## 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} | γ ε Γ_{2}) to be realizable in terms of the linking number. (3) The set of non-adaptable graphs all of whose proper minors are adaptable contains eight specified planar graphs.

Original language | English |
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Pages (from-to) | 87-109 |

Number of pages | 23 |

Journal | Topology and its Applications |

Volume | 112 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2001 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Geometry and Topology