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

Abstract

We exhibit an infinite family of knots in the Poincare 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. In the appendix, it is shown that hyperbolic knots in the Poincare homology sphere with a lens space surgery has either no symmetries or just a single strong involution.

15 pages, 7 figures. Comments welcome. Aside from minor edits, Version 3, rewrite of appendix and minor revisions in the body suggested by the referee, to appear in Pacific J. Math., Version 2 includes an appendix by Neil R. Hoffman which answers Question 10(2) of Version 1

References in corpus (3)

• Knot Floer homology detects fibred knots
• Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology
• Bordered Floer homology and existence of incompressible tori in homology spheres

Cited by in corpus (1)

• Tunnel number one knots satisfy the Berge Conjecture