intrinsically knotted

A graph Γ is called intrinsically knotted if every embeddingMathworldPlanetmathPlanetmath of Γ in 3 contains a nontrivial knot.

Example: K7, the complete graphMathworldPlanetmath 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 Γ is not intrinsically knotted and the graph Γ can be obtained from Γ by deleting or contracting edges, then Γ 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 minimalPlanetmathPlanetmath, 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
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