The non-commutative A -polynomial of twist knots
aa r X i v : . [ m a t h . G T ] J u l THE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS STAVROS GAROUFALIDIS AND XINYU SUN
Abstract.
The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method,and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A -polynomial of twist knots.Our multivariable creative telescoping method allows us to compute linear recursions for sums of theform J ( n ) = P k c ( n, k ) ˆ J ( k ) given a recursion relation for ( ˆ J ( n )) a the hypergeometric kernel c ( n, k ). As anapplication of our method, we explicitly compute the non-commutative A -polynomial for twist knots with − A -polynomial of a knot encodes the monic, linear, minimal order q -differenceequation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 isconjectured to be the better-known A -polynomial of a knot, which encodes important information aboutthe geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easilycomputable for knots with 50 crossings, the A -polynomial is harder to compute and already unknown forsome knots with 12 crossings. Contents
1. Introduction 21.1. The goal 21.2. The Jones polynomial of a knot 21.3. The non-commutative A -polynomial of a knot and its significance 21.4. Computing the non-commutative A -polynomial 21.5. A sample of our results 31.6. Plan of the proof 41.7. Acknowledgement 42. The strategy 43. A brief review of creative telescoping 64. Multi-certificate creative telescoping and Theorem 2 84.1. Multi-certificate Creative Telescoping 84.2. A first reduction to linear algebra 94.3. A second reduction to linear algebra 115. Odds and ends 115.1. A generalization of Theorem 2 115.2. Is there a recursion of the non-commutative A -polynomial with respect to the number of twists? 12Appendix A. A generating functions approach 12Appendix B. The non-commutative A -polynomial for p = ± A -polynomial for p = − , . . . , Date : July 6, 2009.S.G. was supported in part by National Science Foundation.1991
Mathematics Classification.
Primary 57N10. Secondary 57M25.
Key words and phrases: Knots, Jones polynomial, colored Jones function, A -polynomial, C -polynomial, non-commutative A -polynomial, q -difference equations, WZ Algorithm, Creative Telescoping, Gosper’s Algorithm, certificate, multi-certificate. Introduction
The goal.
The purpose of the paper is two-fold: to introduce a multivariable creative telescopingmethod, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A -polynomial of twist knots.Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form J ( n ) = P k c ( n, k ) ˆ J ( k ) given a recursion relation for ( ˆ J ( n )) a the hypergeometric kernel c ( n, k ). Generaltheory implies the existence of a recursion relation for ( ˆ J ( n )). However, in practice the computation is notmanageable for twist knots, and there is no guarantee that the recursion relation will be of minimal order. Ourmethod does not guarantee a minimal order recursion relation either, however (unlike the known methods) itis manageable and produces a minimal order recursion relation the non-commutative A -polynomial for twistknots of − , . . . ,
11 twists. The non-commutative A -polynomial encodes the unique monic, linear, minimalorder q -difference equation satisfied by the sequence of colored Jones polynomials of the knot. Our resultsgive a new proof of the AJ-Conjecture for those knots.1.2. The Jones polynomial of a knot.
In this section we recall the relevant Laurent polynomial invariantsof knots, such as the Jones polynomial and its colored cousins. In 1985 V. Jones introduced the famous
Jonespolynomial of a knot K in 3-space, [Jo]. The Jones polynomial (an element of Z [ q ± ]) is a powerful knotinvariant which amongth other things detects cheirality, and it can be extended to a sequence ( J K ( n ))of Laurent polynomials by taking parallels of a knot K . Technically, J K ( n ) ∈ Z [ q ± ] is the quantumgroup invariant of the 0-framed knot using the n -dimensional representation of SU(2), and normalized by J Unknot ( n ) = 1. For a detailed definition, see [Tu] and also [GL1]. With this normalization, we have that J K (1) = 1, and J K (2) is the Jones polynomial of K .For a given knot K , the sequence of Laurent polynomials ( J K ( n )) is not random. To be precise, ( J K ( n ))is q - holonomic i.e., it satisfies a linear q -difference equation (which of course, depends on the knot) withcoefficients in Q ( q, q n ). This fact, proven in [GL1], is an easy consequence of two facts:(a) J K ( n ) is a finite multisum of a proper q -hypergeometric term, as follows from the state-sum definitionof the colored Jones function; see [GL1].(b) ultisums of proper q -hypergeometric terms are q -holonomic, as follows from the WZ theory of Wilf-Zeilberger; see [WZ].1.3. The non-commutative A -polynomial of a knot and its significance. A q -holonomic sequence isannihilated by a unique monic homogeneous linear q -difference equation of smallest degree, and the corre-sponding monic polynomial in two q -commuting variables E and Q is an invariant (the so-called characteristicpolynomial of the q -holonomic sequence. We define the non-commutative A -polynomial A K ( E, Q, q ) of a knot K to be the characteristic polynomial of ( J K ( n )).In [Ga], it was conjectured by the first author (the so-called AJ Conjecture) that the specialization A K ( E, Q,
1) of A K ( E, Q, q ) should agree with the A -polynomial of a knot. The latter is an importantinvariant that parametrizes the SL(2 , C ) character variety of the knot complement, as viewed from theboundary torus. For a detailed definition of the A -polynomial, its properties and its applications to thegeometry and topology of the knot complement, see [CCGLS]. Thus, A K ( E, Q, q ) can be thought of as adeformation (or quantization) of the character variety.The Jones polynomial of a knot is easily computable via skein theory with knots with, say, 50 crossings;see for example [B-N]. On the other hand, the A -polynomial of a knot is much harder to compute, andat present it is unknown for some knots with 12 crossings. There are two general methods to computethe A -polynomial: an exact (primarily elimination, and Puiseux expansions) developed by Boyd [Bo] and anumerical one developed by Culler [Cu].The non-commutative A -polynomial and its possible relation with the A -polynomial of a knot is animportant ingredient to the Hyperbolic Volume Conjecture and its generalization.1.4. Computing the non-commutative A -polynomial. For theoretical as well as experimental reasonsit would be good to have explicit formulas for the non-commutative A -polynomial. So far, an explicit formulahas been given for torus knots in [Ge] (using properties of the Kauffman bracket skein module of the solid HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 3 torus), as well as for the simplest hyperbolic 4 knot in [GL1] (using an explicit single-sum formula for thecolored Jones function).The WZ algorithm has been implemented (see [PWZ, PR2, PR3]) and together with explicit state-sumformulas for the colored Jones function of an arbitrary planar projection given in [GL1], in principle onecan obtain a linear q -difference equation for the colored Jones function of an arbitrary knot. There are twoproblems with this approach:(a) The number of summation variables in the multisum formulas is generally two less than the numberof crossings, and the q -multisum algorithms appear to be slow for the current machines.(b) There is no guarantee that the various q -multisum algorithms will give a minimal order linear q -difference equation. In fact, in many cases (where symmetry is involved), it has been observed thatthey fail to give the minimal order q -difference equation. See [PR1] for well-known examples of thisfailure.With respect to the first problem, we were unable to use the sofware of [AZ, S] to compute a q -differenceequation for our double sums.One might wonder whether Problem (b) really occurs for the state-sums that originate in knot theory. Asexpected, this problem does occur. The knots 5 and 6 have double-sum formulas for their colored Jonesfunction. An application of the q -multisum package of [PR3] was done by Takata in [Ta] who found out anexplicit inhomogeneous q -difference equation of degree 5 and 5 respectively. On the other hand, as we shallsee, there exist inhomogeneous q -difference equations of degree 3 and 4 respectively.In a different direction, Le used geometric methods of the Kauffman bracket skein module and was ableto prove the AJ Conjecture for most 2-bridge knots, as well as give a linear algebra algorithm that inprinciple computes the non-commutative A -polynomial; see [Le, Thm.1] and [Le, Sec.5.6.3]. The algorithmwas implemented in Maple by the second author, but proved to be too slow to run for the 5 and 6 knots.1.5. A sample of our results.
