  We consider homologically essential simple closed curves on Seifert surfaces
of genus one knots in $S^3$, and in particular those that are unknotted or
slice in $S^3$. We completely characterize all such curves for most twist
knots: they are either positive or negative braid closures; moreover, we
determine exactly which of those are unknotted. A surprising consequence of our
work is that the figure eight knot admits infinitely many unknotted essential
curves up to isotopy on its genus one Seifert surface, and those curves are
enumerated by Fibonacci numbers. On the other hand, we prove that many twist
knots admit homologically essential curves that cannot be positive or negative
braid closures. Indeed, among those curves, we exhibit an example of a slice
but not unknotted homologically essential simple closed curve. We further
investigate our study of unknotted essential curves for arbitrary Whitehead
doubles of non-trivial knots, and obtain that there is a precisely one
unknotted essential simple closed curve in the interior of the doubles'
standard genus one Seifert surface. As a consequence of all these we obtain
many new examples of 3-manifolds that bound contractible 4-manifolds.
