations in theso called “classical case”. When t = −
1, the Kauffman bracket skein relationyields the trace identity for the negative of the trace of sl (2 , C ) charactersof the fundamental group of M :( − tr ρ ( αβ )) + ( − tr ρ ( α ))( − tr ρ ( β )) + ( − tr ρ ( αβ − )) = 0 , as it has been noticed in [2]. On the other hand, the skein relation of Kirbyand Melvin yields, when t = 1, the trace identity for the trace itselftr ρ ( αβ ) − tr ρ ( α ) tr ρ ( β ) + tr ρ ( αβ − ) = 0 . In Chern-Simons theory t = e iπh , where h is interpreted depending on thecontext as either the coupling constant or Planck’s constant. Setting t = 1is equivalent to setting the coupling constant or Planck’s constant equalto zero, and this is predicted to correspond to the classical (nonquantized)situation, that is to the character variety. It is because of this physicalinterpretation, and because it is more natural to work with the trace thanthe negative of the trace, that we have proposed in our previous work thestudy of the skein modules RT t ( M ) of the Jones polynomial. The presentpaper clarifies the relationship between the two types of skein modules: K t ( M ) and RT t ( M ). 2. The main result
Let M be a compact, orientable 3-dimensional manifold, and let L be aframed link in M . Consider a compact orientable 3-dimensional manifold N such that ∂N = − ∂M , and consider the closed manifold M ∪ N obtainedby gluing M and N along their common boundary.The 3-dimensional manifold M ∪ N , being closed, is the boundary of a4-dimensional manifold W , and if we glue 2-handles to W along the compo-nents of L we obtain a 4-dimensional manifold W ′ , in which these 2-handlesdefine homology classes in H ( W ′ , Z ). Denote by tr( L ) the trace of the inter-section matrix of these homology classes. This trace is the sum of [ L j ] · [ L j ]over the components L j of L , where [ L j ] · [ L j ] is the algebraic intersectionnumber of the homology class [ L j ] defined by L j with itself.Note that tr( L ) depends on the choice of N and W , but this fact does notalter the conclusion of the following theorem, and in practical applicationsone should always make the simplest choice. SHAMON ALMEIDA AND R ˘AZVAN GELCA
Additionally, for a link L , we denote by n ( L ) the number of componentsof L . Theorem 2.1.
The equality n X k =1 c k L k = 0 holds in K t ( M ) for some Laurent polynomials c k ∈ C [ t, t − ] and some framedlinks L k in M if and only if the equality n X k =1 ( − n ( L k )+tr( L k ) c k L k = 0 holds in RT t ( M ) .Proof. All we have to show is that the statement of the theorem is invariantunder skein relations. We have to examine three cases.
Case 1.
If two components L α and L β of one of the links L k cross, thenafter resolving the crossing the number of components dropped by 1. Onthe diagonal of the intersection matrix the entries [ L α ] · [ L α ] and [ L β ] · [ L β ]disappear, and the entry[ L α ] · [ L α ] + [ L β ] · [ L β ] + 2[ L α ] · [ L β ] − n ( L k ) + tr( L k ) changes by an even number. Andindeed, the skein relation for two disjoint components that cross is the samefor the Kauffman bracket and for the Jones polynomial in the Reshetikhin-Turaev normalization. Case 2.
If a component L α of some link L k crosses itself, the crossingcan be positive or negative. Let H and V be the diagrams obtained afterresolving the crossing as in Figure 1. If the crossing is positive, then tr( V ) =tr( H ) = tr( L k ) −
1, and V has the same number of components as L k whilethe H term has one component more. Thus when passing from the Kauffmanbracket to the Jones polynomial we keep the same sign in front of H , whilewe change the sign in front of V , exactly as in the skein relation for theJones polynomial in the Reshetikhin-Turaev normalization. If the crossingis negative, then tr( V ) = tr( H ) = tr( L k ) + 1, but this time the numberof components stays the same in H and increases in V . And this is againconsistent with the skein relation. Case 3.
If we remove a trivial component, then the Kauffman bracketis multiplied by − t − t − , while the Jones polynomial is multiplied by t + t − . In this case the number of link components decreases by 1, and sothe exponent of − (cid:3) If we vary N and W we just multiply by a ± AUFFMAN BRACKET VERSUS JONES POLYNOMIAL 5
Remark . If M ⊂ S , we can choose N = S \ M and let W be the 4-dimensional ball that S bounds. Then the intersection matrix whose traceis tr( L ) is just the linking matrix of L .Note that you can swap the two relations in order to pass from RT t ( M )to K t ( M ).If we work over the field of fractions C ( t ), we obtain the following imme-diate corollary. Proposition 2.3.
