Reducible surgery in lens spaces and seiferters

TL;DR

This paper constructs a counterexample to Baker's conjecture by using seiferters to show hyperbolic knots in lens spaces can have non-prime surgeries, challenging previous beliefs about reducible surgeries.

Contribution

It introduces a novel construction method using seiferters to find hyperbolic knots with non-prime surgeries in lens spaces, countering Baker's conjecture.

Findings

Counterexample to Baker's conjecture constructed

Obstruction for small Seifert fibred spaces obtained by surgery

Seiferters can produce non-prime surgeries in lens spaces

Abstract

The Cabling Conjecture states that surgery on hyperbolic knots in $S^3$ 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 fibred space to be obtainable by a surgery with a seiferter.