# intrinsically linked

A graph $\Gamma$ is intrinsically linked  if any embedding   of $\Gamma$ 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 $\Gamma$ is not intrinsically linked and $\Gamma^{\prime}$ can be obtained from $\Gamma$ by edge contractions or deletions, then $\Gamma^{\prime}$ 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 $\triangle$-Y or Y-$\triangle$ transformations.

Title intrinsically linked IntrinsicallyLinked 2013-03-22 18:06:11 2013-03-22 18:06:11 YourInnerNurmo (20577) YourInnerNurmo (20577) 7 YourInnerNurmo (20577) Definition msc 54J05 IntrinsicallyKnotted