# Čech complex

# Čech complex

## cech_persistence

This program computes the persistent homology with coefficient field *Z/pZ* of
a Čech complex defined on a set of input points, using Euclidean distance. The
output diagram contains one bar per line, written with the convention:

`p dim birth death`

where `dim`

is the dimension of the homological feature, `birth`

and `death`

are respectively the birth and death of the feature, and `p`

is the
characteristic of the field *Z/pZ* used for homology coefficients (`p`

must be
a prime number).

**Usage**

`cech_persistence [options] <OFF input file>`

**Allowed options**

`-h [ --help ]`

Produce help message`-o [ --output-file ]`

Name of file in which the persistence diagram is written. Default print in standard output.`-r [ --max-edge-length ]`

(default = inf) Maximal length of an edge for the Čech complex construction.`-d [ --cpx-dimension ]`

(default = 1) Maximal dimension of the Čech complex we want to compute.`-p [ --field-charac ]`

(default = 11) Characteristic p of the coefficient field Z/pZ for computing homology.`-m [ --min-persistence ]`

(default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.

Beware: this program may use a lot of RAM and take a lot of time if `max-edge-length`

is set to a large value.

**Example 1 with Z/2Z coefficients**

`cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 2`

**Example 2 with Z/3Z coefficients**

`cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 3`