ZigzagPersistence class

The class deals with the setting K_1 \rightarrow K_2 \leftarrow K_3 \rightarrow \dots. The Triangle zigzag example illustrates the use of ZigzagPersistence.

class ZigzagPersistence
add(boundary, birth)

Adds a simplex with the given boundary to the complex, i.e. K_{i+1} = K_i \cup \sigma and boundary = \partial \sigma. If a new class is born as a result of the addition, birth is stored with it for future reference.

Returns:

a pair (i, d). The first element is the index i. It is the internal representation of the newly added simplex, and should be used later for removal or when constructing the boundaries of its cofaces. In other words, boundary must consist of these indices. The second element d is the death element. It is None if a birth occurred, otherwise it contains the value passed as birth to add() or remove() when the class that just died was born.

remove(index, birth)

Removes the simplex identified by the given index from the complex. If a new class is born as a result of the removal, birth is stored with it for future reference.

Returns:

None if a birth has occurred, otherwise it contains the value passed as birth to add() or remove() when the class that just died was born.

is_alive(z)

Determines whether a given cycle is alive. The input should be an instance of ZNode.

__iter__()

Iterator over elements of type ZNode, i.e. the cycles stored in the structure.

class ZNode
birth

The birth value associated with the cycle. It is passed to ZigzagPersistence in method add().

__iter__()

Iterator over the individual nodes (simplices) of the cycle (same indices that are passed to and returned by add().

Auxilliary functions

A pair of auxilliary functions is provided to help add and remove entire collections of simplices. Both are pure Python functions defined in bindings/python/dionysus/zigzag.py.

add_simplices(zigzag, simplices, complex, birth, report_local=False)

Adds each simplex in simplices to the zigzag. complex is a dictionary mapping simplices to their indices (in zigzag’s internal representation). All the newly born classes are given the value of birth.

Returns:

list of deaths that occur as a result of simplices’ removal. Each death is a pair of the dimension of the class and the birth value passed when the class was born. By default the deaths equal to birth are not reported unless report_local is set to True.

remove_simplices(zigzag, simplices, complex, birth, report_local=False)

Same parameters and return as in add_simplices() except that simplices are removed from the zigzag and the complex.

ImageZigzagPersistence class

The class deals with the setting

\begin{array}{ccccccc} K_1 & \rightarrow & K_2 & \leftarrow & K_3 & \rightarrow & \dots \\ \uparrow & & \uparrow & & \uparrow & \\ L_1 & \rightarrow & L_2 & \leftarrow & L_3 & \rightarrow & \dots \end{array}

where the vertical maps are inclusions, i.e. L_i \subseteq K_i.

class ImageZigzagPersistence
add(boundary, subcomplex, birth)

Interface is the same as in ZigzagPersistence.add(). The additional parameter subcomplex controls whether the simplex is added to L or not. We always have K_{i+1} = K_i \cup \sigma. If subcomplex is true, then L_{i+1} = L_i \cup \sigma, otherwise L_{i+1} = L_i.

remove(index, birth)

Interface is exactly the same as in ZigzagPersistence.remove().