Surgery on a knot in (Surface x I)

Martin Scharlemann; Abigail Thompson

doi:10.2140/agt.2009.9.1825

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arXiv · 0807.0405

Surgery on a knot in (Surface x I)

Martin Scharlemann·Abigail Thompson

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Abstract

Suppose F is a compact orientable surface, K is a knot in F x I, and N is the 3-manifold obtained by some non-trivial 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 3-manifold 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 non-trivial winding number; or K lies in a ball; or K lies in a fiber; or K is a cabled knot.

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BibTeX RIS

Martin Scharlemann, Abigail Thompson. 2008-07-03. Surgery on a knot in (Surface x I). https://doi.org/10.2140/agt.2009.9.1825

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