Mathematics and Statistics Vol. 6(2), pp. 20 - 24
DOI: 10.13189/ms.2018.060202
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Generating Minimal Boundary Maps

Marian Anton 1,2,*, Landon Renzullo 2
1 Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
2 Department of Math Sciences, Central Connecticut State University, 1615 Stanley St, New Britain, CT 06050, USA


The field of computational topology is evolving rapidly and new algorithms are updated and released at a rapid pace. A good reference for currently available opensource libraries with peer-review publication can be found in [7]. 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 [1] 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 ∆-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 JavaPlex [8]. We note that the generating data of a complex is equivalent in information to its incidence matrix and provide efficient algorithms for converting from an incidence matrix to a Generating Set.

Topological Data Analysis, Boundary Maps, Incidence Matrix, Simplicial Complex

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Marian Anton , Landon Renzullo , "Generating Minimal Boundary Maps," Mathematics and Statistics, Vol. 6, No. 2, pp. 20 - 24, 2018. DOI: 10.13189/ms.2018.060202.

(b). APA Format:
Marian Anton , Landon Renzullo (2018). Generating Minimal Boundary Maps. Mathematics and Statistics, 6(2), 20 - 24. DOI: 10.13189/ms.2018.060202.