  Given $(V_1,V_2)$ a Heegaard splitting of the complement of a composite knot
$K=K_1# K_2$ in $S^3$, where $K_i, i=1,2$ are prime knots, we have a unique, up
to isotopy, decomposing annulus $A$. When the intersection of $A$ and $V_1$ is
a minimal collection of disks we study the components of $V_1-N(A)$ and show
that at most one component is a 3-ball meeting $A$ in two disks. This is a
crucial step in proving the conjecture that a necessary and sufficient
condition for the tunnel number of a connected sum to be less than or equal to
the sum of the tunnel numbers is that one of the knots has a Heegaard splitting
in which a merdian curve is primitive.
