### Abstract

For a graph G, let Γ be either the set Γ_{2} of cycles of G or the set Γ_{2} of pairs of disjoint cycles of G. Suppose that for each γ ε Γ, an embedding φ_{γ} : γ → S^{3} is given. A set {φγ \ γ ε Γ) is realizable if there is an embedding f : G → S^{3} such that the restriction map f\γ is ambient isotopic to φγ for any γ ε Γ. A graph is adaptable if any set {φ_{γ} \ γ ε Γ_{1}] is realizable. In this paper, we have the following three results: (1) For the complete graph K_{5} on 5 vertices and the complete bipartite graph K_{3,3} on 3 + 3 vertices, we give a necessary and sufficient condition for {φγ \ γ ε Γ_{1}} to be realizable in terms of the second coefficient of the Conway polynomial. (2) For a graph in the Petersen family, we give a necessary and sufficient condition for {φ_{γ}

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

Pages (from-to) | 87-109 |

Number of pages | 23 |

Journal | Topology and its Applications |

Volume | 112 |

Issue number | 1 |

Publication status | Published - 2001 |

Externally published | Yes |

### Fingerprint

### Keywords

- Adaptable
- Graph
- Knot
- Link
- Minor
- Petersen family
- Realizable

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*112*(1), 87-109.

**Realization of knots and links in a spatial graph.** / Taniyama, Kouki; Yasuhara, Akira.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 112, no. 1, pp. 87-109.

}

TY - JOUR

T1 - Realization of knots and links in a spatial graph

AU - Taniyama, Kouki

AU - Yasuhara, Akira

PY - 2001

Y1 - 2001

N2 - For a graph G, let Γ be either the set Γ2 of cycles of G or the set Γ2 of pairs of disjoint cycles of G. Suppose that for each γ ε Γ, an embedding φγ : γ → S3 is given. A set {φγ \ γ ε Γ) is realizable if there is an embedding f : G → S3 such that the restriction map f\γ is ambient isotopic to φγ for any γ ε Γ. A graph is adaptable if any set {φγ \ γ ε Γ1] is realizable. In this paper, we have the following three results: (1) For the complete graph K5 on 5 vertices and the complete bipartite graph K3,3 on 3 + 3 vertices, we give a necessary and sufficient condition for {φγ \ γ ε Γ1} to be realizable in terms of the second coefficient of the Conway polynomial. (2) For a graph in the Petersen family, we give a necessary and sufficient condition for {φγ

AB - For a graph G, let Γ be either the set Γ2 of cycles of G or the set Γ2 of pairs of disjoint cycles of G. Suppose that for each γ ε Γ, an embedding φγ : γ → S3 is given. A set {φγ \ γ ε Γ) is realizable if there is an embedding f : G → S3 such that the restriction map f\γ is ambient isotopic to φγ for any γ ε Γ. A graph is adaptable if any set {φγ \ γ ε Γ1] is realizable. In this paper, we have the following three results: (1) For the complete graph K5 on 5 vertices and the complete bipartite graph K3,3 on 3 + 3 vertices, we give a necessary and sufficient condition for {φγ \ γ ε Γ1} to be realizable in terms of the second coefficient of the Conway polynomial. (2) For a graph in the Petersen family, we give a necessary and sufficient condition for {φγ

KW - Adaptable

KW - Graph

KW - Knot

KW - Link

KW - Minor

KW - Petersen family

KW - Realizable

UR - http://www.scopus.com/inward/record.url?scp=0012034826&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0012034826&partnerID=8YFLogxK

M3 - Article

VL - 112

SP - 87

EP - 109

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 1

ER -