T-dualizing the deformed and resolved conifold

In a previous paper, we used T-duality to construct a new type of 1/4-BPS solution describing a pair of NS5-branes intersecting in 1 + 3 dimensions and localized in all other directions except for a single transverse circle. This led to an explicit solution to a sourced Monge-Ampere equation, of which there are few known examples. In this paper, we refine this formalism and apply it to two important generalizations: the resolved and deformed conifolds. In doing so, we construct two new solutions describing, respectively, a pair of NS5-branes separated in a transverse direction and a pair of NS5-branes with a smooth 'diamond' profile. We show how the parameter of the resolved conifold (size of S²) maps to a transverse separation of the NS5-branes, while the modulus of the deformed conifold (size of S³) maps to the deformation parameter of the diamond web.