Handle decompositions of rational balls and Casson-Gordon invariants

TL;DR

This paper explores the handle decompositions of rational homology balls bounding rational homology spheres, utilizing Casson-Gordon invariants and Levine-Tristram signatures to establish bounds and provide explicit examples.

Contribution

It introduces a method to estimate the complexity of rational homology balls using Casson-Gordon invariants and computes explicit examples with Levine-Tristram signatures.

Findings

Lower bounds on handle complexity of rational homology balls

Bounds on fusion number of ribbon knots

Explicit examples demonstrating the bounds

Abstract

Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine-Tristram signatures to compute these bounds and produce explicit examples.