The vector spaces K t ( M ) and RT t ( M ) are isomorphic.Proof. As isotopy classes of framed links span K t ( M ), we can find a basisconsisting of framed links. But then this basis is a spanning set for RT t ( M ).It is either a basis, or it contains a basis. If it is not a basis, then the basisit contains is a spanning set for K t ( M ), a contradiction. Thus any basisof framed links of K t ( M ) is a basis of framed links of RT t ( M ). Hence theconclusion. (cid:3) Applications and examples
Let us introduce the polynomials T n ( ξ ) = 2 cos[ n arccos( ξ/ n ∈ Z ,which is a normalized version of the Chebyshev polynomial polynomial of thefirst kind, and S n ( ξ ) = sin[( n +1) arccos( ξ/ / sin arccos( ξ/ n ∈ Z , whichis a normalized version of the Chebyshev polynomial of the second kind. Fora framed knot K in some compact, oriented, 3-dimensional manifold M anda positive integer m , we let K m be the framed link consisting of m parallelcopies of K , where in order to produce the parallel copies K is pushed inthe direction of the framing. Given a framed link L = L ∪ L ∪ · · · ∪ L k and a k -tuple of positive integers, ( j , j , ..., j k ), we can construct the link L j ∪ L j ∪ · · · ∪ L j k k by taking parallel copies of each component.In particular, for a knot K we can construct the skeins T n ( K ) and S n ( K )in either K t ( M ) or RT t ( M ).3.1. The product-to-sum formula and Weyl quantization.
Let usgive another reason for our focus on the skein modules of the Jones poly-nomial in the Reshetikhin-Turaev normalization. If a manifold is a cylinderover a surface, then the operation of gluing one cylinder on top of the otherinduces an algebra structure on the skein module; this is the skein algebraof the surface. Of particular interest is the skein algebra of the the torus, K t ( T ). As a module, it is free with basis ( p, q ) T , p, q ∈ Z , p ≥
0, where( p, q ) T = T n (( p/n, q/n )), with n the greatest common divisor of p and q and( p/n, q/n ) the curve of slope q/p on the torus whose framing is parallel tothe torus.As shown in [6] and [13], for both the Kauffman bracket and the Jonespolynomial in the Reshetikhin-Turaev normalization, the multiplication isgiven by the product-to-sum formula( p, q ) T ( r, s ) T = t ps − qr ( p + r, q + s ) T + t − ps + qr ( p − r, q − s ) T . SHAMON ALMEIDA AND R ˘AZVAN GELCA
For a manifold with boundary, the operation of gluing a cylinder over theboundary to the 3-dimensional manifold induces a module structure on itsskein module, over the skein algebra of the boundary. A situation that wasinvestigated in [6] and [12] is that where the manifold is the solid torus. Let α be the curve that is the core of the solid torus (the image of (1 ,
0) underthe inclusion of the boundary). The following result was proved in [12] and[13].
Proposition 3.1.
In the case of the Jones polynomial in the Reshetikhin-Turaev normalization, the action of the skein algebra of the cylinder overthe torus on the skein module of the solid torus is given by ( p, q ) T S j − ( α ) = t − pq [ t jq S j − p − ( α ) + t − jq S j + p − ( α )] , (3.1)A consequence of Theorem 2.1 is the following. Proposition 3.2.
For the Kauffman bracket, the action of the skein algebraof the cylinder over the torus on the skein module of the solid torus is givenby ( p, q ) T S j − ( α ) = ( − q t − pq [ t jq S j − p − ( α ) + t − jq S j + p − ( α )] . Proof.
