# intrinsically linked

A graph $\mathrm{\Gamma}$ is *intrinsically linked ^{}* if any embedding

^{}of $\mathrm{\Gamma}$ in ${\mathbb{R}}^{3}$ contains a nontrivial link.

*Example:*${K}_{6}$, the complete graph

^{}on 6 vertices, was proven to be intrinsically linked by Conway and Gordon.

The property of being

*not*intrinsically linked is inherited by minors. That is, if $\mathrm{\Gamma}$ is not intrinsically linked and ${\mathrm{\Gamma}}^{\prime}$ can be obtained from $\mathrm{\Gamma}$ by edge contractions or deletions, then ${\mathrm{\Gamma}}^{\prime}$ is also not intrinsically linked.

By the Robertson-Seymour Theorem (formerly Wagner’s Conjecture), the obstruction set for this property must be finite. This means that the set of minor minimal

^{}intrinsically linked graphs is finite. In fact, there are 7 graphs in this set; it is known as the Petersen family, and it consists of graphs which can be obtained from ${K}_{6}$ by repeated $\mathrm{\u25b3}$-Y or Y-$\mathrm{\u25b3}$ transformations.

Title | intrinsically linked |
---|---|

Canonical name | IntrinsicallyLinked |

Date of creation | 2013-03-22 18:06:11 |

Last modified on | 2013-03-22 18:06:11 |

Owner | YourInnerNurmo (20577) |

Last modified by | YourInnerNurmo (20577) |

Numerical id | 7 |

Author | YourInnerNurmo (20577) |

Entry type | Definition |

Classification | msc 54J05 |

Related topic | IntrinsicallyKnotted |