Heegaard structure respects complicated JSJ decompositions

Let $M$ be a 3-manifold with torus boundary components $T_1$ and $T_2$. Let $ϕ\colon T_1 \to T_2$ be a homeomorphism, $M_ϕ$ the manifold obtained from $M$ by gluing $T_1$ to $T_2$ via the map $ϕ$, and $T$ the image of $T_1$ in $M_ϕ$. We show that if $ϕ$ is "sufficiently complicated" then any incompressible or strongly irreducible surface in $M_ϕ$ can be isotoped to be disjoint from $T$. It follows that every Heegaard splitting of a 3-manifold admitting a "sufficiently complicated" JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.

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