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2009 Surgery on a knot in $(\mathrm{surface} \times I)$
Martin Scharlemann, Abigail A Thompson
Algebr. Geom. Topol. 9(3): 1825-1835 (2009). DOI: 10.2140/agt.2009.9.1825


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 KM yields a reducible manifold, then either

* the projection KMS1 has nontrivial winding number,

* K lies in a ball,

* K lies in a fiber, or

* K is cabled.


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Martin Scharlemann. Abigail A Thompson. "Surgery on a knot in $(\mathrm{surface} \times I)$." Algebr. Geom. Topol. 9 (3) 1825 - 1835, 2009.


Received: 5 June 2009; Revised: 10 August 2009; Accepted: 11 August 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1197.57011
MathSciNet: MR2550096
Digital Object Identifier: 10.2140/agt.2009.9.1825

Primary: 57M27

Rights: Copyright © 2009 Mathematical Sciences Publishers


Vol.9 • No. 3 • 2009
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