### Abstract

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.

Original language | English |
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Pages (from-to) | 309-324 |

Number of pages | 16 |

Journal | Topology and its Applications |

Volume | 128 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - 2003 Feb 15 |

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### Keywords

- C-move
- Finite type invariant
- Spatial theta-curve
- Vassiliev invariant

### ASJC Scopus subject areas

- Geometry and Topology