The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two

Kenneth L. Baker

Vol. 305 (2020), No. 1, 1–27

DOI: 10.2140/pjm.2020.305.1

Abstract

We exhibit an infinite family of knots in the Poincaré homology sphere with tunnel number $2$ that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. Additionally, we update (and correct) our earlier work on Hedden’s almost-simple knots. In the appendix, it is shown that a hyperbolic knot in the Poincaré homology sphere with a lens space surgery has either no symmetries or just a single strong involution.

Keywords

Poincaré homology sphere, Dehn surgery, lens space, tunnel number

Mathematical Subject Classification 2010

Primary: 57M27

Milestones

Received: 16 May 2017

Revised: 27 June 2019

Accepted: 2 October 2019

Published: 17 March 2020