Let p = np ′ , and q = nq ′ , with p ′ , q ′ coprime. Then ( p, q ) T is a lin-ear combination of links, each of which having the number of componentscongruent to n modulo 2. Each of these components is a copy of the curveof slope q/p on the torus. Because we work with the blackboard framingof the torus, each component contributes ( p ′ − q ′ + q ′ = p ′ q ′ to tr( L ),and so modulo 2, each term of ( p, q ) T contributes np ′ q ′ to the trace. And S j − ( α ) contributes nothing to the exponent of -1 in the formula from The-orem 2.1. Also, modulo 2, the number of link components in S k ( α ) is k .Thus when switching from the Jones polynomial picture to the Kauffmanbracket picture, (3.1) becomes( − np ′ q ′ + n + j − ( p, q ) T S j − ( α ) = t − pq [ t jq ( − j − np ′ − S j − p − ( α )+( − j + np ′ − t − jq S j + p − ( α )] , An easy case check shows that if p ′ , q ′ are coprime then p ′ q ′ + 1 − p ′ ≡ q ′ (mod 2), so np ′ q ′ + n − np ′ ≡ nq ′ (mod 2) and the formula is proved. (cid:3) We point out that for a curve γ , in the setting of the Reshetikhin-Turaevtheory S j ( γ ) corresponds to γ colored by the j + 1-dimensional irreduciblerepresentation V j +1 (which we denote by V j +1 ( γ )), while in the setting ofthe Kauffman bracket it corresponds to the coloring of the curve by the j thJones-Wenzl idempotent. So the relation from Proposition 3.1 has the nicerform ( p, q ) T V j ( α ) = t − pq [ t jq V j − p ( α ) + t − jq V j + p ( α )] . This equation has been related by the second author and A. Uribe [12] tothe action of the Heisenberg group on theta functions discovered by A. Weil[20], and as such to the Weyl quantization of the moduli space of SU (2) AUFFMAN BRACKET VERSUS JONES POLYNOMIAL 7 connections on the tours, and this gives a second reason for our focus on theskein modules of the Jones polynomial. More explicitly, the moduli spacein question is the “pillow case” obtained by factoring the complex plane C by the maps z z + m + ni , m, n ∈ Z and z
7→ − z . To perform geometricquantization we let Planck’s constant be the reciprocal of an even integer h = (2 r ) − , and let ζ j be the sections of the Chern-Simons line bundle overthe moduli space that are lifted to the plane as the entire functions as ζ j = √ re − j π r ( θ j − θ j ) , θ j ( z ) = ∞ X n = −∞ e − π (2 rn +2 jn )+2 πiz ( j +2 rn ) . Then we let C ( p, q ) be the operator associated by peforming equivariantWeyl quantization to the function 2 cos(2 π ( px + qy )) on the pillow case(here z = x + iy ). A computation with integrals yields C ( p, q ) ζ j = t − pq [ t jq ζ j − p + t − jq ζ j + p ] , where t = e iπ r , which has been interpreted as saying that the Weyl quantization and thequantum group quantization of the moduli space of flat SU (2) connectionson the torus coincide. So this result puts the accent on the use of the skeinmodules RT t ( M ).3.2. The skein module of the complement of the (2 p + 1 , torusknot. Let us now show an example that arises in the search for patterns inskein modules. Computations with skeins have exponential complexity, andthese computations are expected to yield complicated results. Sometimes,for apparently no reason, the result of a lengthy computation produces asimple formula. This is the case with the following example, which we willexamine, for comparison, in both situations. The Kauffman bracket skeinmodule of the complement S \ N ( T p +1 , ) of a regular neighborhood of the(2 p + 1 ,
2) torus knot T p +1 , is free with basis x n y k , n ≥
0, 0 ≤ k ≤ p , as itwas shown by D. Bullock in [3], where x and y are depicted in Figure 2 andare endowed with the blackboard framing. Then RT t ( S \ T p +1 , ) is also x y Figure 2. free, with the same basis. Indeed, using Theorem 2.1 and Bullock’s resultwe conclude that every skein in RT t ( S \ T p +1 , ) is a linear combination ofthe elements x n y k , n ≥ , ≤ k ≤ p . And any nontrivial linear combinationequal to 0 in RT t ( S \ T p +1 , ) would yield a nontrivial linear combinationequal to zero in K t ( S \ T p +1 , ), which is impossible. SHAMON ALMEIDA AND R ˘AZVAN GELCA
For the Kauffman bracket skein module, the following surprising formulawas discovered by J. Sain in [10] t − i − S p + i ( y ) + t i +1 S p − i − ( y ) = ( − i S i ( x )( t − S p ( y ) + tS p − ( y )) , which allows the reduction of higher “powers” of y to lower powers. ByTheorem 2.1 in the skein module RT t ( S \ N ( T p +1 , )) we have the slightlymore elegant formula t − i − S p + i ( y ) + t i +1 S p − i − ( y ) = S i ( x )( t − S p ( y ) + tS p − ( y )) . The colored Jones polynomials and the noncommutative A-polynomial of a knot. If K ⊂ S is a framed knot with framing zero, then,in RT t ( S ), the skein S n ( K ) is equal to the colored Jones polynomial of K corresponding to the coloring of K by the n + 1st irreducible representationof the the quantum group of SU (2) multiplied by the empty link: S n ( K ) = J ( K, n ) ∅ . Theorem 