# Brieskorn spheres bounding rational balls

## TL;DR

This paper identifies new examples of Brieskorn spheres that bound rational homology balls but not integral ones, expanding understanding of their topological properties and providing explicit handlebody diagrams.

## Contribution

It introduces new families of Brieskorn spheres bounding rational homology balls and provides explicit handlebody diagrams, building on prior work involving the figure-eight knot.

## Key findings

- New families of Brieskorn spheres bounding rational homology balls
- Explicit handlebody diagrams with 3-handles included
- Extension of Fintushel and Stern's argument to additional cases

## Abstract

Fintushel and Stern showed that the Brieskorn sphere $\Sigma(2,3,7)$ bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls: the families $\Sigma(2,4n+1,12n+5)$ and $\Sigma(3,3n+1,12n+5)$ for $n$ odd. We also provide handlebody diagrams for a rational homology ball containing a rationally slice disk for the figure-eight knot, as well as for a rational homology ball bounded by $\Sigma(2,3,7)$. These handle diagrams necessarily contain 3-handles.