# Generating minimal boundary maps

## TL;DR

This paper introduces algorithms for constructing boundary maps of simplicial complexes using a combinatorial structure called 'Generating Set', improving efficiency and scalability over previous methods.

## Contribution

It refines existing approaches to generate boundary maps, enabling easier handling of relations and extension to $ riangle$-complexes, with efficient conversion algorithms from incidence matrices.

## Key findings

- Algorithms efficiently generate boundary maps with many relations.
- Faces are generated only once, even if shared among simplices.
- The generating data is equivalent to the incidence matrix, with conversion algorithms provided.

## Abstract

In this paper we examine the descriptive potential of a combinatorial data structure known as "Generating Set" in constructing the boundary maps of a simplicial complex. By refining the approach of \cite{Dumas} in generating these maps, we provide algorithms that allow for relations among simplices to be easily accounted for. In this way we explicitly generate faces of a complex only once, even if a face is shared among multiple simplices. The result is a useful interface for constructing complexes with many relations and for extending our algorithms to $\Delta$-complexes. Once we efficiently retrieve the representatives of "living" simplices i.e., of those that have not been related away, the construction of the boundary maps scales well with the number of relations and provides a simpler alternative to JPlex. We finish by noting that the generating data of a complex is equivalent in information to its incidence matrix and we provide efficient algorithms for converting from an incidence matrix to a Generating Set.