A wild knot and its complement

Both (blue and red) curves represent non-trivial elements in the fundamental group of the complement. That is, none of them can be shrunk to a point.

To see this, consider the linking number of these curves with the knot itself: it's $2$ for the red one and $1$ for the blue one (assuming consistent orientations). Since the linking number is invariant under deformations, it has to be zero for nullhomotopic curves.

This being said, your initial statement is not what makes wild knots weird compared to tame knots: indeed, the complement of any knot is not simply connected (since a curve such as the blue one here will always have linking number $\pm 1$ with the knot; it's called a meridian curve). What you probably intended to say is that

The fundamental group of the complement of a wild knot may be not finitely generated.