On nonsimple knots in lens spaces with tunnel number one

Details On nonsimple knots in lens spaces with tunnel number one On nonsimple knots in lens spaces with tunnel number one

2009-08-12

arxiv.org/abs/0908.1765v1

Mathematics

Geometric Topology

10 pages, 2 figures

Scientific paper

A knot k in a closed orientable 3-manifold is called nonsimple if the exterior of k possesses a properly embedded essential surface of nonnegative Euler characteristic. We show that if k is a nonsimple prime tunnel number one knot in a lens space M (where M does not contain any embedded Klein bottles), then k is a (1,1) knot. Elements of the proof include handle addition and Dehn filling results/techniques of Jaco, Eudave-Munoz and Gordon as well as structure results of Schultens on the Heegaard splittings of graph manifolds.

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