Taut foliations from double-diamond replacements

Suppose M is an oriented 3-manifold with connected boundary a torus, and suppose M contains a properly embedded, compact, oriented, surface R with a single boundary component that is Thurston norm minimizing in H₂(M,∂M). We define a readily recognizable type of sutured manifold decomposition, which for notational reasons we call double-diamond taut, and show that if R admits a double-diamond taut sutured manifold decomposition, then for every boundary slope except one, there is a co-oriented taut foliation of M that intersects ∂M transversely in a foliation by curves of that slope. In the case that M is the complement of a knot κ in S³, the exceptional filling is the meridional one; in particular, restricting attention to rational slopes, it follows that every manifold obtained by non-trivial Dehn surgery along κ admits a co-oriented taut foliation. As an application, we show that if R is a Murasugi sum of surfaces R1 and R₂, where R₂ is an unknotted band with an even number 2m ≥ 4 of half-twists, then every manifold obtained by non-trivial surgery on κ = ∂R admits a co-oriented taut foliation.