A note on cabled slice knots and reducible surgeries

TL;DR

This paper investigates the conditions under which slice knots can admit reducible Dehn surgeries, providing new restrictions and Heegaard Floer theoretic criteria related to cabled knots.

Contribution

It establishes that slice knots cannot have reducible surgeries with more than two summands and introduces a necessary Heegaard Floer condition for positive cable knots to be slice.

Findings

Slice knots cannot admit reducible surgeries with more than two summands.

A necessary Heegaard Floer condition for positive cable knots to be slice.

Restrictions on the types of reducible surgeries for slice knots.

Abstract

We consider the question of when a slice knot admits a reducible Dehn surgery. By analyzing the correction terms associated to such a surgery, we show that slice knots cannot admit surgeries with more than two summands. We also give a necessary Heegaard Floer theoretic condition for a positive cable of a knot to be slice.