Surgery on a knot in (surface × I)

Scharlemann, Martin; Thompson, Abigail A

doi:10.2140/agt.2009.9.1825

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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

DOI: 10.2140/agt.2009.9.1825

Abstract

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

Keywords

Dehn surgery, taut sutured manifold

Mathematical Subject Classification 2000

Primary: 57M27

References

Publication

Received: 5 June 2009

Revised: 10 August 2009

Accepted: 11 August 2009

Published: 2 October 2009

Authors

Martin Scharlemann
Mathematics Department University of California, Santa Barbara Santa Barbara, CA 93117 USA
http://www.math.ucsb.edu/~mgscharl/
Abigail A Thompson
Mathematics Department University of California, Davis Davis, CA 95616 USA
http://www.math.ucdavis.edu/~thompson/

Volume 9, issue 3 (2009)