Divisibility of great webs and reducible Dehn surgery

TL;DR

This paper investigates the conditions under which reducible Dehn surgeries on knots in the three-sphere produce manifolds with multiple summands, establishing bounds related to the knot's bridge number and specific knot classes.

Contribution

It introduces combinatorial graph techniques to analyze reducible surgeries and proves bounds on surgery slopes based on the knot's bridge number, excluding certain cases.

Findings

If a reducible surgery yields more than two summands, the slope is bounded by the knot's bridge number.

No such surgeries occur for knots with bridge number at most five.

Positive braid closures do not admit reducible surgeries producing multiple summands.

Abstract

We use the combinatorial techniques of graphs of intersection to study reducible Dehn surgeries on knots in the three-sphere. In particular, in the event that a reducible surgery on a knot K in the three-sphere of slope r produces a manifold with more than two connected summands, we show that r is bounded in absolute value by the bridge number of K. As a consequence, this possibility is ruled out for knots with bridge number at most five and for positive braid closures.