# intrinsically knotted

A graph $\mathrm{\Gamma}$ is called *intrinsically knotted* if every embedding^{} of $\mathrm{\Gamma}$ in ${\mathbb{R}}^{3}$ contains a nontrivial knot.

*Example:* ${K}_{7}$, the complete graph^{} on 7 vertices, was proven to be intrinsically knotted by Conway and Gordon.

The property of being *not* intrinsically knotted is *inherited by minors*. That is, if a graph $\mathrm{\Gamma}$ is not intrinsically knotted and the graph ${\mathrm{\Gamma}}^{\prime}$ can be obtained from $\mathrm{\Gamma}$ by deleting or contracting edges, then ${\mathrm{\Gamma}}^{\prime}$ is also not intrinsically knotted.

According to the Robertson-Seymour Theorem (also known as Wagner’s Conjecture), this means that the obstruction set for this property must be finite. Thus there are only a finite number of intrinsically knotted graphs which are minor minimal^{}, that is, for which any graph obtained by edge deletion or contraction is not intrinsically knotted. As of the creation of this article (06/01/2008), this set is still not known.

Title | intrinsically knotted |
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Canonical name | IntrinsicallyKnotted |

Date of creation | 2013-03-22 18:06:08 |

Last modified on | 2013-03-22 18:06:08 |

Owner | YourInnerNurmo (20577) |

Last modified by | YourInnerNurmo (20577) |

Numerical id | 5 |

Author | YourInnerNurmo (20577) |

Entry type | Definition |

Classification | msc 54J05 |

Related topic | IntrinsicallyLinked |