intrinsically linked

A graph $\mathrm{\Gamma }$ is intrinsically linked if any embedding of $\mathrm{\Gamma }$ in ${ℝ}^3$ 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 $\mathrm{△}$-Y or Y-$\mathrm{△}$ 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