Heegaard splittings and virtually Haken Dehn filling II
  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. 
Joseph D. Masters, William Menasco and Xingru Zhang