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On Jones Knot Invariants and Vassiliev Invariants 1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), 293-299 ON JONES KNOT INVARIANTS AND VASSILIEV INVARIANTS JUN ZHU (Received May 1996) Abstract. We show that the n -th derivative of a quantum group invariant, evaluated at 1, is a Vassiliev invariant while the derivative of the Jones poly­ nomial, evaluated at a real number ^ 1, is not a Vassiliev invariant. The coeffi­ cients of the classical Conway polynomial are known to be Vassiliev invariants. W e show that the coefficients of the Jones polynomial are not Vassiliev invari­ ants. 1. Introduction Let S 1 be the unit circle in the complex plane with a given orientation. A singular knot of order n is a piecewise linear immersion L : S 1 — > M3 which has exactly n transverse double points. Two singular knots L and L' are equivalent if there exists an isotopy ht : M3 R 3, t G [0,1] such that ho = id, h\L = L' and the double points of htL are all transverse for every t G [0,1]. We denote K x , K + and the singular knots identical outside a small ball around a crossing and different inside a ball as shown in Figure 1. F i g u r e 1. Related knot diagrams. Let ICi be the set of equivalence classes of singular knots with exactly i double points. In particular, /Co = /C, the set of all knot types. Let’s abbreviate Uj>* by & > i and denote the Q-vector space generated by a set A by Q(-A). Then the n-th finite type space Fn is defined as the vector space generated by the set /C>o subject to the following relations (1) K X = K + - K _ for K x G JC> i (2) K = 0 for K e J C > n+l. Any element in the dual space Hom(jPn,Q ) of Fn, but not in Hom(Fn_i, Q ) is called a Vassiliev invariant of order n or finite type invariant of order n. 1991 A M S Mathematics Subject Classification: Primary 57M25; Secondary 57M15, 57N99. K ey words and phrases: Knot, Jones Polynomial and Vassiliev invariant. 294 JUN ZHU In [4], J. Birman and X.S. Lin established a fundamental relationship between Jones knot invariants and Vassiliev invariants via a substitution t = ex. We show here that this relation can be rearranged into a different form (Theorem 1), probably more natural, from which one can prove that every derivative of a quantum group invariant, evaluated at 1, is a Vassiliev invariant. But the derivative of the Jones polynomial, evaluated at a real number ^ 1, is not a Vassiliev invariant. Bar-Natan [1] observed that every coefficient of the Conway polynomial is of finite type, it is a little bit surprising that every coefficient of the Jones polynomial is not of finite type. This new result is proved in Theorem 3. For more definitions and notations in this paper, we refer to [1], [3] or [4]. Acknowledgement. I wish to thank professor D. Rolfsen for guidance and pro­ fessor K. Lam for encouragement. Thanks also go to J. Birman and X.S. Lin for their helpful comments on this work. 2. Relationship Between Quantum Group Invariants and Vassiliev Invariants Recall that a quantum group invariant or a generalized Jones invariant is a knot or link invariant obtained from a trace function on a “/2-matrix representation” of the family of braid groups { B n \ n — 1,2,3 • • • }. The Jones polynomial, the HOM- FLY polynomial and the Kauffman polynomial are all quantum group invariants. See [3] for more details. Theorem 2.1. Let J(t) = J2^=-ooc^ n be a quantum group invariant and J{t) = ££M £)(t_ 1)m m=0 be the Taylor series of J(t) at 1, where J^m\ 1) denotes the m -th derivative eval­ uated at 1. Then the constant term is 1 and the coefficient J m^ of (t — l ) m is a Vassiliev invariant of order m for m > 1. Proof. First we note that a Vassiliev invariant of order m times a nonzero constant is also a Vassiliev invariant of order m, and that the sum of a Vassiliev invariant of order m and a Vassiliev invariant of order m! is a Vassiliev invariant of order less than or equal to maIt is easy to see that the constant term is j«»(i) = j(i) = i, so we assume m > 1. By the definition of derivative, we have oo j(m )(i) _ Cnn(n - 1 ) ■ ■ ■ (n - (m - 1)). n = — oo Since n(n-!)