2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61326The $C_k$-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 $\leq k-1$. 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 $\leq k-1$. However the group is not necessarily abelian. In fact, we show that it is nonabelian for $k\geq 12$. 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 $k\geq 12$ and $m\geq 2$.LaTeX, 15 pages with 12 figuresGeometric Topology57M25 (Primary) 57M27 (Secondary)text