Stabilizing Heegaard splittings of toroidal 3-manifolds

Let $T$ be a separating incompressible torus in a 3-manifold $M$. Assuming that a genus $g$ Heegaard splitting $V{\cup }_{S}W$ can be positioned nicely with respect to $T$ (e.g. $V{\cup }_{S}W$ is strongly irreducible), we obtain an upper bound on the number of stabilizations required for $V{\cup }_{S}W$ to become isotopic to a Heegaard splitting which is an amalgamation along $T$. In particular, if $T$ is a canonical torus in the JSJ decomposition of $M$, then the number of necessary stabilizations is at most $4g-4$. As a corollary, this establishes an upper bound on the number of stabilizations required for $V \cup_S W$andanyHeegaardsplittingobtainedbyaDehntwistof$V \cup_S W$along$T$ to become isotopic.