The main goal of the paper is to give an explicit formula for the non-commutative A -polynomial of twist knots with p twists, where p = − , . . . , twist knots K p for integer p , shown in Figure 1. The planar projection of K p has 2 | p | + 2crossings, 2 | p | of which come from p full twists, and 2 come from the negative clasp . ... full twistsp Figure 1.
The twist knot K p , for integers p . For small p , these knots may be identified with ones from Rolfsen’s table (see [Rf] and [B-N]) as follows: K = 3 , K = 5 , K = 7 , K = 9 ,K − = 4 , K − = 6 , K − = 8 , K − = 10 . Let E and Q denote the operators that act on a sequence ( J ( n )) of Laurent polynomials J ( n ) ∈ Z [ q ± ] by:(1) ( EJ )( n ) = J ( n + 1) , ( QJ )( n ) = q n J ( n ) . Note that EQ = qQE . Let ( A nhp ( E, Q, q ) , B p ( q n , q )) denote the inhomogeneous non-commutative A -polynomialof K p . That is, A nhp ( E, Q, q ) is monic and minimal degree (with respect to E ) that satisfies the equation(2) A nhp ( E, Q, q ) J p = B p ( q n , q ) , STAVROS GAROUFALIDIS AND XINYU SUN where B p ( q n , q ) ∈ Q ( q n , q ). To convert the inhomogeneous equation above to a homogeneous one, see Section3. Theorem 1. (a) For p = ± A nh ( E, Q, q ) = q n +2 ( q n −
1) + ( q n +1 − EB nh ( q n , q ) = ( q n +1 − q n A nh − ( E, Q, q ) = q n +2 ( q n − q n +3 − − ( q n +1 + 1)( q n +4 − q n +3 − q n +3 − q n +1 − q n +1 + 1)( q n +1 − E + q n +2 ( q n +1 − q n +2 − E B nh − ( q n , q ) = q n +1 ( q n +3 − q n +1 + 1)( q n +1 − (b) For p = ± A nh ( E, Q, q ) = q n +9 ( − q n )( q n +4 − q n +5 − − q n +5 ( q n +1 − q n +2 − q n +5 − q n +6 − q n +5 − q n +5 + q n +4 − q n +3 − q n +2 − q n +2 + q n +1 + q n +1 − E + q ( q n +2 − q n +1 − q n +4 − q n +9 − q n +7 − q n +7 + q n +6 + q n +5 + q n +5 − q n +4 + 2 q n +2 + q n +2 − E + ( q n +3 − q n +1 − q n +2 − E ,B ( q n , q ) = ( q n +1 + 1)( q n +1 − q n +2 + 1)( q n +5 − q n +3 − q n +4 ,A nh − ( E, Q, q ) = ( − q n )( q n +5 − q n +6 − q n +7 − q n +8 − q n +5 ( q n +1 − q n +6 − q n +7 − q n +2 − q n +9 − q n +8 − q n +8 + q n +7 + q n +7 + q n +6 − q n +6 − q n +6 − q n +5 − q n +5 − q n +4 + q n +3 + q n +3 − q n +3 − q n +2 − q n +2 − q n +1 + q + 1) E + q ( q n +2 − q n +1 − q n +4 − q n +7 − − q n +3 − q n +2 − q n +3 + q n +5 +2 q n +8 + 3 q n +8 − q n +7 − q n +6 − q n +6 − q n +6 + q n +5 − q n +2 + q n +5 + q n +4 − q n +10 − q n +14 − q n +11 − q n +15 − q n +11 + q n +12 + q n +16 + q n +13 + q n +13 + q n +9 + q n +9 + 2 q n +4 − q n +14 + 3 q n +9 + 2 q n +8 − q n +10 +2 q n +12 ) E − ( q n +3 − q n +6 − q n +2 − q n +1 − q n +16 + q n +15 − q n +14 − q n +13 − q n +13 − q n +12 + q n +10 + q n +10 − q n +9 − q n +9 − q n +8 − q n +8 − q n +7 + q n +6 + q n +6 + q n +5 − q n +3 − q n +3 + 1) E + q n +12 ( q n +4 − q n +1 − q n +2 − q n +3 − E ,B − ( q n , q ) = ( q n +3 + 1)( q n +5 − q n +7 − q n +6 ( q n +2 + 1)( q n +1 + 1)( q n +1 − q n +3 − . The formulas quickly become too lengthy to type. For more information, see Appendix B for p = ± p = − , . . . ,
11. Theorem 1 gives a new proof of the AJ-Conjecture for twistknots with − , . . . ,
11 twists.1.6.
Plan of the proof.
In Section 2 we outline the main strategy. The idea is to use the recursion relationof the cyclotomic function ( ˆ J p ( n )) of the twist knot K p (from [GS1]) as well as a single-sum relation between J p ( n ) and ˆ J p ( n ), together with some new ideas of Creative Telescoping and some guessing. In Section 3 wereview the method of Creative Telescoping and in Section 4 we present a multi-certificate version that takesinto account the product of a hypergeometric summand with a q -holonomic one.We conclude with three appendices: in Appendix A we present an alternative method that uses generatingfunctions (that was kindly communicated to us by Zeilberger). In Appendix B we give the non-commutativepolynomial of twist knots K p for p = − ,
3, and in Appendix C we give the A -polynomial of the same knots.1.7. Acknowledgement.
