Ck-MOVES ON SPATIAL THETA-CURVES AND VASSILIEV INVARIANTS

The Ck-equivalence is an equivalence relation generated by Ck-moves defined by Habiro. Habiro showed that the set of Ck-equivalence classes of the knots forms an abelian group under the connected sum and it can be classified by the additive Vassiliev invariant of order ≤ k−1. We see that the set of Ck-equivalence classes of the spatial θ-curves forms a group under the vertex connected sum and that if the group is abelian, then it can be classified by the additive Vassiliev invariant of order ≤ k − 1. However the group is not necessarily abelian. In fact, we show that it is nonabelian for k ≥ 12. As an easy consequence, we have the set of Ck-equivalence classes of m-string links, which forms a group under the composition, is nonabelian for k ≥ 12 and m ≥ 2. 1. Ck-moves and Vassiliev invariants of spatial θ-curves A tangle T is a disjoint union of properly embedded arcs in the unit 3-ball B. A local move is a pair of tangles (T1, T2) with ∂T1 = ∂T2 such that for each component t of T1 there exists a component u of T2 with ∂t = ∂u. Two local moves (T1, T2) and (U1, U2) are equivalent if there is an orientation preserving self-homeomorphism ψ : B → B such that ψ(Ti) and Ui are ambient isotopic in B 3 relative to ∂B for i = 1, 2. Here ψ(Ti) and Ui are ambient isotopic in B 3 relative to ∂B if ψ(Ti) is deformed to Ui by an ambient isotopy of B that is pointwisely fixed on ∂B. Let (T1, T2) be a local move, t1 a component of T1 and t2 a component of T2 with ∂t1 = ∂t2. Let N1 and N2 be regular neighbourhoods of t1 and t2 in (B 3 − T1) ∪ t1 and (B − T2) ∪ t2 respectively such that N1 ∩ ∂B 3 = N2 ∩ ∂B . Let α be a disjoint union of properly embedded arcs in B × [0, 1] as illustrated in Fig. 1.1. Let ψi : B 2 × [0, 1] → Ni be a homeomorphism with ψi(B 2 × {0, 1}) = Ni ∩ ∂B 3 for i = 1, 2. Suppose that 2000 Mathematics Subject Classification: Primary 57M25; Secondary 57M27