Filtered Complexes

Detailed Description

Author
Clément Maria

A simplicial complex \(\mathbf{K}\) on a set of vertices \(V = \{1, \cdots ,|V|\}\) is a collection of simplices \(\{\sigma\}\), \(\sigma \subseteq V\) such that \(\tau \subseteq \sigma \in \mathbf{K} \rightarrow \tau \in \mathbf{K}\). The dimension \(n=|\sigma|-1\) of \(\sigma\) is its number of elements minus \(1\).

A filtration of a simplicial complex is a function \(f:\mathbf{K} \rightarrow \mathbb{R}\) satisfying \(f(\tau)\leq f(\sigma)\) whenever \(\tau \subseteq \sigma\). Ordering the simplices by increasing filtration values (breaking ties so as a simplex appears after its subsimplices of same filtration value) provides an indexing scheme.

Implementations

Simplex tree

There are two implementation of complexes. The first on is the Simplex_tree data structure. The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes. The data structure is described in [6]

Simplex_tree_representation.png
Simplex tree representation

Examples

Here is a list of simplex tree examples :

Simplex tree construction with \(\mathbb{Z}/3\mathbb{Z}\) coefficients on weighted graph Klein bottle file:

$> ./simplex_tree_from_cliques_of_graph ../../data/points/Klein_bottle_complex.txt 3
Insert the 1-skeleton in the simplex tree in 0.000404 s.
max_dim = 3
Expand the simplex tree in 3.8e-05 s.
Information of the Simplex Tree:
Number of vertices = 10 Number of simplices = 98

Hasse complex

The second one is the Hasse_complex. The Hasse complex is a data structure representing explicitly all co-dimension 1 incidence relations in a complex. It is consequently faster when accessing the boundary of a simplex, but is less compact and harder to construct from scratch.

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