Surgery on a knot in (Surface x I)
Abstract
Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3manifold obtained by some nontrivial surgery on K. If F x {0} compresses in N, then there is an annulus in F x I with one end K and the other end an essential simple closed curve in F x {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 a knot K in M yields a reducible manifold, then either: the projection of K to S^1 has nontrivial winding number; or K lies in a ball; or K lies in a fiber; or K is a cabled knot.
 Publication:

arXiv eprints
 Pub Date:
 July 2008
 DOI:
 10.48550/arXiv.0807.0405
 arXiv:
 arXiv:0807.0405
 Bibcode:
 2008arXiv0807.0405S
 Keywords:

 Mathematics  Geometric Topology;
 57M25;
 57M27
 EPrint:
 Revised to include reference to Yi Ni's related "Dehn surgeries that yield fibred 3manifolds", arXiv:0712.4387