■■■ (n-{m -l)) = nm - I V i I n”1"1 + ( V ] ij I nm~2 H-----+ (— — l)!n, ON JONES KNOT INVARIANTS AND VASSILIEV INVARIANTS 295 we have oo j"*( i) = £ cnn ) n= — oo oo + (-l)m_1(m - 1)! ^ 2 cnn. 71— — OO On the other hand, substituting t = ex in J(t) — J2^L-oo c^ n and expanding ex in Taylor series, we have OO OO OO m OO / OO J (t )= £ c « r * = £ C"£^r*m = E £ c» n= —oo 7i= —oo m = 0 m = 0 \ n——oo By a theorem of Birman and Lin, see [4, Theorem 1], the coefficient of x m 71— — OO 71 — — OO is a Vassiliev invariant of order m for every m > 1. Therefore, by the remark at the beginning of this proof, as a linear combination of Vassiliev invariants of order < m, is a Vassiliev invariant of order less than or equal to m. Since there is only one term in the summation having order m, — is a Vassiliev invariant of order m. The proof is completed. □ Remark 2.2. The theorem has been obtained by J. Birman and X.S. Lin [3, Corallary 4.3] in the case of HOMFLY polynomial. Corollary 2.3. The m -th derivative of a quantum group invariant evaluated at 1 is a Vassiliev invariant of order m. In particular, the m -th derivative of the Jones polynomial, evaluated at 1, is a Vassiliev invariant of order m. Proof. It is clear from the proof of the above theorem. □ Corollary 2.4. For every quantum group invariant J(t), J^l) = 0 and J"( 1) = av2, where a is a constant and v2 is the first nontrivial Vassiliev invariant. F i g u r e 2. Torus knot of type (2,2n + 1). 296 JUN ZHU P roof. The first equation follows from the fact that there are no nontrivial Vassiliev invariants of order 1. The second one follows from that there is only one nontrivial Vassiliev invariant v2 of order less than or equal to 2, up to constant multiplication. □ In connection with Theorem 1, we may ask if Theorem holds for Taylor series expansion in powers of (t — c) with 1. The answer is no, because of the following theorem. Theorem 2.5. The Jones polynomial evaluated at any nonzero complex number, except \,uj and uj2 , is not of finite type, where uj is a primitive root of x3 = 1. P roof. Let J(t) be the Jones polynomial. We extend the Jones polynomial to singular knots as follows: JKx (t) — JK+{t) ~ where the K x , K + and denote the singular knots identical outside a small ball around a crossing and different inside a ball as shown in Figure 1. Let i^2n+i be the torus knot of type (2,2n+ 1) as shown in Figure 2. According to V. Jones [6], Jk2„+1(«) = i ^ ( l - t2n+2 - t 3 + t2n+3). Let K 2n denote the singular knot shown in Figure 3, then we have 2n +1 2n F igure 3. A singular torus knot. ON JONES KNOT INVARIANTS AND VASSILIEV INVARIANTS 297 JK2u(t) = 5 ^ ( - l)P( S ) jK 2{2n_p)+1{t) p= 0 VP ' 2r\ /9rA = ^(_1)P j (1 _ t2(2n-p)+2 _ t3 + t2(2n-P)+3) p=0 \P / = r h [ % } ~ l)P0 t2n~r - ) <3(2" " rt+2 - ^ ( - i ) p (2n\ 2n-p+3 + ^ (-l)pf2nV <2n'-ri+3l P=o V P / p=o \ P / J 1 {(1 - t)2n - t2( 1 - i3)2n - t3( 1 - *)2n + t3(l - i3)2n} l - * 2 1 (1 - t)2n{ 1 - *3 - i2(l + i + i2)2n(l - t)} l ^ i 2 1 1 _ 2 (i - *)2n(l - *3)(l- *2(1 + t + £2)2n-1). It follows that if t is not a root of the above term, then JK2n(t) is nonzero. Hence, to prove the theorem we only need to deal with these roots. To this end we consider some other singular knots. Let K 2n+1 be the knot as shown in Figure 4. 2n+2 2n*1 F i g u r e 4. A singular torus knot with 2n + 1 double point. Similarly to the above computation, we obtain = T^ f(i-t)’*«(i-*s)(i -t2(i+t+*2n . It is easy to see that —1 is the only common zero of (l — t2(l + t + f2)2n_1) and (l — i2(l + 1 + t2)2ny Since t= — 1 is a zero of order 1 for (l — t2 (l + 1 + t2)2n_1), JK2n(—1) ^ 0. Therefore, given any to which is not a cube root of unity, for every positive integer n there exists a singular knot K with more than n double points such that Jk:(£o) 0- Hence the Jones polynomial evaluated at to is not of finite type. The proof is completed. □ 298 JUN ZHU R em ark 2.6. The Jones polynomial evaluated at any cube root of unity is known to be the constant 1.
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