Geodesic
Surgery on a knot in (surface × I)
Scharlemann, Martin1 ; Thompson, Abigail A2
1Mathematics Department, University of California, Santa Barbara, Santa Barbara, CA 93117, USA
2Mathematics Department, University of California, Davis, Davis, CA 95616, USA
Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1825-1835
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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.

DOI : 10.2140/agt.2009.9.1825
Keywords: Dehn surgery, taut sutured manifold

Scharlemann, Martin  1   ; Thompson, Abigail A  2

1 Mathematics Department, University of California, Santa Barbara, Santa Barbara, CA 93117, USA
2 Mathematics Department, University of California, Davis, Davis, CA 95616, USA 
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Scharlemann, Martin; Thompson, Abigail A. Surgery on a knot in (surface × I). Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1825-1835. doi: 10.2140/agt.2009.9.1825

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