# intrinsically knotted

A graph $\Gamma$ is called intrinsically knotted if every embedding   of $\Gamma$ 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 $\Gamma$ is not intrinsically knotted and the graph $\Gamma^{\prime}$ can be obtained from $\Gamma$ by deleting or contracting edges, then $\Gamma^{\prime}$ 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 IntrinsicallyKnotted 2013-03-22 18:06:08 2013-03-22 18:06:08 YourInnerNurmo (20577) YourInnerNurmo (20577) 5 YourInnerNurmo (20577) Definition msc 54J05 IntrinsicallyLinked