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 poly­nomial, evaluated at a real number ^ 1, is not a Vassiliev invariant. The coeffi­cients of the classical Conway polynomial are known to be Vassiliev invariants.We show that the coefficients of the Jones polynomial are not Vassiliev invari­ants.

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 pro­fessor 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 m a I t 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

ooj(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

) oo

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 invariantsof 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.

we have

j"*( i) = £ cnn

F ig u r e 2. Torus knot of type (2,2n + 1).

296 JUN ZHU

Proof. 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 \ , u j and uj2 , is not of finite type, where uj is a primitive root of x3 = 1.

Proof. 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 ) j K 2{2n_p)+1{t) p= 0 V P '

2r\ /9rA= ^ ( _ 1 ) P j ( 1 _ t 2 ( 2 n - p ) + 2 _ t 3 + t 2 ( 2 n - P ) + 3 )

p=0 \P /

= r h [ % } ~ l)P0 t2n~r - ) <3(2" " rt+2

- ^ ( - i ) p ( 2n\ 2n- p+3 + ^ ( - l ) p f 2nV < 2n'-ri+3lP=o V P / p=o \ P / J1

l - * 2

1l ^ i 2

1

{(1 - t)2n - t2( 1 - i3)2n - t3( 1 - *)2n + t3(l - i3)2n}

(1 - t)2n{ 1 - *3 - i2(l + i + i2)2n(l - t)}

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 - t 2(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

Rem ark 2.6. The Jones polynomial evaluated at any cube root of unity is known to be the constant 1. Hence we have completely determined whether the Jones polynomial, evaluated at a complex number, is of finite type.

Rem ark 2.7. A similar proof shows that the one-variable HOMFLY polynomials evaluated at a complex number outside the unit circle are not of finite type.

3. The Coefficients o f the Jones Polynom ial

Bar-Natan [1] observed that every coefficient of the Conway polynomial is a Vassiliev invariant. In contrast, we have

Theorem 3.1. Every coefficient of the Jones polynomial is not of finite type.

Proof. First we prove the constant term Co of the Jones polynomial J — Yl^L-oo cn̂ n is not of finite type.

Notice that if the minimal degree of the Jones polynomial J (K ) for a knot K is greater than zero, then c$(K) = 0. Denote the torus knot of type (2, 2n + 1) by K 2n+i, see Figure 2.

According to V. Jones [6], we have

= j-T ta(l - t 2" +2- t 3 + t2n+3).

Hence co(K2n+i) = 0 for n > 1 and co(K\) = 1.Now extend cq to singular knots as usual, i.e.,

co(tfx) = c o { K + ) -c o ( K -) .

where K x , K + and K - denote the singular knots identical outside a small ball around a crossing and different inside a ball as shown in the Figure 1.

Let K 2n be the singular knot as shown in Figure 3. From the above observation, we have

co (K2") = f > l ) ”(2n)co(tf2(2n_p)+1) = (-1)2" = 1 / 0 .p=0 \P /

Since n is arbitrary, we see that cq is not of finite type.Now we turn to cr for r > 0. It is well known that for the trefoil knot K we have

J{K ) = t + t3 - t A.

Let K r# K 2n+\ be the connected sum of K 2n+i and r copies of K . By a property of the Jones polynomial, we have J (K r# K 2n+\) — J (K )r J (K 2n+i). It follows that the minimal degree of J (K r# K 2n+i) > r + 1 for n > 1, hence we have

cr(K r# K 2n+1) = 0 for n > 1.As before we extend cr to singular knots, since cr(K r# K i) = cr(K r) = 1, we have

^ j.

cr(Kr# K 2n) = Cr(Kr#Kv, 2— ri+i). = ( - 1 ) 2" + o.p=0 p

Hence cr is not of finite type.Applying the above argument to the mirror images of knots we can prove that

cr is also not of finite type for r < 0. This completes the proof. □

ON JONES KNOT INVARIANTS AND VASSILIEV INVARIANTS 299

Rem ark 3.2. Theorem 3 is also true for the one-variable HOMFLY polynomials.

References

1. Dror Bar-Natan, On the Vassiliev invariants, Topology 34 (1995), 423-472.2. J. Birman, Braids, Links and Mapping Class Group, Ann. Math. Stud. Vol. 84,

1974.3. J. Birman, New points of view in knot theory, Bull. AMS 28 (1993), 153-287.4. J. Birman and Xiao-Song Lin, Knot polynomials and Vassiliev invariants,

Invent. Math. I l l (1993), 225-270.5. V. Jones, A polynomial invariant for knots via von Neumann algebra,

Bull. AMS 12 (1985), 103-111.6. V. Jones, Hecke algebra representations of braid groups and link polynomials,

Ann. Math. 126 (1987), 335-388.7. L. Kauffman, State models and the Jones polynomial, Topology 26 (1987),

395-407.8. D. Rolfsen, Knots and Links, Publish or Perish, Berkeley, CA, 1976.9. T. Stanford, Finite Type Invariants of Knots, Links and Graphs, Columbia

University, preprint, 1992.10. R. Trapp, Twist sequences and Vassiliev invariants, Jour, of Knot Theory and

its Ramification, Vol. 3, No.3, (1994) 391-405.11. V.A. Vassiliev, Cohomology of knot spaces, in Theory of Singularities and Appli­

cations (V.I. Arnold, ed.), Amer. Math. Soc., Providence, RI, 1990, pp. 23-69.

Jun ZhuMathematics DepartmentUniversity of British ColumbiaVancouver B.C.C A N A D A V 6T 1Z2zhuj@math.ubc.ca