C_k-moves on spatial theta-curves and Vassiliev invariants

The C_k-equivalence is an equivalence relation generated by C_k-moves defined by Habiro. Habiro showed that the set of C_k-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 C_k-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 C_k-equivalence classes of m-string links, which forms a group under the composition, is nonabelian for k≥12 and m≥2.