Bridge Number and the Curve Complex
arXiv:math/0603102
Abstract
We show that there are hyperbolic tunnel-number one knots with arbitrarily high bridge number and that "most" tunnel-number one knots are not one-bridge with respect to an unknotted torus. The proof relies on a connection between bridge number and a certain distance in the curve complex of a genus-two surface.
13 pages, 3 figures. References have been added and the order of exposition changed slightly