Surgery on a knot in $(\mathrm{surface} \times I)$

Abstract

Suppose $F$ is a compact orientable surface, $K$ is a knot in $F\times I$, and ${\left(F\times I\right)}_{surg}$ is the $3$–manifold obtained by some nontrivial surgery on $K$. If $F\times \left\{0\right\}$ compresses in ${\left(F\times I\right)}_{surg}$, then there is an annulus in $F\times I$ with one end $K$ and the other end an essential simple closed curve in $F\times \left\{0\right\}$. Moreover, the end of the annulus at $K$ determines the surgery slope.

An application: Suppose $M$ is a compact orientable $3$–manifold that fibers over the circle. If surgery on $K\subset M$ yields a reducible manifold, then either

* the projection $K\subset M\to S^1$ has nontrivial winding number,

* $K$ lies in a ball,

* $K$ lies in a fiber, or

* $K$ is cabled.