### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**C _{k}-moves on spatial theta-curves and Vassiliev invariants.** / Yasuhara, Akira.

Research output: Contribution to journal › Article

_{k}-moves on spatial theta-curves and Vassiliev invariants',

*Topology and its Applications*, vol. 128, no. 2-3, pp. 309-324. https://doi.org/10.1016/S0166-8641(02)00131-1

_{k}-moves on spatial theta-curves and Vassiliev invariants. Topology and its Applications. 2003 Feb 15;128(2-3):309-324. https://doi.org/10.1016/S0166-8641(02)00131-1

}

TY - JOUR

T1 - Ck-moves on spatial theta-curves and Vassiliev invariants

AU - Yasuhara, Akira

PY - 2003/2/15

Y1 - 2003/2/15

N2 - 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.

AB - 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.

KW - C-move

KW - Finite type invariant

KW - Spatial theta-curve

KW - Vassiliev invariant

UR - http://www.scopus.com/inward/record.url?scp=0038012591&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038012591&partnerID=8YFLogxK

U2 - 10.1016/S0166-8641(02)00131-1

DO - 10.1016/S0166-8641(02)00131-1

M3 - Article

AN - SCOPUS:0038012591

VL - 128

SP - 309

EP - 324

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 2-3

ER -