The authors wish to thank D. Zeilberger for encouragement and enlighteningconversations. 2.
The strategy
In this section we will describe our strategy to obtain a formula for A p ( E, Q, q ). HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 5 (a) We consider the cyclotomic function ˆ J K ( n ) ∈ Z [ q ± ] introduced by Habiro in [Ha], who used thenotation J K ( P ′′ n ).(b) The relation between the cyclotomic and the colored Jones functions is given by:(3) J K ( n ) = n X k =0 c ( n, k ) ˆ J K ( k ) , where the cyclotomic kernel c ( n, k ) is a proper q -hypergeometric term given for 0 ≤ k ≤ n by: c ( n, k ) = { n − k }{ n − k + 1 } . . . { n + k − }{ n + k }{ n } (4) = ( − k q − k ( k +1) / ( q − n ; q ) k ( q n ; q ) k , where { n } = q n/ − q − n/ , and the quantum factorial is defined by:( x ; q ) n = (1 − x ) · · · (1 − xq n − ) if n > n = 0; − xq − ) ··· (1 − xq n ) if n < J K ( n )) is q -holonomic, as shown in [GL1], and its characteristic polynomial C K ( E, Q, q ) is defined to be the non-commutative C -polynomial of a knot. Let us abbreviate ˆ J K p ( n ), J K p ( n ), A K p ( E, Q, q ) and C K p ( E, Q, q ) for the twist knots K p by ˆ J p ( n ), J p ( n ), A p ( E, Q, q ) and A p ( E, Q, q ) respectively.(d) In [GS1] we gave an explicit formula for C p ( E, Q, q ).(e) Using the explicit formula for C p ( E, Q, q ) as well as the relation (3) and a version of creative tele-scoping (and some guessing), we deduce a linear q -difference equation for ( J p ( n )), which specializesto the A -polynomial of K p when q = 1.(f) Since the A -polynomial of K p is irreducible (see [HS1]), and the non-commutative A -polynomial of K p specializes to the A -polynomial, it follows that our q -difference equation is indeed of minimalorder. This computes A p ( E, Q, q ).For completeness and concreteness, we give a formula for ˆ J p ( n ), using [Ma, Thm.5.1] (compare also with[Ga, Sec.3]): ˆ J p ( n ) = n X k =0 q n ( n +3) / pk ( k +1)+ k ( k − / ( − n + k +1 ( q k +1 − q ; q ) n ( q ; q ) n + k +1 ( q ; q ) n − k (5) = n X k =0 q n ( n +3) / pk ( k +1)+ k ( k − / ( − n + k +1 ( q k +1 − q n − k +1 ; q ) k ( q ; q ) n + k +1 . Observe that since ( q n − k +1 ; q ) k = 0 for k > n >
0, we can assume that the k -summation in the aboveequation is for 0 ≤ k < + ∞ .Equations (3), (4) and (5) imply that J p ( n ) is given by a double-sum formula of a proper q -hypergeometricsummand. As explained earlier, the qMultisum.m implementation of the WZ algorithm given in [PR2, PR3]and used in [Ta] is slow to run, and gives q -difference equations of higher than actual degree. An applicationof our multicertificate version of Creative Telescoping is the following theorem. STAVROS GAROUFALIDIS AND XINYU SUN
Theorem 2.
The minimal inhomogeneous recursion for J p ( n ) for − ≤ p ≤ is an explicit linear q -difference equation of order ( p − if < p ≤ | p | if − ≤ p < . The inhomogeneous recursion is given by Theorem 1 for p = − , . . . ,
2, Appendix B for p = ± p = − , . . . , A brief review of creative telescoping
In this section we recall briefly some key ideas of Zeilberger on recursion relations of combinatorial sums.An excellent reference is [PWZ]. For a longer introduction, see also [GS1, Sec.3].A term is F ( n, k ) called hypergeometric if both F ( n +1 ,k ) F ( n,k ) and F ( n,k +1) F ( n,k ) are rational functions over n and k . In other words,(6) F ( n + 1 , k ) F ( n, k ) ∈ Q ( n, k ) , F ( n, k + 1) F ( n, k ) ∈ Q ( n, k ) . Examples of hypergeometric terms are F ( n, k ) = ( an + bk + c )! (for integers a, b, c ), and ratios of products ofsuch. The latter are actually called proper hypergeometric . A key problem is to construct recursion relationsfor sums of the form:(7) S ( n ) = X k F ( n, k ) , where F ( n, k ) is a proper hypergeometric term. The summation set can be the set of all integers or aninterval thereof. Let us first suppose that summation is over entire set of integers. Sister Celine [Fa] (seealso [PWZ]) proved the following: Theorem 3.
