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.