Constructing Rational Homology 3-Spheres That Bound Rational Homology 4-Balls

TL;DR

This paper introduces three new large families of plumbed 3-manifolds that bound rational homology 4-balls, expanding known examples and providing explicit surgeries on torus knots that achieve this property.

Contribution

The paper constructs new families of plumbed 3-manifolds bounding rational homology 4-balls using novel operations, and identifies specific surgeries on torus knots with this property.

Findings

New families of plumbed 3-manifolds bounding rational homology 4-balls

Explicit descriptions of surgeries on certain torus knots

Confirmation of the slice-ribbon conjecture for complex arborescent knots

Abstract

We present three large families of new examples of plumbed 3-manifolds that bound rational homology 4-balls. These are constructed using two operations, also defined here, that preserve the lack of a lattice embedding obstruction to bounding rational homology balls. Apart from in the cases shown in this paper, it remains open whether these operations are rational homology cobordisms in general. The families of new examples include a multitude of families of rational surgeries on torus knots, and we explicitly describe which positive torus knots we now know to have a surgery that bounds a rational homology ball. While not the focus of this paper, we implicitly confirm the slice-ribbon conjecture for new, more complicated, examples of arborescent knots, including many Montesinos knots.