Intrinsically Linked

$\begin{figure}\begin{center}\BoxedEPSF{IntrinsicallyLinked.epsf}\end{center}\end{figure}$

A Graph is intrinsically linked if any embedding of it in 3-D contains a nontrivial Link. A Graph is intrinsically linked Iff it contains one of the seven Petersen Graphs (Robertson et al. 1993).

The Complete Graph (left) is intrinsically linked because it contains at least two linked Triangles. The Complete k-Partite Graph $K_{3,3,1}$ (right) is also intrinsically linked.

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 217-221, 1994.

Robertson, N.; Seymour, P. D.; and Thomas, R. ``Linkless Embeddings of Graphs in 3-Space.'' Bull. Amer. Math. Soc. 28, 84-89, 1993.