Virtually Haken surgeries on once-punctured torus bundles
  We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken.   We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of ``most'' commensurability classes of punctured torus bundles.   In particular, we define an integer-valued complexity function on monodromies $f$ (essentially the length of the LR-factorization of $f_*$ in $PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$ has complexity at most 5, then   $M$ is finitely covered by an element of $\mathcal{C}$.   If the monodromy has complexity at most 12, then, with at most 36 exceptions,   $M$ is finitely covered by an element of $\mathcal{C}$.   We also give a method for computing ``algebraic boundary slopes'' in certain finite covers of punctured torus bundles. 
Joseph D. Masters