Library for canonical stratification of simplicial complexes. This code is a companion to the publication "Algorithmic Canonical Stratifications of Simplicial Complexes" (https://arxiv.org/abs/1808.06568).
There are no special module dependency for the core library.
However, "benchmark.py" requires scipy and numpy for producing Delaunay triangulation of 2-sphere and 3-ball.
To install:
$ pip install /path/to/CanonicalStratification
There are two example files, example_fromfile.py and example_fromlist.py to try the stratification on a pre-made simplicial complex.
Also, benchmark.py will let you try the stratification on 2-sphere and 3-ball with an arbitrary number of vertices.
You must first create a simplicial complex using the SimplicialComplex object. This can be done through a python list or a file, and is discussed later. But first we describe how a simplex is defined.
We assume that all vertices in the input dataset are indexed. A simplex in the complex is defined by a set of indices for vertices. For example, a simplex [0,1,2] consists of vertices indexed 0, 1 and 2.
SimplicialComplex object is constructed by adding these simplices one by one. For convenience, adding a simples adds all the implied simplices as well. Implied simplices of a simplex are all subsets of the simplex; mathematically they are the simplices in a closed sieve with the simplex at the top. For example, a simplex [0,1,2] implies simplices [0,1],[0,2],[1,2],[0],[1] and [2]. These are added automatically.
SimplicialComplex object can be loaded from an iterable of lists. Each list in the iterable corresponds to a simplex, and contain the vertices of the simplex. For example, the following specifies two simplicies [0,1,2] and [0,3]
simplices = [[0,1,2],[0,3]]
And a complex can be construced by.
sc = SimplicialComplex.fromList(simplices)
Note that this complex will have 9 simplices instead of 2 because of the implied simplices (see above).
Aditionally, the function takes any iterable of lists as an input. So, for example, you can input a generator of lists to this function as well.
SimplicialComplex object can be loaded from a csv file. Each line in the file corresponds to a simplex, and contain the vertices of the simplex. For example, a file with the following entries specifies two simplicies [0,1,2] and [0,3]
0,1,2
0,3
And a complex can be construced by.
sc = SimplicialComplex.fromFile(filename)
Where filename is the string containing path to the file.
Note that this complex will have 9 simplices instead of 2 because of the implied simplices (see above).