Distance and bridge position

David Bachman and Saul Schleimer

Vol. 219 (2005), No. 2, 221–235

DOI: 10.2140/pjm.2005.219.221

Abstract

J. Hempel’s definition of the distance of a Heegaard surface generalizes to a notion of complexity for any knot that is in bridge position with respect to a Heegaard surface. Our main result is that the distance of a knot in bridge position is bounded above by twice the genus, plus the number of boundary components, of an essential surface in the knot complement. As a consequence knots constructed via sufficiently high powers of pseudo-Anosov maps have minimal bridge presentations which are thin.

Keywords

Heegaard splitting, curve complex

Mathematical Subject Classification 2000

Primary: 57M25, 57M27

Milestones

Received: 29 August 2003

Revised: 2 October 2003

Published: 1 April 2005