Realization of knots and links in a spatial graph

Research output: Contribution to journalArticle

23 Citations (Scopus)

Abstract

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

Original languageEnglish
Pages (from-to)87-109
Number of pages23
JournalTopology and its Applications
Volume112
Issue number1
Publication statusPublished - 2001
Externally publishedYes

Fingerprint

Spatial Graph
Knot
Graph in graph theory
Conway Polynomial
Cycle
Necessary Conditions
Complete Bipartite Graph
Sufficient Conditions
Complete Graph
Disjoint
Restriction
Coefficient

Keywords

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

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

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

In: Topology and its Applications, Vol. 112, No. 1, 2001, p. 87-109.

Research output: Contribution to journalArticle

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