Representation of types and 3-manifolds. (Q1880065)

If \(X\) is a connected compact orientable irreducible 3-manifold with torus boundary and \(r\) is a slope in \(\partial X\), then \(X(r)\) denotes the closed manifold obtained by Dehn-filling \(X\) along \(r\). If \(X\) is a knot exterior admitting 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, as shown in [\textit{C. McA. Gordon}, Lectures at Knots '96, Ser. Knots Everything. 15, 263--290 (1997; Zbl 0940.57022)]. This summand \(M\) has special Heegaard splittings, whose structure is studied in this paper. In particular, the genus two case is classified, showing that no hyperbolic manifold can occur as \(M\).