Suppose we have a fibred knot $K$ with a fiber surface $F$ and let $c$ be an unknot disjoint from $F$ (but not homotopically trivial in the complement of $F$). Is it possible that every twist along $c$ leaves $K$ fibred with $F$ still being the fiber surface?



Can one twist fibred knots and still get fibred knots?
Such examples were [constructed by Morton](http://www.ams.org/mathscinet-getitem?mr=728587). He showed that one can find unknotted curves lying on fiber surfaces *with zero framing*. Twisting about them preserves fiberedness and the fact that it is a knot in $S^3$. Also, curves on the fiber can be pushed disjoint from the fiber meeting your requirement.


**Edit:** As Danny points out in the comments [John Stallings originally came up with this construction.](http://www.maths.ed.ac.uk/~aar/papers/stallfib2.pdf) 


