Reducible surgeries and Heegaard Floer homology

TL;DR

This paper uses Heegaard Floer homology to analyze reducible surgeries on knots, establishing that knots with L-space surgeries have at most one reducible surgery, especially for knots of genus up to two.

Contribution

It proves that knots with L-space surgeries have at most one reducible surgery, extending the result to knots of genus at most two.

Findings

p-surgery on non-cable knots with L-space surgery is reducible only if p=2g(K)-1

Knots with L-space surgeries have at most one reducible surgery

The result applies to knots of genus up to two

Abstract

In this paper, we use Heegaard Floer homology to study reducible surgeries. In particular, suppose K is a non-cable knot in the three-sphere with an L-space surgery. If p-surgery on K is reducible, we show that p equals 2g(K)-1. This implies that any knot with an L-space surgery has at most one reducible surgery, a fact that we show additionally for any knot of genus at most two.