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SURGERY ON A KNOT IN SURFACE I MARTIN SCHARLEMANN AND ABIGAIL THOMPSON
 

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 3-manifold obtained by some non-trivial 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 3-manifold that fibers
over the circle. If surgery on K M yields a reducible manifold, then either
the projection K M S1 has non-trivial winding number,
K lies in a ball,
K lies in a fiber, or
K is cabled
The study of Dehn surgery on knots in 3-manifolds 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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics