  We report on numerical simulations of critical phenomena near the threshold
of black hole formation in the collapse of axisymmetric gravitational waves in
vacuum. We discuss several new features of our numerical treatment, and then
compare results obtained from families of quadrupolar and hexadecapolar initial
data. Specifically, we construct (nonlinear) initial data from quadrupolar and
hexadecapolar, time-symmetric wavelike solutions to the linearized Einstein
equations (often referred to as Teukolsky waves), and evolve these using a
shock-avoiding slicing condition. While our degree of fine-tuning to the onset
of black-hole formation is rather modest, we identify several features of the
threshold solutions formed for the two families. Both threshold solutions
appear to display an at least approximate discrete self-similarity with an
accumulation event at the center, and the characteristics of the threshold
solution for the quadrupolar data are consistent with those found previously by
other authors. The hexadecapolar threshold solution appears to be distinct from
the quadrupolar one, providing further support to the notion that there is no
universal critical solution for the collapse of vacuum gravitational waves.