2.1 shows that if we evaluate S n ( K ) in the Kauffman bracket skeinmodule K t ( S ) instead, we obtain ( − n J ( K, n ) ∅ , because the trace of eachterm of S n ( K ) is zero and the number of componets is congruent to n modulo2. In other words, the n th colored Jones polynomial is equal to ( − n timesthe n th colored Kauffman bracket: J ( K, n ) = ( − n h S n ( K ) i a fact that is being used widely (see for example [17]).There are two versions of the definition of the noncommutative gener-alization of the A-polynomial of a knot and the aim of this paragraph isto give a better understanding of the relationship between the two. Thefirst was defined by the second author in joint work with Ch. Frohman andW. Lofaro in [7] and is based on the Kauffman bracket. The constructionuses the action of the Kauffman bracket skein algebra of the cylinder overthe torus, K t ( T ), on the Kauffman bracket skein module K t ( S \ ( N ( K )) ofthe complement of the regular neighborhood of a knot K , which arises fromgluing the cylinder to the knot complement. The annihilator of the emptylink, which is a left ideal in K t ( T ), is called the peripheral ideal of the knotand is denoted by I t ( K ). It consists of the linear combinations of framedcurves on the boundary torus that become equal to zero when “pushed”inside the skein module of the knot complement. If we extend this ideal toa left ideal in the ring C t [ l, l − , m, m − ] = C (cid:10) l, l − , m, m − (cid:11) / ( lm = t ml )using the inclusion of K t ( T ) into this latter ring defined by (1 , l + l − ,(0 , (0 ,
1) (see [6]), then restrict it to C t [ l, m ], we obtain what was calledthe non-commutative A-ideal of K [7]. The reason for the definition is thatfor t = − AUFFMAN BRACKET VERSUS JONES POLYNOMIAL 9 and [9] that every element in the non-commutative A-ideal yields a recursiverelation for the colored Kauffman brackets < S n ( K ) > = ( − n J ( K, n ).The second construction of the noncommutative generalization of the A-polynomial has its origin in [8] and is based on quantum groups, beingtherefore related to the Jones polynomial in the Reshetikhin-Turaev nor-malization. The idea is to view the family of colored Jones polynomials asa function f : Z → C [ t, t − ], f ( n ) = J ( K, n ) and consider the operators L and M on such functions Lf ( n ) = f ( n + 1) and M f ( n ) = t n f ( n ). Theseoperators satisfy LM = t M L , and so they generate the ring C t [ L, L − , M, M − ] = C (cid:10) L, L − , M, M − (cid:11) / ( LM = t M L ) . The recurrence ideal of the knot K is the left ideal consisting of the poly-nomials P ( L, M ) satisfying P ( L, M ) f = 0, where f is the function definedabove. It has been shown in [8] that this ideal is always nonzero. The twoconstructions are related because any recursive relation for < S n ( K ) > =( − n J ( K, n ) can be transformed into a recursive relation for J ( K, n ), soevery element in the peripheral ideal defined in [7] can be transformed intoan element in the recurrence ideal, but this transformation is somewhat adhoc because it requires several sign adjustments.However, if we use for the definition of the noncommutative A-ideal theskein modules of the Jones polynomial, thus working instead with the actionof RT t ( T ) on RT t ( S \ N ( K )), then ideal resulting from extending the pe-ripheral ideal to C t [ l, l − , m, m − ] and then restricting to C t [ l, m ] is automat-ically included in the recurrence ideal under the identification l = L, m = M ;no more change of signs. Moreover, Theorem 2.1 implies that to pass fromthe peripheral ideal for the case of the Kauffman bracket to that for the caseof the Jones polynomial in the Reshetikhin-Turaev normalization, one hasto substitute each ( p, q ) T by ( − p ( p, q ) T .3.4. The skein module of the complement of the figure-eight knot.
We illustrate the facts that we have just discussed with the example of thefigure-eight knot. Let therefore K be the figure-eight knot and let N ( K )be one of its open regular neighborhoods. Consider the left action of theskein algebra of the torus, RT t ( T ), on RT t ( S \ N ( K )) defined by gluingthe cylinder over the torus to the boundary of the knot complement suchthat the curve (1 ,
0) is identified with the longitude and the curve (0 ,
1) isidentified with the meridian of the knot. To understand the RT t ( T )-modulestructure of RT t ( S \ N ( K )), we need to explicate the action of the elements( p, q ) T from the boundary.The Kauffman bracket skein module of the figure-eight knot comple-ment was found by D. Bullock and W. Lofaro in [4] to be free with basis x n , x n y, x n y where n ≥
0, or equivalently x n , x n y, x n z , where n ≥
0, theframed curves x, y and z being shown in Figure 3 and being endowed withthe blackboard framing. For the same reason as in the case of the torusknots discussed above, RT t ( S \ N ( K )) is also free, with the same basis. zy x Figure 3.
