A Tool for Integer Homology Computation: Lambda-At Model

TL;DR

This paper introduces the lambda-AT-model, a new method for computing integer homology that avoids Smith Normal Form, providing efficient calculation of Betti numbers, torsion primes, and homology generators for complex structures.

Contribution

The paper formalizes the lambda-AT-model and presents an algorithm to compute homological invariants without Smith Normal Form, reducing computational costs for torsion analysis.

Findings

Successfully computes Betti numbers and torsion primes.

Identifies minimal lambda to optimize torsion subgroup calculations.

Applicable to various structured objects and digital images.

Abstract

In this paper, we formalize the notion of lambda-AT-model (where $\lambda$ is a non-null integer) for a given chain complex, which allows the computation of homological information in the integer domain avoiding using the Smith Normal Form of the boundary matrices. We present an algorithm for computing such a model, obtaining Betti numbers, the prime numbers p involved in the invariant factors of the torsion subgroup of homology, the amount of invariant factors that are a power of p and a set of representative cycles of generators of homology mod p, for each p. Moreover, we establish the minimum valid lambda for such a construction, what cuts down the computational costs related to the torsion subgroup. The tools described here are useful to determine topological information of nD structured objects such as simplicial, cubical or simploidal complexes and are applicable to extract such an information from digital pictures.