should be taken into account. Considerable work has been done to investigate the minimal genus of a given class in $H_{2}(X)$, first by proving the Thom conjecture (Kronheimer–Mrowka [@KM]) and then its generalizations by Morgan–Szabó–Taubes [@MST] and Ozsváth–Szabó [@OS6]. When considering a configuration of surfaces, the sum of their genera is closely related to the number of their geometric intersections, as shown by Gilmer [@GILMER]. He showed that the the minimal number of such intersections can be estimated using the Casson–Gordon invariant. This estimate has been improved by Strle [@SASO] for configurations of $n=b_{2}^{+}(X)$ algebraically disjoint surfaces of positive self-intersection by an application of the Seiberg–Witten equations on a cylindrical end manifold.

Multiple plumbings of two trivial disk bundles over spheres have been investigated by Sunukjian in his thesis [@NS]. He calculated the Heegaard-Floer homology of the boundary of such plumbings in cases where the two spheres are plumbed either once or zero times algebraically and $n$ times geometrically.

We investigate the double plumbing ${N_{m,n}}$ of two disk bundles with Euler classes $m$ and $n$ over spheres, which represents the simplest case of a configuration of two surfaces with algebraic (and geometric) intersection 2. We calculate the $d_{b}$–invariants of the 