ABSTRACT
This chapter discusses a sympathetic example or problem concerning unknot diagrams and ways of unknotting them. It considers an arbitrary diagram of the trivial knot. However, one can consider not only planar knot link diagrams, but also spherical diagrams. Namely, one can think of the sphere as the plane compactified by the point at infinity. The chapter describes the notion of fundamental group for arbitrary topological spaces and shows how to calculate it for link complements. The link complement fundamental group is a very strong invariant. For instance, it recognises trivial links among links with the same number of components. This result follows from M. Dehn's theorem. Thus, Dehn's theorem reduces the trivial link recognition problem to the free group recognition problem. The statement of Dehn's theorem shows that the fundamental group is rather a strong invariant.
