Handle additions producing essential surfaces

Ruifeng Qiu and Shicheng Wang

Vol. 229 (2007), No. 1, 233–255

DOI: 10.2140/pjm.2007.229.233

Abstract

We construct a small, hyperbolic 3-manifold M with the property that, for any integer g ≥ 2, there are infinitely many separating slopes r in ∂M such that the 3-manifold M(r) obtained by attaching a 2-handle to M along r contains an essential separating closed surface of genus g. The resulting manifolds M(r) are still hyperbolic. This contrasts sharply with known finiteness results on Dehn filling and with the known finiteness result on handle addition for the cases g = 0,1. Our 3-manifold M is the complement of a hyperbolic, small knot in a handlebody of genus 3.

Keywords

Hyperbolic knot, small knot, handle additions

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 57M50

Milestones

Received: 13 June 2005

Revised: 14 January 2006

Published: 1 January 2007