Given a proper hypergeometric term F ( n, k ) , there exist a natural number I ∈ N and a set offunctions a i ( n ) ∈ Q ( n ) , ≤ i ≤ I , such that I X i =0 a i ( n ) F ( n + i, k ) = 0 . (8)The important part of the above theorem is that the functions a i ( n ) are independent of k . Therefore ifwe take the sum over k on both sides, we get I X i =0 a i ( n ) X k F ( n + i, k ) = 0 . (9)In other words, we have:(10) I X i =0 a i ( n ) S ( n + i ) = 0 . So, Equation (8) produces a recursion relation, which is inhomogeneous if we are summing over an interval.How can we find functions a i ( n ) that satisfy Equation (8)? The idea is simple: divide Equation (8) by F ( n, k ), and use (6) to convert the divided equation into a linear equation over the field Q ( n, k ), withunknowns a i ( n ) for i = 0 , . . . , I . Clearing denominators, we get linear equation over Q ( n )[ k ] with the sameunknowns a i ( n ). Thus, the coefficients of every power of k must vanish, and this gives a linear system ofequations over Q ( n ) with unknowns a i ( n ). If there are more unknowns than equations, one is guaranteed tofind a nonzero solution. By a counting argument, one may see that if we choose I high enough (this dependson the complexity of the term F ( n, k )), then we have more equations than unknowns. HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 7 Although it can be numerically challenging to find a i ( n ) that satisfy Equation (8), it is routine to checkthe equation once a i ( n ) are given. Indeed, one only need to divide the equation by F ( n, k ), and then checkthat a function in Q ( n, k ) is identically zero. The latter is computationally easy task in the field Q ( n, k ).This algorithm produces a recursion relation for S ( n ). However, it is known that the algorithm does notalways yield a recursion relation of the smallest order.Applying Gosper’s algorithm, Wilf and Zeilberger invented another algorithm, the WZ algorithm , alsocalled creative telescoping . Instead of looking for 0 on the right-hand side of Equation (8), they insteadlooked for a function G ( n, k ) such that N X i =0 a i ( n ) F ( n + i, k ) = G ( n, k + 1) − G ( n, k ) . (11)Summing over k , and using telescoping cancellation of the terms in the right hand side, we get a recursionrelation for S ( n ). How to find the a i ( n ) and G ( n, k ) that satisfy (11)? The idea is to look for a rationalfunction Cert( n, k ) (the so-called certificate of (11)) such that(12) G ( n, k ) = Cert( n, k ) F ( n, k ) . Dividing out (11) by F ( n, k ) and proceeding as before, one reduces this to a problem of linear algebra. Asbefore, given a i ( n ) and Cert( n, k ), it is routine to check whether (11) holds.Now, let us rephrase the above equations using operators. We define operators E , E k , n and k that acton a function F ( n, k ) by:( EF )( n, k ) = F ( n + 1 , k ) , ( E k F )( n, k ) = F ( n, k + 1) , (13) ( nF )( n, k ) = nF ( n, k ) , ( kF )( n, k ) = kF ( n, k ) . (14)The operators E and n (and also E k , k ) do not commute. Instead, we have: En = ( n + 1) E, E k k = ( k + 1) E k . On the other hand, n, E commute with k, E k . Then we can rewrite Equation (11) as I X i =0 a i ( n ) E i ! F ( n, k ) = ( E k − G ( n, k ) = ( E k − n, k ) F ( n, k ) . (15)Implementation of the algorithms are available in various platforms, such as, Maple and Mathematica. See,for example, [Z2] and [PR2].Let us mention now how one deals with boundary terms. In the applications below, one considers notquite the unrestricted sums of Equation (7), but rather restricted ones of the form:(16) S ′ ( n ) = ∞ X k =0 F ( n, k ) , where F ( n, k ) is a proper hypergeometric term. When we apply the Creating Telescoping summation to(11), we are left with some boundary terms R ( n ) ∈ Q ( n ). In that case, Equation (10) becomes: I X i =0 a i ( n ) E i ! S ′ ( n ) = R ( n ) . This is an inhomogeneous equation of order I which can be converted into a homogeneous recursion of order I + 1 by following trick: apply the operator ( E −
1) 1 R ( n ) STAVROS GAROUFALIDIS AND XINYU SUN on both sides of the recursion. We get (cid:18) R ( n + 1) E − R ( n ) (cid:19) I X i =0 a i ( n ) E i ! S ′ ( n ) = 0 , i.e., a I ( n + 1) R ( n + 1) E I +1 + I X i =1 (cid:18) a i − ( n + 1) R ( n + 1) − a i ( n ) R ( n ) (cid:19) E i − a ( n ) R ( n ) ! S ′ ( n ) = 0 . In Quantum Topology we are using q -factorials rather than factorials. The previous results translate withoutconceptual difficulty to the q -world, although the computer implementation is slower. A term F ( n, k ) is called q - hypergeometric if F ( n + 1 , k ) F ( n, k ) , F ( n, k + 1) F ( n, k ) ∈ Q ( q, q n , q k ) . Examples of q -hypergeometric terms are the quantum factorials of linear forms in n, k , and ratios of prod-ucts of quantum factorials and q raised to quadratic functions of n and k . The latter are called q - properhypergeometric .Sister Celine’s algorithm and the WZ algorithm work equally well in the q -case. In either algorithms, wecan replace the operators E, n, E k , k of (13) by the operators E, Q, E k , Q k defined by:( EF )( n, k ) = F ( n + 1 , k ) , ( E k F )( n, k ) = F ( n, k + 1) , (17) ( QF )( n, k ) = q n F ( n, k ) , ( Q k F )( n, k ) = q k F ( n, k ) . (18)Observe that E, Q (and also E K , Q k ) q -commute, i.e., we have:(19) EQ = qQE, E k Q k = qQ k E k . On the other hand,
E, Q commute with E k , Q k . With these modifications, and with the replacement of thefield Q ( n ) by Q ( q, q n ), the rest of the proofs still apply naturally. The implementations of the q -case include[PR3], [Ko] and [Z2].4. Multi-certificate creative telescoping and Theorem 2
Multi-certificate Creative Telescoping.
In a nut-shell, the method of creative telescoping works asfollows. To find the recursion such that m X i =0 a i ( n ) X k ≥ F ( n + i, k ) = b ( n ) , it suffices to find a rational function Cert( n, k ) ∈ Q ( q, q n , q k ) such that G ( n, k ) := Cert( n, k ) F ( n, k ) satisfies: m X i =0 a i ( n ) E i ! F ( n, k ) = m X i =0 a i ( n ) F ( n + i, k ) = G ( n, k + 1) − G ( n, k ) = ( E k − G ( n, k ) . If this can be done, we sum both sides for 0 ≤ k < + ∞ , and we obtain: m X i =0 a i ( n ) X k ≥ F ( n + i, k ) = G ( n, . For twist knots K p , we have from Equation (3):(20) J p ( n ) = n X k =0 c ( n, k ) ˆ J p ( k ) , HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 9 where c ( n, k ) is proper q -hypergeometric (given by (4)), and ˆ J p ( n ) satisfies a linear q -difference equation ofdegree | p | from [GS1].Without loss of generality, suppose that p >
0. Suppose the minimal order recursion of ˆ J p ( k ) is:(21) p X i =0 r i ( k ) E ik ! ˆ J p ( k ) = 0 , with r p ( k ) = 1 and r i ( k ) ∈ Q ( q, q k ) for i = 0 , . . . , p . The idea is to look for p certificates { C ( n, k ) , . . . C p − ( n, k ) } ,such that(22) m X i =0 a i ( n ) E i ! c ( n, k ) ˆ J p ( k ) = ( E k − p − X j =0 C j ( n, k ) E jk c ( n, k ) ˆ J p ( k ) . A first reduction to linear algebra.
