Let be a closed, irreducible, genus two 3–manifold, and a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in . Then each component manifold of has handle number at most one, ie admits a Heegaard splitting obtained by attaching a single 1–handle to one or two components of . This result also holds for a decomposition of along a maximal collection of incompressible tori.
"Genus two 3–manifolds are built from handle number one pieces." Algebr. Geom. Topol. 1 (2) 763 - 790, 2001. https://doi.org/10.2140/agt.2001.1.763