On finiteness of certain Vassiliev invariants

The best known examples of Vassiliev invariants are the coefficients of a Jones-type polynomial expanded after exponential substitution. We show that for a given knot, the first N Vassiliev invariants in this family determine the rest for some integer N . Supported in part by NSF Grant No. DMS-9504471 Supported in part by NSF Grant #23068 2 Kauffman, L. H. et. al. In this paper we prove a finiteness property for certain families of Vassiliev knot invariants. Let VK(t) denote the Jones polynomial [4] of a classical knot or a link K (i.e. smoothly embedded circle(s) in the 3-space). It is known [2, 1] that the coefficients of powers of x in VK(e ) are Vassiliev invariants of finite type. More specifically, if we write VK(e ) = ∑ ∞ n=1 vn(K)x , then vn(K) is a Vassiliev invariant of type n. Let us call {vn(K)} the Jones-Vassiliev invariants. More generally, if PK(α, z) denotes the HOMFLY polynomial [3, 4] and YK(α, z) the Kauffman polynomial [5], we obtain a Laurent polynomial (possibly times t) by substituting t−t for z and t for α, where a is a nonnegative half-integer. Again substituting e for t, the nth coefficient of the power series in x is a degree n Vassiliev invariant [2, 1]. Let us call these Vassiliev invariants respectively pa,n(K) and fa,n(K). We prove the following. Theorem 1 Given a projection of a link K, vn(K) is a linear combination of the quantities v0(K), . . . , vN (K), where N is the number of crossings in the projection. The coefficients of the linear combination depend on the following data: n, N , the number of separable pieces in K, the writhe of the projection, and the number of components of the link obtained by parallel smoothings at positive crossings and transverse smoothings at negative crossings that are defined in [10]. Theorem 2 Given a projection of a link K and a half-integer a, pa,n(K) is a linear combination of pa,0(K), . . . pa,N (K), where N is (2a− 1)(s− 1) + c, s is the number of Seifert circles in the diagram, and c is the crossing number of the link. The coefficients of the linear combination depend only on n, N , and the writhe of the diagram. Theorem 3 Given a projection of a link K and a half-integer a, fa,n(K) is a linear combination of fa,0(K), . . . fa,N(K) where N is the greatest integer less than 3ac, for c the crossing number, and the linear combination depend only on n, N , the number of positive and the number of negative crossings in the diagram, and the number of connected components of the diagram. These theorems follow from bounds on the degrees of the link polynomials together with the following straightforward lemma. Vassiliev invariants 3 Lemma 1 If F (t) is of the form t times a degree N polynomial P (t) in t, so that F (t) = tP (t), for some M ∈ R, and the largest and smallest powers of t in F are L and M respectively (so that N = L −M), then there exist linear functions fL,N,j(x0, . . . , xN ), for each positive integer j, such that d/dxF (e)|x=0 = fL,N,j(F (1), d/dx(F (e ))|x=0, . . . , d /dx (F (e))|x=0). Proof. Consider the linear ODE Def(x) = 0, where D = d/d(e) = ed/dx. Notice that changing variables via t = e this is just (d/dt)tf(ln t) = 0, so in particular f(x) = F (e) is a solution. Since this is a linear N + 1st order differential equation in x, any solution is a linear combination of the N+1 fundamental solutions with the coefficients being certain linear combinations of the initial data. By the initial data we mean the first N derivatives of the solution evaluated at x = 0, which in this case are F (1), d/dx(F (e))|0, . . . , d /dx (F (e))|0. By differentiating this linear combination j times, we see that (d/dx)F (e)|0 is a linear combination of the initial data too. 2 Proof (of Theorem 1). By [6, 10, 11] the span of the Jones polynomial (the highest degree minus the lowest degree) is bounded by c + g − 1, where c is the crossing number and g is the number of disconnected components of the projection. Further, the upper and lower bounds depend on the quantities given in the statement of the theorem. But notice that VK(t) is divisible by (t + t), since this is true of a link with g or more unlinked unknots, and VK(t) is a linear combination of such by the skein relations. The result of dividing VK(t) by this factor is a Laurent polynomial (possibly times t ) with span bounded by exactly c. By the lemma, the jth derivative of this Laurent polynomial evaluated at t = 1 can be written explicitly as a linear combination of the various kth derivatives for 0 ≤ k ≤ c. Since the jth derivative of VK(t) is a linear combination of the kth derivative of this polynomial for k ≤ j, and vice versa, this gives the result. 