A fibered knot is a knot $K$ in the $3$-sphere whose complement is a surface bundle over a circle. If $S$ is the fiber, the fundamental group of $S$ is free (of even rank), and the fundamental group of the complement is an HNN extension $$\pi_1(S) \to \pi_1(S^3-K) \to Z$$ where the $Z$ is generated by the meridian of the knot.

Since surgery on $K$ recovers the $3$-sphere, the group $\pi_1(S^3-K)$ has the interesting property that it is normally generated by (the conjugacy class of) the meridian.

What I didn't realize until recently is that there are many examples of "non-geometric" automorphisms $\phi$ of free groups $F$ for which the associated HNN extension $F \to G \to Z$ is normally generated by the conjugacy class of the monodromy. One simple example is the case $F = \langle a,b,c \rangle$ and $\phi$ is the automorphism $a \to c^{-1}abac, b \to bac, c\to bc$.

Is there any systematic way of generating such examples? Is there a classification? One reason to be interested is that such examples can be used to construct smooth $4$-manifolds which are topologically $S^4$ but not obviously diffeomorphically $S^4$.

Edit: a link to the construction is http://lamington.wordpress.com/2009/11/09/4-spheres-from-fibered-knots/ (this explains the construction in the case of a fibered knot, but the group-theoretic condition is the only important ingredient).

wasa deliberate joke, for the reason Henry gives, and I have changed it back. Greg, what gives?! $\endgroup$ – Danny Calegari Nov 15 '09 at 0:27