# Can one twist fibred knots and still get fibred knots?

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?

• What do you mean by "twist along $c$"? – Marco Golla Sep 14 '15 at 10:06
• @MarcoGolla technically it means 'perform a $1/n$ surgery on $c$ for some non-zero integer $n$'. I imagine it as cutting the solid torus (exterior of $c$) along a meridional disc, twisting and regluing. – shestipalov Sep 21 '15 at 10:31

Such examples were constructed by Morton. 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.