Fundamental group. The knot group

Let us now discuss a sympathetic example (or problem) concerning unknot diagrams and ways of unknotting them.

First, consider an arbitrary diagram of the trivial knot. Let us try to unknot it by using Reidemeister moves. In the “good” case this can be done only by decreasing the number of vertices (we mean the “decreasing” version of the first two Reidemeister moves and the third move).