Constructing Complicated Spheres

TL;DR

This paper introduces an infinite series of complex 4-sphere examples, called Akbulut-Kirby spheres, designed to test advanced homology algorithms used in analyzing solid materials, proteins, and digital data.

Contribution

It provides a new family of complicated 4-sphere test cases constructed through handlebody methods based on finitely presented groups.

Findings

The Akbulut-Kirby spheres serve as challenging benchmarks for homology algorithms.

They demonstrate the need for robust algorithms capable of handling complex topological structures.

The construction method links group theory with topological sphere examples.

Abstract

Fast and efficient homology algorithms are in demand in the applied sciences for analyzing solid materials and proteins, processing digital imaging data, or pattern classification among others. Recent advances employ discrete Morse theory as a preprocessor. Research in this area has lead to the need to find complicated test examples. We present an infinite series of examples that have been constructed to test some of the latest algorithms under development. This family of 4-spheres (known as the Akbulut-Kirby spheres) is based on a handlebody construction via finitely presented groups.