Study of multi black hole and ring singularity apparent horizons
  We study critical black hole separations for the formation of a common
apparent horizon in systems of $N$ - black holes in a time symmetric
configuration. We study in detail the aligned equal mass cases for $N=2,3,4,5$,
and relate them to the unequal mass binary black hole case. We then study the
apparent horizon of the time symmetric initial geometry of a ring singularity
of different radii. The apparent horizon is used as indicative of the location
of the event horizon in an effort to predict a critical ring radius that would
generate an event horizon of toroidal topology. We found that a good estimate
for this ring critical radius is $20/(3\pi) M$. We briefly discuss the
connection of this two cases through a discrete black hole 'necklace'
configuration.