2 4 Kauffman, L. H. et. al. Corollary 1 If a knot K has a projection with N crossings, and vk(K) = 0 for 1 ≤ k ≤ N , then the Jones polynomial of K is 1. 2 Proof (of Theorem 2). It was proved in [9] that dmax(z) ≤ c− s+ 1 and w − s+ 1 ≤ dmin(α) ≤ dmax(α) ≤ w + s− 1 where dmin and dmax denote the lowest and highest degrees of the respective variable in PK(α, z), and c, s and w are respectively the crossing number, number of Seifert circles and writhe of the projection of a knot K. Notice here that the negative powers of z arise from the loop value (the contribution of a disjoint trivial circle in the skein computations, see [5]) δ = (t − t)/(t − t) which is a power of t times a polynomial in t. Thus terms with negative powers of z have lower dmax(t) and higher dmin(t) than the degree of α would indicate. So the largest power of t which can occur would come from the product of the largest power of α with the largest power of z, and the smallest power of t comes from the smallest power of α times the largest power of z. Since the largest power of z occurs multiplied by the lowest and highest power of α, we have after substitution that a · dmin(α) − dmax(z)/2 ≤ dmin(t) ≤ dmax(t) ≤ a · dmax(α) + dmax(z)/2 and hence we get the highest and lowest degrees bounded by aw ±N/2. From this and the lemma the result follows. 2 Proof (of Theorem 3). The skein definition of the Kauffman polynomial (Dubrovnik version) is given as follows. Let D be the polynomial defined by the skein relation DK+ −DK− = z(DK= −DK)() DK<+> = αDK DK<−> = α DK Vassiliev invariants 5 where K+, K− are link diagrams with positive and negative crossings at a single particular crossing point respectively (and the rest of their diagrams coincide), K= and K)( are link diagrams obtained by two ways of smoothings at the crossing. In the second and third equalities K<+> (resp. K<−>) denotes the link diagram K with a small positive (resp. negative) kink added. Now the Kauffman polynomial [7] is defined by YK(α, z) = α D(α, z) where w(K) is the writhe of the diagram K. Note that the loop value is μ = z(α− α) + 1. As in the case of HOMFLY, the negative terms of z come from the loop value, and do not contribute to our estimates of the degrees in t. Now we estimate the degrees with respect to α, μ, and z. Each branch of the skein tree of DK (corresponding to a state σ, which is a choice of smoothings of some of the crossings that results in a collection of framed unlinked unknots) will contribute (a linear combination of) some terms of αμz. Some of them may cancel out each other but we only need an estimate of bounds of the degrees so that we pick the highest and lowest possible degrees among them. The degree of z is the number of smoothings we performed in the skein tree. The degree of μ is the number of components of the state, and the degree of α is the the writhe of the state. Thus we obtain the following estimates: dmax(z) ≤ max{n(σ)} ≤ c dmax(α) ≤ max{w(σ)} ≤ c+ dmin(α) ≥ min{w(σ)} ≥ c− dmax(μ) ≤ max{l(σ)} − 1 ≤ c+ g − 1 where w(σ) and l(σ) are the writhe and number of components of σ respectively, n(σ) is the number of crossings smoothed to get to σ, and c+, c−, and g are the number of positive crossings, the number of negative crossings, and the number of connected components of the diagram of K respectively. Thus after substitution z = t − t and α = t, we get dmax(t) ≤ a · dmax(α) + dmax(z)/2 + (a− 1/2)dmax(μ) ≤ ac+ + c/2 + (a− 1/2)(c+ g − 1) 6 Kauffman, L. H. et. al. dmin(t) ≥ a · dmin(α)− dmax(z)/2− (a− 1/2)dmax(μ) ≥ ac− − c/2− (a− 1/2)(c+ g − 1). The result then follows from the Lemma. This gives a bound of N ≤ 3ac+ (2a− 1)(g − 1), but the same argument as in the proof of Theorem 1 gives