Reducible surgery in lens spaces and seiferters

Abstract

The Cabling Conjecture states that surgery on hyperbolic knots in [Formula: see text] never produces reducible manifolds. In contrast, there do exist hyperbolic knots in some lens spaces with non-prime surgeries. Baker constructed a family of such hyperbolic knots and posed a conjecture that his examples encompass all hyperbolic knots in lens spaces with non-prime surgeries. Using the idea of seiferters we construct a counterexample to this conjecture. In the process of construction, we also derive an obstruction for a small Seifert fibered space to be obtainable by a surgery with a seiferter.