  We construct a new set of asymptotically flat, static vacuum solutions to the
Einstein equations in dimensions 4 and 5, which may be interpreted as a
superposition of positive and negative mass black holes. The resulting
spacetimes are axisymmetric in 4-dimensions and bi-axisymmetric in
5-dimensions, and are regular away from the negative mass singularities, for
instance conical singularities are absent along the axes. In 5-dimensions, the
topologies of signed mass black holes used in the construction may be either
spheres $S^3$ or rings $S^1 \times S^2$; in particular, the negative mass
static black ring solution is introduced. A primary observation that
facilitates the superposition is the fact that, in Weyl-Papapetrou coordinates,
negative mass singularities arise as overlapping singular support for a
particular type of Green's function. Furthermore, a careful analysis of conical
singularities along axes is performed, and formulas are obtained for their
propagation across horizons, negative mass singularities, and corners. The
methods are robust, and may be used to construct a multitude of further
examples. Lastly, we show that balancing does not occur between any two signed
mass black holes of the type studied here in 4 dimensions, while in 5
dimensions two-body balancing is possible.
