  This paper studies the question of whether minimal genus Heegaard splittings
of exterior spaces of knots which are connected sums are weakly reducible or
not. Furthermore it is shown that the Heegaard splittings of the knots used by
Morimoto to show that tunnel number can be sub-additive are all strongly
irreducible. These are the first examples of strongly irreducible minimal genus
Heegaard splittings of composite knots. We also give a characterization of when
is a set of primitive annuli on a handlebody simultaneously primitive. This
characterization is different from that given in [Go].
