Can Dehn surgery yield three connected summands?

TL;DR

This paper investigates the possibility of Dehn surgery on knots in the 3-sphere producing manifolds with three or more connected summands, providing bounds and supporting the Cabling Conjecture.

Contribution

It refines bounds on surgery parameters for producing multiple summands, strengthening evidence for the Cabling Conjecture.

Findings

Dehn surgery cannot produce more than two summands under certain conditions

Bounds on surgery parameters are sharpened in relation to bridge number

Supports the Cabling Conjecture through new inequalities

Abstract

A consequence of the Cabling Conjecture of Gonzalez-Acu\~{n}a and Short is that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of the bridge number by a result of Sayari. Here this bound is sharpened, providing further evidence in favour of the Cabling Conjecture.