According to theorems of C. Gordon, J. Luecke, and W. Parry, if a knot exterior $X$ has two distinct planar boundary slopes $r_1,r_2$, then at least one of the manifolds $X(r_1),X(r_2)$ has a connected summand $M$ with nontrivial torsion in first homology. The 3-manifolds $M$ obtained in this way, which we call {\it t-manifolds}, have special Heegaard splittings, or {\it 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.