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

Unknotted Curves on Seifert Surfaces