### Abstract

We show that the Vassiliev invariants of the knots contained in an embedding of a graph G into R^{3} satisify certain equations that are independent of the choice of the embedding of G. By a similar observation we define certain edge-homotopy invariants and vertex-homotopy invariants of spatial graphs based on the Vassiliev invariants of the knots contained in a spatial graph. A graph G is called adaptable if, given a knot type for each cycle of G, there is an embedding of G into R^{3} that realizes all of these knot types. As an application we show that a certain planar graph is not adaptable.

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

Pages (from-to) | 191-205 |

Number of pages | 15 |

Journal | Pacific Journal of Mathematics |

Volume | 200 |

Issue number | 1 |

Publication status | Published - 2001 Sep |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*200*(1), 191-205.

**Vassiliev invariants of knots in a spatial graph.** / Ohyama, Yoshiyuki; Taniyama, Kouki.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 200, no. 1, pp. 191-205.

}

TY - JOUR

T1 - Vassiliev invariants of knots in a spatial graph

AU - Ohyama, Yoshiyuki

AU - Taniyama, Kouki

PY - 2001/9

Y1 - 2001/9

N2 - We show that the Vassiliev invariants of the knots contained in an embedding of a graph G into R3 satisify certain equations that are independent of the choice of the embedding of G. By a similar observation we define certain edge-homotopy invariants and vertex-homotopy invariants of spatial graphs based on the Vassiliev invariants of the knots contained in a spatial graph. A graph G is called adaptable if, given a knot type for each cycle of G, there is an embedding of G into R3 that realizes all of these knot types. As an application we show that a certain planar graph is not adaptable.

AB - We show that the Vassiliev invariants of the knots contained in an embedding of a graph G into R3 satisify certain equations that are independent of the choice of the embedding of G. By a similar observation we define certain edge-homotopy invariants and vertex-homotopy invariants of spatial graphs based on the Vassiliev invariants of the knots contained in a spatial graph. A graph G is called adaptable if, given a knot type for each cycle of G, there is an embedding of G into R3 that realizes all of these knot types. As an application we show that a certain planar graph is not adaptable.

UR - http://www.scopus.com/inward/record.url?scp=0037819686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037819686&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0037819686

VL - 200

SP - 191

EP - 205

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -