  The class of connected LOG (Labelled Oriented Graph) groups coincides with
the class of fundamental groups of complements of closed, orientable
2-manifolds embedded in S^4, and so contains all knot groups. We investigate
when Campbell and Robertson's generalized Fibonacci groups H(r,n,s) are
connected LOG groups. In doing so, we use the theory of circulant matrices to
calculate the Betti numbers of their abelianizations. We give an almost
complete classification of the groups H(r,n,s) that are connected LOG groups.
All torus knot groups and the infinite cyclic group arise and we conjecture
that these are the only possibilities. As a corollary we show that H(r,n,s) is
a 2-generator knot group if and only if it is a torus knot group.
