  We determine the PSL_2(C) and SL_2(C) character varieties of the
once-punctured torus bundles with tunnel number one, i.e. the once-punctured
torus bundles that arise from filling one boundary component of the Whitehead
link exterior. In particular, we determine `natural' models for these algebraic
sets, identify them up to birational equivalence with smooth models, and
compute the genera of the canonical components. This enables us to compare
dilatations of the monodromies of these bundles with these genera. We also
determine the minimal polynomials for the trace fields of these manifolds.
Additionally we study the action of the symmetries of these manifolds upon
their character varieties, identify the characters of their lens space
fillings, and compute the twisted Alexander polynomials for their
representations to SL_2(C).
