Vol. 305, No. 1, 2020

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

Kenneth L. Baker

Appendix: Neil R. Hoffman

Vol. 305 (2020), No. 1, 1–27
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
Authors
Kenneth L. Baker
Department of Mathematics
University of Miami
Coral Gables, FL
United States
Neil R. Hoffman
Department of Mathematics
Oklahoma State University
Stillwater, OK
United States