Our goal in this section, stated in Proposition 4.1 below, is totranslate the functional equation (22) into a system of linear equations with unknowns a i ( n ) ∈ Q ( q, q n ) and C j ( n, k ) ∈ Q ( q, q n , q k ) for i = 0 , . . . , m and j = 0 , . . . , p −
1. Since c ( n, k ) is proper q -hypergeometric, wehave: Ec ( n, k ) c ( n, k ) = s ( n, k ) ∈ Q ( q, q n , q k ) , E k c ( n, k ) c ( n, k ) = t ( n, k ) Q ( q, q n , q k ) . Observe that 0 = p X i =0 r i ( k ) ˆ J p ( k + i )= p X i =0 r i ( k ) c ( n, k + p ) c ( n, k + i ) ˆ J p ( k + i ) c ( n, k + i )= p X i =0 r i ( k ) c ( n, k + p ) c ( n, k + i ) ˆ J p ( k + i ) c ( n, k + i )= p X i =0 r i ( k ) c ( n, k + p ) c ( n, k + i ) E ik ! ˆ J p ( k ) c ( n, k ) . So if we define R i ( n, k ) = r i ( k ) c ( n, k + p ) c ( n, k + i )= r i ( k ) p − i − Y j =0 t ( n, k + i + j ) ∈ Q ( q, q n , q k ) , then we obtain that(23) p X i =0 R i ( n, k ) E ik ! ˆ J p ( k ) c ( n, k ) = 0 . Notice that since r p ( k ) = 1, it follows that R p ( n, k ) = 1 too. Proposition 4.1.
Equation (22) is equivalent to the following system of linear equations: (24) m X i =0 a i ( n ) i − Y j =0 s ( n + j, k ) = − p − X j =0 C p − ( n, k − j + 1) R j ( n, k − j ) and (25) C j − ( n, k + 1) = C j ( n, k ) + C p − ( n, k + 1) R j ( n, k ) , ≤ j ≤ p − , in the unknowns a i ( n ) ∈ Q ( q, q n ) for i = 0 , . . . , m and C j ( n, k ) ∈ Q ( q, q n , q k ) for j = 0 , . . . , p − .Proof. For convenience we define C − ( n, k ) = 0. Then using the commutation relation( E k − C j ( n, k ) = C j ( n, k + 1) E k − C j ( n, k )and Equation (23) we obtain that: m X i =0 a i ( n ) E i ! c ( n, k ) ˆ J p ( k ) = ( E k − p − X j =0 C j ( n, k ) E ik c ( n, k ) ˆ J p ( k )= p − X j =0 C j ( n, k + 1) E i +1 k − p − X j =0 C j ( n, k ) E ik c ( n, k ) ˆ J p ( k )= p X j =1 C j − ( n, k + 1) E ik − p − X j =0 C j ( n, k ) E ik c ( n, k ) ˆ J p ( k )= − C p − ( n, k + 1) p − X j =0 R j ( n, k ) E jk + p − X j =1 C j − ( n, k + 1) E ik − p − X j =0 C j ( n, k ) E ik c ( n, k ) ˆ J p ( k )= p − X j =0 ( − C p − ( n, k + 1) R j ( n, k ) + C j − ( n, k + 1) − C j ( n, k )) E jk (cid:17) c ( n, k ) ˆ J p ( k ) . So m X i =0 a i ( n ) i − Y j =0 s ( n + j, k ) c ( n, k ) ˆ J p ( k )= m X i =0 a i ( n ) E i ! c ( n, k ) ˆ J p ( k )= p − X j =0 ( − C p − ( n, k + 1) R j ( n, k ) + C j − ( n, k + 1) − C j ( n, k )) E jk c ( n, k ) ˆ J p ( k ) . If we divide both sides by c ( n, k ), which is hypergeometric, we obtain a new recursion on ˆ J p ( k ) of order p − J p ( k ) satisfies a minimal order recursion of degree p , the last equality implies that the coefficient ofeach E ik is 0 for all i . Hence ( − C p − ( n, k + 1) R j ( n, k ) + C j − ( n, k + 1) − C j ( n, k ) = 0 if 1 ≤ j ≤ p − , P mi =0 a i ( n ) c ( n + i,k ) c ( n,k ) = − C p − ( n, k + 1) R ( n, k ) − C ( n, k ) if j = 0 . HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 11 The first equation implies (25). In particular, C ( n, k + p −
1) = C p − ( n, k ) + p − X j =0 C p − ( n, k + j + 1) R p − j − ( n, k + j ) . Therefore, m X i =0 a i ( n ) c ( n + i, k ) c ( n, k ) = m X i =0 a i ( n ) i − Y j =0 s ( n + j, k ) = − p − X j =0 C p − ( n, k − j + 1) R j ( n, k − j ) , which proves (24) and concludes the proof of the proposition. (cid:3) A second reduction to linear algebra.
Proposition 4.1 reduces the problem of finding p certificates C j ( n, k ) to a problem of finding a single certificate C p − ( n, k ). Since C p − ( n, k ) ∈ Q ( q, q n , q k ) is a rationalfunction, we can write it in the form:(26) C p − ( n, k ) = N p ( n, k ) D p ( n, k ) , where N p ( n, k ) = P r d i =0 d i ( n ) q ki and D p ( n, k ) = P r e i =0 e i ( n ) q ki , in which d i ( n ) and e i ( n ) are in Q ( q, q n ). Bymaking the proper choices of D p ( n, k ) and the values of m and r d , we can clear denominators and convertEquation (25) as a linear equation in Q ( q, q n )[ q k ] with unknowns a i ( n ), d i ( n ), and e i ( n ).Setting every coefficient of every power of q k to zero, we obtain a system of linear equations in theunknowns a i ( n ), d i ( n ), and e i ( n ) and coefficients in the field Q ( q, q n ). A nontrivial solution is guaranteedby Sister Celine’s method for the case of J p ( n ). At any rate, we can solve the system of equations usingsoftware like Maple or Mathematica.Now comes the tricky part, and an educated guess for the case of twist knots. Since c ( n + i, k ) c ( n, k ) = i − Y j =0 s ( n + i, k ) , and R j ( n, k ) are all polynomials in Q ( q, q n )[ q k ], the most natural choice of D p ( n, k ) is the one such that D p ( n, k − j + 1) divides the polynomial R j ( n, k − j ) for all j . Let D p ( n, k ), the denominator of the certificate C p − ( n, k ), be ( q pk Q p − i =1 − p (1 − q k − n − i ) if p > , Q | p | i =1 (1 − q k − n + | p |− i ) if p < . While there is no guarantee that this will give a minimal order inhomogeneous recursion relation for J p ( n ), itdoes work for − ≤ p ≤
11. To show that the recursion relation for ( J p ) is of minimal order (and incidentallyto check the AJ Conjecture of [Ga]), we can set q = 1. An explicit computation shows that the L -degree of A nhp ( L, M, q ) does not drop when we specialize to q = 1 and moreover we have A nhp ( L, M,
1) = A p ( L, M ) F p ( M ) , where A p ( L, M ) is the A -polynomial of the twist knot K p , and F p ( M ) ∈ Q ( M ). A p ( L, M ) has beencomputed by Hoste-Shanahan in [HS1], and has been shown to be irreducible in [HS2]. It follows that A nhp ( E, Q, q ) does not have any right factors and concludes the proof of Theorem 2. (cid:3) Odds and ends
A generalization of Theorem 2.
