### Abstract

We define A_{k}-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let A_{k}-equivalence denote an equivalence relation generated by A_{k}-moves and ambient isotopy. A_{k}-equivalence implies A_{k-1}-equivalence. Let script F Sign be an A_{k-1}-equivalence class of the embeddings of a finite graph into the 3-sphere. Let script G Sign be the quotient set of script F Sign under A_{k}-equivalence. We show that the set script G Sign forms an abelian group under a certain geometric operation. We define finite type invariants on script F Sign of order (n; k). And we show that if any finite type invariant of order (1; k) takes the same value on two elements of script F Sign, then they are A_{k}-equivalent. A_{k}-move is a generalization of C_{k}-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to C_{k}-move and ambient isotopy if and only if any Vassiliev invariant of order ≤ k - 1 takes the same value on them. The 'if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be C_{k}-equivalent.

Original language | English |
---|---|

Pages (from-to) | 183-200 |

Number of pages | 18 |

Journal | Pacific Journal of Mathematics |

Volume | 211 |

Issue number | 1 |

Publication status | Published - 2003 Sep |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*211*(1), 183-200.

**Local moves on spatial graphs and finite type invariants.** / Taniyama, Kouki; Yasuhara, Akira.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 211, no. 1, pp. 183-200.

}

TY - JOUR

T1 - Local moves on spatial graphs and finite type invariants

AU - Taniyama, Kouki

AU - Yasuhara, Akira

PY - 2003/9

Y1 - 2003/9

N2 - We define Ak-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let Ak-equivalence denote an equivalence relation generated by Ak-moves and ambient isotopy. Ak-equivalence implies Ak-1-equivalence. Let script F Sign be an Ak-1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let script G Sign be the quotient set of script F Sign under Ak-equivalence. We show that the set script G Sign forms an abelian group under a certain geometric operation. We define finite type invariants on script F Sign of order (n; k). And we show that if any finite type invariant of order (1; k) takes the same value on two elements of script F Sign, then they are Ak-equivalent. Ak-move is a generalization of Ck-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to Ck-move and ambient isotopy if and only if any Vassiliev invariant of order ≤ k - 1 takes the same value on them. The 'if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be Ck-equivalent.

AB - We define Ak-moves for embeddings of a finite graph into the 3-sphere for each natural number k. Let Ak-equivalence denote an equivalence relation generated by Ak-moves and ambient isotopy. Ak-equivalence implies Ak-1-equivalence. Let script F Sign be an Ak-1-equivalence class of the embeddings of a finite graph into the 3-sphere. Let script G Sign be the quotient set of script F Sign under Ak-equivalence. We show that the set script G Sign forms an abelian group under a certain geometric operation. We define finite type invariants on script F Sign of order (n; k). And we show that if any finite type invariant of order (1; k) takes the same value on two elements of script F Sign, then they are Ak-equivalent. Ak-move is a generalization of Ck-move defined by K. Habiro. Habiro showed that two oriented knots are the same up to Ck-move and ambient isotopy if and only if any Vassiliev invariant of order ≤ k - 1 takes the same value on them. The 'if' part does not hold for two-component links. Our result gives a sufficient condition for spatial graphs to be Ck-equivalent.

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UR - http://www.scopus.com/inward/citedby.url?scp=0242592464&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0242592464

VL - 211

SP - 183

EP - 200

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -