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Computing homology of subcomplexes of the 3-torus

Abspoel, M.A. (2009) Computing homology of subcomplexes of the 3-torus. Bachelor's Thesis, Mathematics.

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Suppose one has a collection of large geometric objects and one wishes to differentiate between them. When taking measurements on an object, one can probably only get a finite approximation of the object. It is not always possible to fully reconstruct what the original object looked like from this approximation. However, we can compute certain characteristics of the object. This thesis deals with computing some of these characteristics, called homology groups. The approximations we use are simplicial complexes. We use a fast incremental algorithm by Delfinado and Edelsbrunner for computing simplicial homology over Z2 for subcomplexes of the 3-sphere. These complexes include simplicial approximations of objects realized in ordinary 3-dimensional space. The algorithm outputs the Betti numbers of a complex. These numbers are the ranks of the homology groups, and they uniquely identify the homology group. Many approximating data sets, however, are periodic. We therefore adapt the algorithm to work with subcomplexes of the 3-torus. To use the algorithm, we need to expand subcomplexes of the 3-torus to complete triangulations of the 3-torus. A method is developed to accomplish this. The adapted algorithm is shown to be approximately correct: the true first and second Betti numbers can be at most three higher than the computed ones. For large simplicial complexes this incremental algorithm might be more practical than the already available slower methods which provide a more accurate output.

Item Type: Thesis (Bachelor's Thesis)
Supervisor name: Vegter, G.
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 07:28
Last Modified: 17 Apr 2019 12:35

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