Is there any known example of fibered knot which is topologically slice but not (expected to be) smoothly slice?



Fibered example of topologically slice knots
Such a knot would yield a counterexample to one of two important conjectures in the area. A preliminary definition: a slice knot is homotopically ribbon if the inclusion of the knot into the slice disk complement induces a surjection on the fundamental group. It's easy to see that a ribbon knot is homotopically ribbon; note that homotopically ribbon doesn't require any smoothness (but you still want local flatness to avoid trivialities). This criterion is viewed as the right substitute for ribbon in the topological setting. 


The remarkable theorem of Casson-Gordon (A loop theorem for duality spaces and fibred ribbon knots; Invent. Math. **74**, 119-137 (1983)) says that a fibered knot is homotopically ribbon *if and only if* its monodromy (filled in over a disk) extends over a handlebody. The if part constructs a smooth slice smooth homotopy ball with boundary the 3-sphere (if your knot lay in the 3-sphere.) So in particular, a fibered knot that is topologically homotopy ribbon is smoothly homotopically ribbon!!! So an example of the sort that you ask for would either be slice but not homotopically ribbon (contradicting the conjecture that this doesn't happen) or would give a smooth slice counterexample to the 4-dimensional Poincaré conjecture. 


So to summarize, no such example is known.


PS If there were an MO question asking for "your favorite paper that people don't think about enough", the Casson-Gordon paper mentioned above might top my list.