From the work in [11] one can infer that the action of the algebra K t ( T )on K t ( S \ N ( K )) is determined by the following(1 , q ) T ∅ = t q [( t S q ( x ) + t − S − q ( x )) Y + ( t S q ( x ) + t − S − q ( x )) Z − ( t S − − q ( x ) + t − S − q ( x ))] , (1 , q ) T Y = t q +4 [ − ( t S q ( x ) + t − S − − q ( x )) Y − ( t S q +2 ( x )+ t − S − q ( x )) Z + ( t S − − q ( x ) + t − S q ( x ))] , (1 , q ) T Z = t q − [ − ( t S q ( x ) + t − S − q ( x )) Y − ( t S − q ( x )+ t − S − q ( x )) Z + t q − ( t − S − q ( x ) + t S − q ( x ))] . where Y = t y + 1, Z = t − z + 1 and q ∈ Z . We point out that the actionof K t ( T ) on elements of the form x n , as well as on x n y and x n z for n > , q ) T , the product-to-sumformula allows the computation of the action of a general ( p, q ) T , albeitwithout a nice closed form formula.Applying Theorem 2.1, and noticing that the computation consists of justcounting link components modulo 2, we obtain the following result. Theorem 3.3.
The action of RT t ( T ) on RT t ( S \ N ( K )) is determined by (1 , q ) T ∅ = t q [( t S q ( x ) + t − S − q ( x )) Y + ( t S q ( x ) + t − S − q ( x )) Z +( t S − − q ( x ) + t − S − q ( x ))] , (1 , q ) T Y = t q +4 [( t S q ( x ) + t − S − − q ( x )) Y + ( t S q +2 ( x )+ t − S − q ( x )) Z + ( t S − − q ( x ) + t − S q ( x ))] , (1 , q ) T Z = t q − [( t S q ( x ) + t − S − q ( x )) Y + ( t S − q ( x )+ t − S − q ( x )) Z + t q − ( t − S − q ( x ) + t S − q ( x ))] . where Y = t y − , Z = t − z − and q ∈ Z . Using this module structure, after a tedious computation, one obtains thefollowing example of an element in the peripheral ideal of K in the version AUFFMAN BRACKET VERSUS JONES POLYNOMIAL 11 that uses the Jones polynomial in the Reshetikhin-Turaev normalization: t − (2 , T − t (2 , − T − t (1 , T + t (1 , T + ( t − t + t − + t − )(1 , T +( − t + t + t − )(1 , T + ( t − t − t + t − − t − )(1 , − T +( − t − t )(1 , − T + t − (1 , − T + t (0 , T + ( − t + t − t − )(0 , T +( − t + t − t − t − )(0 , T + ( t − t + 1 + t − )(0 , T . This should be contrasted with t − (2 , T − t (2 , − T + t (1 , T − t (1 , T + ( − t + t − t − − t − )(1 , T +( t − t − t − )(1 , T + ( − t + 2 t + t − t − + t − )(1 , − T +( t + t )(1 , − T − t − (1 , − T + t (0 , T + ( − t + t − t − )(0 , T +( − t + t − t − t − )(0 , T + ( t − t + 1 + t − )(0 , T which was obtained in [11] as an element of the peripheral ideal definedusing the Kauffman bracket. The former gives rise to the following recursiverelation for colored Jones polynomials y n = J ( K , n ) of the figure-eight knot( t n +6 − t − n +2 ) y n +2 + ( − t n +24 + t n +16 + t n +20 − t n +12 + t n +8 + t n +4 − t n +12 + t n +8 + t n − + t − n +8 − t − n +4 − t − n + t − n − − t − n − − t − n +4 − t − n − + t − n − ) y n +1 + ( t n +22 − t n +18 + t n +14 − t n +6 − t n +18 + t n +14 − t n +10 − t n +6 + t n +2 + t n +14 − t n +10 + t n +2 + t n − + t − n +10 − t − n +6 + t − n − + t − n − − t − n +6 + t − n +2 − t − n − − t − n − + t − n − − t − n − + t − n − − t − n − + t − n − ) y n + ( t n +4 − t n +16 − t n +4 + t n +12 − t n +8 − t n +4 + t n − t n − − t − n +8 + t − n +4 + t − n − + t − n +8 − t − n + t − n − + t − n − + t − − − t − n − ) y n − + ( t n +6 + t − n − ) y n − = 0 . If we use the construction based on the Kauffman bracket, we obtain arecursive relation for ( − n J ( K, n ) instead.
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