Volume 9, issue 3 (2009)

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

Martin Scharlemann and Abigail A Thompson

Algebraic & Geometric Topology 9 (2009) 1825–1835

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 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 3–manifold 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.

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Dehn surgery, taut sutured manifold
Mathematical Subject Classification 2000
Primary: 57M27
Received: 5 June 2009
Revised: 10 August 2009
Accepted: 11 August 2009
Published: 2 October 2009
Martin Scharlemann
Mathematics Department
University of California, Santa Barbara
Santa Barbara, CA 93117
Abigail A Thompson
Mathematics Department
University of California, Davis
Davis, CA 95616