  We use Heegaard splittings to give a criterion for a tunnel number one knot
manifold to be non-fibered and to have large cyclic covers. We also show that
such a knot manifold (satisfying the criterion) admits infinitely many
virtually Haken Dehn fillings.
  Using a computer, we apply this criterion to the 2 generator, non-fibered
knot manifolds in the cusped Snappea census.
  For each such manifold M, we compute a number c(M), such that, for any
n>c(M), the n-fold cyclic cover of M is large.
