Exceptional surgeries in 3-manifolds

TL;DR

This paper demonstrates that certain 3-manifolds containing hyperbolic knots also admit non-hyperbolic surgeries, including toroidal surgeries, advancing understanding of exceptional surgeries in 3-manifold topology.

Contribution

It extends previous results by showing the existence of hyperbolic knots with non-hyperbolic surgeries in specific 3-manifolds, using work of Ikeda and Adams-Reid.

Findings

Existence of hyperbolic knots with non-hyperbolic surgeries in certain 3-manifolds

Identification of toroidal surgeries as a particular case

Open questions and conjectures about reducible surgeries

Abstract

Myers shows that every compact, connected, orientable $3$--manifold with no $2$--sphere boundary components contains a hyperbolic knot. We use work of Ikeda with an observation of Adams-Reid to show that every $3$--manifold subject to the above conditions contains a hyperbolic knot which admits a non-trivial non-hyperbolic surgery, a toroidal surgery in particular. We conclude with a question and a conjecture about reducible surgeries.