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

TL;DR

This paper constructs an infinite family of knots in the Poincare homology sphere with tunnel number two that admit lens space surgeries, challenging existing conjectures and analyzing their symmetry properties.

Contribution

It introduces new examples of knots with lens space surgeries in the Poincare homology sphere that are not doubly primitive, providing counterexamples to previous conjectures.

Findings

Infinite family of knots with lens space surgeries in the Poincare homology sphere.

These knots have tunnel number two and are not doubly primitive.

Hyperbolic knots with lens space surgeries have limited symmetries.

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.