In fact, the multi-certificate proof of Theorem 2 implies the followingresult.
Theorem 4. If c ( n, k ) is proper q -hypergeometric term and ( ˆ J ( n )) is q -holonomic, and J ( n ) = n X k =0 c ( n, k ) ˆ J ( k ) , then J ( n ) is q -holonomic. A linear q -difference equation for ( J ( n )) can be constructed from a linear q -difference equation for ( ˆ J ( n )) and c ( n, k ) . A software package that accompanies the proof Theorem 2 was developed by the second author.
Remark . Our proof of Theorem 2 reduces to solving a system of 2 | p | linear equations over the field Q ( q, q n ). When − ≤ p ≤
11, this system can be solved explicitly by symbolic software. Le’s algorithm forcomputing the non-commutative A -polynomial of a 2-bridge knot, also requires a system of linear equations(2 | p | )! over the field Q ( q, q n ) in the case of twist knots; see [Le]. However, an implementation of Le’s algorithmexceeded the capacity of our symbolic software for p = 1 and p = − Is there a recursion of the non-commutative A -polynomial with respect to the number oftwists? Recall that A p ( L, M ) denotes the A -polynomial of the twist knot K p . In [HS1], Hoste-Shanahanuse a trace identity in SL(2 , C ) in order to give a second order linear recursion relation for the sequence ( A p ).There is supporting evidence that A nhp ( L, M,
1) is annihilated by the following operator(27) M ( M − ( M + 1) ( L + M ) − ( M − ( M + 1) ( M − LM + 2 LM + L M + M + 2 LM + 2 LM + L − L ) P + P if p > − ( M − ( M + 1) ( M − LM + 2 LM + L M + M + 2 LM + 2 LM + L − L ) P + M ( M − ( M + 1) ( L + M ) P if p < , where P A nhp ( L, M,
1) = A nhp +1 ( L, M, . Equation (27) may be proven using the recursion on ˆ J p ( k ) and its simplification when q = 1; see [GS1,Thm.2]. Unfortunately, there is equally strong evidence that the sequence A nhp ( E, Q, q ) does not satisfy alinear recursion with respect to p . Appendix A. A generating functions approach
In this appendix we present an alternative approach to get a recursion relation for J K ( n ) given Equation(3) and a recursion relation for ˆ J K ( n ). This idea was communicated to us by D. Zeilberger, and may beuseful in its own right. We were not able to compute the non-commutative A -polynomial for twist knotsthis way.To explain the idea, let us recall first that a sequence ( a ( n )) of rational numbers is holonomic iff thegenerating series F ( z ) = ∞ X n =0 a ( n ) z n is holonomic, i.e., it is annihilated by an element of the Weyl algebra Q h z, d/dz i ; see [Z1]. The q -analogueof this is the following. Consider a sequence ( a ( n )) with a ( n ) ∈ Q ( q ), and the generating series(28) F ( z, q ) = ∞ X n =0 a ( n ) z n ∈ Q ( q )[[ z ]]There are two operators Q and Z that act on the elements F ( z, q ) of Q ( q )[[ z ]] by:( QF )( z, q ) = F ( qz, q ) , ( ZF )( z, q ) = zF ( z, q )It is easy to see that QZ = qZQ , and that ( a ( n )) is q -holonomic iff the generating series F ( z, q ) is q -holonomic.Now, let us consider two sequences ( J ( n )) and ( ˆ J ( n )) of rational functions that are related by: HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 13 (29) J ( n ) = n X k =0 c ( n, k ) ˆ J ( k )where the kernel c ( n, k ) is given by (4). c ( n, k ) can be slightly simplified into q − nk ( q ; q ) n + k ( q ; q ) n − k − (1 − q n ) . We willabsorb the factor − q n in the colored Jones function and define γ ( n, k ) := q − nk ( q ; q ) n + k ( q ; q ) n − k − = c ( n, k )(1 − q n )ˇ J ( n ) := n X k =0 γ ( n, k ) ˆ J ( k )= ˆ J ( n )(1 − q n ) . Proposition A.1.
Let (30) H ( k, z ) = ∞ X i =0 γ ( k + i, k ) z i , then we have: (31) H ( k, z ) = ( − k q − k ( k +1)2 ( q ; q ) k +1 z k ( qz ; q ) k ( z ; q ) k +2 . Proof.
We will use the idea of the WZ-algorithm to the hypergeometric summand:(32) H ( k, i, z ) = γ ( k + i, k ) z i . We claim that:(33) ( z − q k +1 ) H ( k + 1 , i, z ) + (1 − q k +2 )(1 − q k +3 ) q k +1 (1 − zq k +2 ) H ( k, i, z ) = G ( k, i, z ) − G ( k, i − , z ) , where(34) G ( k, i, z ) = − z (1 − q k + i +1 )( zq k +2 − − zq k + i +4 + q k + i +5 q k + i +1 (1 − zq k +2 ) H ( k, i, z ) . Equation (33) can be verified by dividing both sides by H ( k, i, z ), and then it reduces to an identity in thefield Q ( z, q, q k ) which can be readily checked. Now summing both sides of Equation (32) over k , and we getthe desired result. (cid:3) Consider the generating function of ˇ J ( n ):(35) F ( z, q ) = ∞ X n =0 ˇ J ( n ) z n . Proposition A.2.
We have: (36) F ( z, q ) = ∞ X k =0 ( − k q − k ( k +1)2 ( q ; q ) k +1 ( qz ; q ) k ( z ; q ) k +2 ˆ J p ( k ) . Proof.
