Nonsimple tunnel number one knots in lens spaces

Geometry & Topology
Event time: 
Tuesday, March 3, 2009 - 11:30am to Monday, March 2, 2009 - 7:00pm
431 DL
Mike Williams
Speaker affiliation: 
UC Santa Barbara
Event description: 

Let M be a closed, orientable 3-manifold. A knot K in M has
tunnel number one if the exterior of K in M admits a genus two Heegaard splitting. A knot in M is a (1,1) knot if M admits a genus one Heegaard splitting that cuts K into two unknotted arcs in the Heegaard solid tori. It is easy to show that if K is a (1,1) knot in M, then K has tunnel number one. We call a knot K in M nonsimple if the exterior of K possesses an essential surface of nonnegative Euler characteristic. For example, the nonsimple knots in the 3-sphere are the torus and satellite knots. We show that if K is a nonsimple 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 W. Jaco, M. Eudave-Munoz and C. Gordon as well as structure results of J. Schultens on the Heegaard splittings of graph manifolds.