# Homology spheres and property R

## TL;DR

This paper constructs infinitely many homology spheres with pairs of knots whose 0-surgeries yield $S^1 imes S^2$, answering a long-standing question in topology.

## Contribution

It introduces a new infinite family of homology spheres with specific knot surgery properties, solving a problem posed in 1978.

## Key findings

- Existence of infinitely many such homology spheres.
- Identification of knots with 0-surgeries resulting in $S^1 imes S^2$.
- Resolution of Kirby and Melvin's question from 1978.

## Abstract

We present infinitely many homology spheres which contain two distinct knots whose 0-surgeries are $S^1 \times S^2$. This resolves a question posed by Kirby and Melvin in 1978.