Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (II)

TL;DR

This paper presents improved algorithms for deforming and contracting simply connected discrete closed manifolds, incorporating a tree structure and direct mapping procedures, with potential applications in topological data analysis.

Contribution

It introduces new algorithms with a tree-supported shrinking process and direct component mapping, enhancing applicability over previous methods.

Findings

Algorithms successfully contract manifolds in various cases

Tree structure supports the shrinking process effectively

Potential applications in topological data analysis

Abstract

In an exploration paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}, we designed algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. However, the algorithms could not guarantee to be applicable to every case. This paper will be the continuation of the exploration. This paper contains two main procedures: (1) A shrinking procedure to contract a simply connected closed manifold. Unlike ones in the previous paper, we added a tree structure to support the process. (2) A more direct procedure for mapping a component from a separated simply connected closed manifold to a disk. We also discuss the practical use of these algorithms in topological data analysis. We think that we have an algorithmic solution, but careful detailed analysis should be done next.