We will interchange the order of summation and use Proposition A.1. We get: F ( z, q ) = ∞ X n =0 ˇ J ( n ) z n = ∞ X n =0 n X k =0 γ ( n, k ) ! ˆ J p ( k ) z n = ∞ X k =0 ˆ J p ( k ) ∞ X n = k γ ( n, k ) z n ! , = ∞ X k =0 ˆ J p ( k ) z k ∞ X i =0 γ ( k + i, k ) z i ! , = ∞ X k =0 H ( k, z ) ˆ J p ( k ) z k = ∞ X k =0 ( − k q − k ( k +1)2 ( q ; q ) k +1 ( qz ; q ) k ( z ; q ) k +2 ˆ J ( k ) . (cid:3) One can use a q -difference equation for ˆ J ( n ) and Proposition (28) to get a q -difference equation for F ( z, q ).This will be explored in another publication. Appendix B. The non-commutative A -polynomial for p = ± A -polynomial for p = ±
3. The reader maycompare the size of the output with Theorem 1.
HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 15 A nh ( E, Q, q ) = q n +20 ( − q n )( q n +8 − q n +7 − q n +6 − q n +9 − − q n +14 ( q n +1 − q n +2 − q n +7 − q n +8 − q n +9 − − − q n + q n + q n + q n + q n +3 + q n − q n − q n − q n − q n + q n +3 + q n − q n − q n − q n +2 − q n − q n − q n +4 + q n − q n + q n − q n +5 + q n + q n − q n ) E + q n +9 ( q n +2 − q n +1 − q n +4 − q n +8 − q n +9 − − − q n − q + q n − q n − q n + 2 q n − q n +2 q n − q n +9 − q n − q n − q n − q n − q n − q n + 2 q n + 2 q n + q n − q n + q n + q n +4 q n +12 + q n + q n − q n − q n + q n + q n − q n + 2 q n − q n − q n + q n +2 − q n + q n + q n + 5 q n + 3 q n + 2 q n + 2 q n − q n − q n +4 + 2 q n +2 q n + 2 q n + q n − q n − q n + 3 q n +16 − q n + q n + q n ) E − q ( q n +3 − q n +1 − q n +2 − q n +6 − q n +9 − − q n + q n − q n + 2 q n +9 − q n + q n − q n +22 − q n − q n − q n +22 − q n + q n − q n − q n − q n + q n +21 + 2 q n + 2 q n − q n +12 + 2 q n − q n + q n − q n − q n − q n + 2 q n + q n +15 + q n +25 + 2 q n + q n + 2 q n − q n − q n − q n +2 q n +20 − q n − q n + 2 q n + q n +4 + q n +13 − q n − q n + 2 q n +19 + 2 q n − q n − q n +12 + q n − q n +16 +4 q n − q n + q n +26 − q n ) E + q ( q n +4 − q n +1 − q n +2 − q n +3 − q n +8 − − − q + q n + q n +17 + q n +9 + q n + q n − q n +21 − q n − q n + q n + q n + q n − q n − q n − q n +21 + 2 q n + q n − q n + q n + q n +4 − q n − q n + q n +5 + q n +12 + q n +10 ) E + ( q n +5 − q n +1 − q n +2 − q n +3 − q n +4 − E ,B ( q n , q ) = q n +12 ( q n +1 + 1)( q n +2 + 1)( q n +3 + 1)( q n +4 + 1)( q n +1 − q n +3 − q n +5 − q n +7 − q n +9 − , A nh − ( E, Q, q ) = q n +18 ( − q n )( q n +7 − q n +8 − q n +9 − q n +10 − q n +11 − − q n +14 ( q n +1 − q n +2 − q n +8 − q n +9 − q n +10 − q n +11 − q + q n +12 + q n − q n +7 − q n + q n − q n +2 + q − q n + q n +14 − q n − q n +9 − q n − q n − q n − q n − q n − q n + q n − q n − q n + q n − q n + q n − q n + q n − q n + q n − q n + q n +3 − q n − q n +4 + q n + q n ) E + q n +9 ( q n +2 − q n +1 − q n +4 − q n +9 − q n +10 − q n +11 − − q n +24 + 3 q n + q + q n − q n − q n − q n − q n +19 + 4 q n + 3 q n − q n +9 + q n − q n + q n + q − q n + q n + q n +17 − q n + 2 q n + q n + q n + 3 q n − q n +9 − q n +23 + 3 q n +16 − q n − q n + q n − q n +10 − q n + q n + 2 q n + 2 q n +21 − q n − q n − q n − q n + 3 q n + 3 q n +12 + 4 q n + 2 q n − q n + 2 q n − q n + 2 q n + q n +22 − q n +12 + 2 q n +20 − q n − q n +16 + 3 q n + q n − q n + q n − q n + 6 q n − q n + q n + 2 q n − q n +13 − q n + 4 q n − q n − q n +20 − q n + q n + 2 q n +2 q n − q n + 3 q n − q n + 2 q n +21 + q n − q n + q n ) E − q ( q n +3 − q n +1 − q n +2 − q n +6 − q n +10 − q n +11 − q n +24 − q n − q n − q n − q n − q n + 7 q n +19 + q n − q n − q n +9 + q n + 4 q n − q n + q n − q n +17 +2 q n + 7 q n − q n − q n − q n +21 + q n +23 − q n +16 − q n − q n − q n − q n + q n + q n − q n +21 + q n − q n + q n +32 − q n − q n − q n + 3 q n − q n +28 + q n − q n − q n + q n +12 − q n − q n + q n − q n + q n +32 + 2 q n − q n − q n − q n +22 − q n +28 + 3 q n − q n +32 − q n +22 − q n +4 q n +12 − q n +20 + 4 q n + 2 q n +31 − q n +28 − q n +30 + 3 q n − q n − q n +29 − q n +15 + 7 q n + q n − q n + 2 q n + 5 q n +13 + 6 q n − q n + 6 q n + 5 q n + 3 q n +30 − q n + q n + 3 q n +25 + q n +3 q n + 2 q n − q n − q n +27 − q n + q n + q n +30 − q n +16 +3 q n − q n +21 − q n + q n − q n − q n + 2 q n ) E + q ( q n +4 − q n +1 − q n +2 − q n +3 − q n +8 − q n +11 − q n + q n + q − q n + 3 q n − q n − q n + q n − q n + 2 q n +2 q n − q n +12 + 3 q n +9 − q n + q n + 4 q n − q n +27 + q n + q n − q n + q n +17 + 3 q n + q n − q n + q n − q n +11 +3 q n + q n + q n + 3 q n + 2 q n − q n + q n − q n − q n + 2 q n − q n − q n − q n +34 − q n +20 − q n − q n +28 − q n +32 + q n + q n + q n + q n +22 − q n − q n + 2 q n +9 +2 q n +30 − q n +33 + 6 q n + 3 q n + 2 q n + 2 q n +29 + 4 q n +15 − q n − q n +19 − q n − q n − q n +13 − q n + 3 q n +29 − q n + 3 q n +16 +2 q n + q n − q n − q n − q n − q n +19 + 2 q n +31 + q n +2 q n − q n ) E − ( q n +5 − q n +1 − q n +2 − q n +3 − q n +4 − q n +10 − − q n + q n − q n + q n +19 − q n + q n − q n + q n +9 + q n +10 + q n + q n − q n + q n − q