Computing Homological Information Based on Directed Graphs within Discrete Objects

N-dimensional discrete objects can be interpreted as cubical complexes which are suitable for the study of their homology groups in order to understand the original discrete object. The classic approach consists in computing the Normal Smith Form of some matrices associated to the cubical complex. F...

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Published in:2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing pp. 571 - 578
Main Authors: Gonzalez-Lorenzo, Aldo, Bac, Alexandra, Mari, Jean-Luc, Real, Pedro
Format: Conference Proceeding
Language:English
Published: IEEE 01.09.2014
Subjects:
arbitrary dimension
computational topology
Context
cubical complexes
Discrete Morse Theory
discrete objects
Equations
Face
graphs
homology
Large scale integration
Scientific computing
simplicial complexes
Three-dimensional displays
Vectors
ISBN:9781479984473, 1479984477
Online Access:Get full text
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Abstract N-dimensional discrete objects can be interpreted as cubical complexes which are suitable for the study of their homology groups in order to understand the original discrete object. The classic approach consists in computing the Normal Smith Form of some matrices associated to the cubical complex. Further approaches deal mainly with a pre-processing of the matrices in order to reduce their size. In this paper we propose a new approach, initially based on Discrete Morse Theory, which computes some homological information (Betti numbers and representative cycles) without calculating the Normal Smith Form. It works on any dimension, and it can also be applied to any kind of regular cell complex.
AbstractList N-dimensional discrete objects can be interpreted as cubical complexes which are suitable for the study of their homology groups in order to understand the original discrete object. The classic approach consists in computing the Normal Smith Form of some matrices associated to the cubical complex. Further approaches deal mainly with a pre-processing of the matrices in order to reduce their size. In this paper we propose a new approach, initially based on Discrete Morse Theory, which computes some homological information (Betti numbers and representative cycles) without calculating the Normal Smith Form. It works on any dimension, and it can also be applied to any kind of regular cell complex.
Author Gonzalez-Lorenzo, Aldo
Real, Pedro
Mari, Jean-Luc
Bac, Alexandra
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  givenname: Alexandra
  surname: Bac
  fullname: Bac, Alexandra
  organization: Aix-Marseille Univ., Marseille, France
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  givenname: Jean-Luc
  surname: Mari
  fullname: Mari, Jean-Luc
  organization: Aix-Marseille Univ., Marseille, France
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  givenname: Pedro
  surname: Real
  fullname: Real, Pedro
  organization: Inst. of Math. IMUS, Univ. of Seville, Seville, Spain
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Snippet N-dimensional discrete objects can be interpreted as cubical complexes which are suitable for the study of their homology groups in order to understand the...
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StartPage 571
SubjectTerms arbitrary dimension
computational topology
Context
cubical complexes
Discrete Morse Theory
discrete objects
Equations
Face
graphs
homology
Large scale integration
Scientific computing
simplicial complexes
Three-dimensional displays
Vectors
Title Computing Homological Information Based on Directed Graphs within Discrete Objects
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