 
Summary: SURGERY ON A KNOT IN SURFACE × I
MARTIN SCHARLEMANN AND ABIGAIL THOMPSON
Abstract. Suppose F is a compact orientable surface, K is a knot in F × I,
and (F × I)surg is the 3manifold obtained by some nontrivial surgery on
K. If F × {0} compresses in (F × I)surg, then there is an annulus in F × I
with one end K and the other end an essential simple closed curve in F × {0}.
Moreover, the end of the annulus at K determines the surgery slope.
An application: suppose M is a compact orientable 3manifold that fibers
over the circle. If surgery on K M yields a reducible manifold, then either
· the projection K M S1 has nontrivial winding number,
· K lies in a ball,
· K lies in a fiber, or
· K is cabled
The study of Dehn surgery on knots in 3manifolds has a long and rich history,
interacting in a deep way with
· sophisticated combinatorics ([GL], [CGLS]),
· the theory of character varieties ([CGLS], [BGZ]), and
· sutured manifold theory ([Ga1], [Sch])
It is pleasing then to find a result that is simple to state, easy to understand and
yet has so far escaped explicit notice. Yi Ni has pointed out that there is at least