n − q n + q n + q n − q n + q n − q n − q n +30 − q n − q n +33 − q n − q n − q n + q n − q n − q n +21 − q n + q n +16 − q n +15 + q n − q n ) E + q n +30 ( q n +6 − q n +1 − q n +2 − q n +3 − q n +4 − q n +5 − E , HE NON-COMMUTATIVE A -POLYNOMIAL OF TWIST KNOTS 17 B − ( q n , q ) = q n +15 ( q n +1 + 1)( q n +2 + 1)( q n +3 + 1)( q n +4 + 1)( q n +5 + 1)( q n +1 − q n +3 − q n +5 − q n +7 − q n +9 − q n +11 − . For a computer data of the non-commutative A -polynomial of twist knots, see [GS2]. Appendix C. The A -polynomial for p = − , . . . , A -polynomial A p ( L, M ) of the twist knot K p , taken from [HS1]. p A p ( L, M )1 L + M − − L + LM + M + 2 LM + L M + LM − LM − L + L + 2 L M + LM + 2 L M − LM − L M + 2 LM + L M + 2 LM + M − LM − L − L − L M + L M − LM − L M + 3 LM + 3 L M + M + 3 LM + 6 L M + 3 L M + L M + 3 L M + 3 L M − L M − L M + LM − L M − LM + L M L − L + L − L M + 4 L M − L M + 2 L M + 3 L M + 5 L M + 5 L M + LM + L M + 6 L M − LM − L M − L M − L M + 6 L M + L M + L M + 5 L M +5 L M + 3 LM + 2 L M − L M + 4 LM − L M + M − LM + L M − − L + 2 L − L + 5 L M − L M + L M + 3 L M − L M − L M − L M + 4 L M − LM − L M + 5 LM + 12 L M + 10 L M + M + 4 LM + 10 L M + 20 L M +10 L M + 4 L M + L M + 10 L M + 12 L M + 5 L M − L M − L M + 4 L M − L M − L M − L M + 3 L M + LM − L M + 5 L M − LM + 2 L M − L M References [AZ] M. Apagodu and D. Zeilberger,
Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening ofWilf-Zeilberger theory , Adv. in Appl. Math. (2006) 139–152.[B-N] D. Bar-Natan, KnotAtlas , http://katlas.math.toronto.edu/wiki .[Bo] D.W. Boyd, Mahler’s measure and special values of L -functions , Experiment. Math. (1998) 37–82.[CCGLS] D. Cooper, D, M. Culler, H. Gillet, D. Long and P. Shalen, Plane curves associated to character varieties of -manifolds , Invent. Math. (1994) 47–84.[Cu] M. Culler, A table of A -polynomials , ∼ culler/Apolynomials .[Fa] Fasenmyer, Sister Mary Celine, Some generalized hypergeometric polynomials , Ph.D. dissertation, University of Michi-gan, November, 1945.[GL1] S. Garoufalidis, T.T.Q. Le,
The colored Jones function is q -holonomic Geom. and Topology (2005) 1253–1293.[GL2] and T.T.Q. Le, Asymptotics of the colored Jones function of a knot , preprint 2005 math.GT/0508100 .[Ga] ,
On the characteristic and deformation varieties of a knot , Proceedings of the CassonFest, Geometry andTopology Monographs (2004) 291–309.[GS1] and X. Sun, The C -polynomial of a knot , Algebr. Geom. Topol. (2006) 1623–1653.[GS2] and , Computer data on the non-commutative A -polynomial of twist knots , 2008.[Ge] R. Gelca, On the relation between the A -polynomial and the Jones polynomial , Proc. Amer. Math. Soc. (2002)1235–1241.[Ha] K. Habiro, On the quantum sl invariants of knots and integral homology spheres Geom. Topol. Monogr. (2002)55–68.[HS1] J. Hoste and P. Shanahan, A formula for the A -polynomial of twist knots , J. Knot Theory and its Rami. (2004)193–209.[HS2] and , Trace fields of twist knots , J. Knot Theory and its Rami. (2001) 625–639.[Jo] V. Jones, Hecke algebra representation of braid groups and link polynomials , Annals Math. (1987) 335–388.[Ki] F. Kirwan,
Complex algebraic curves , London Mathematical Society Student Texts, , Cambridge University Press,1992.[Ko] T. H. Koornwinder, On Zeilberger’s algorithm and its q-analogue , J. Comp. and Appl. Math., (1993) 91–111.[Le] T.T.Q. Le, The Colored Jones Polynomial and the A -Polynomial of Two-Bridge Knots , Advances in Math. (2006) 782–804.[Ma] G. Masbaum, Skein-theoretical derivation of some formulas of Habiro , preprint, December 2002.[PR1] P. Paule and A. Riese,
A Mathematica q -Analogue of Zeilberger’s Algorithm Based on an Algebraically MotivatedApproach to q -Hypergeometric Telescoping , in Special Functions, q -Series and Related Topics, Fields Inst. Commun., (1997) 179–210.[PR2] , Mathematica software: [PR3] , Mathematica software: [PWZ] M. Petkovˇsek, H.S. Wilf and D. Zeilberger, A = B , A.K. Peters, Ltd., Wellesley, MA 1996.[Rf] D. Rolfsen, Knots and links, Publish or Perish, 1976.[S] C. Schneider, Symbolic Summation Assists Combinatorics , Sem.Lothar.Combin. (2007) 1–36.[Ta] T. Takata, The colored Jones polynomial and the A-polynomial for twist knots , preprint 2004 math.GT/0401068 .[Th] W. Thurston,
The geometry and topology of 3-manifolds , Lecture notes, Princeton 1977.[Tu] V. Turaev,
The Yang-Baxter equation and invariants of links , Inventiones Math. (1988) 527–553.[WZ] H. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and q ) multisum/integral iden-tities , Inventiones Math. (1992) 575–633.[Z1] D. Zeilberger, A holonomic systems approach to special functions identities , J. Comput. Appl. Math. (1990)321–368.[Z2] , Maple software: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA, ∼ stavros E-mail address : [email protected] Department of Mathematics, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118, USA, ∼ xsun1 E-mail address ::