Representation of types and 3-manifolds

According to theorems of C. Gordon, J. Luecke, and W. Parry, if a knot exterior X has two distinct planar boundary slopes r1,r2, then at least one of the manifolds X(r1),X(r2) has a connected summand M with nontrivial torsion in first homology. The 3-manifolds M obtained in this way, which we call t-manifolds, have special Heegaard splittings, or t-manifold structures. In this paper we study the topology of t-manifolds from the point of view of the homology presentation matrices induced by their t-manifold structures, classify all genus two t-manifold structures, and show that, under some conditions, one of the Dehn fillings of X is a connected sum of t-manifolds and (at most) one prime non t-manifold summand