Theory

Several parameters

Thus far the persistent homology we have considered is derived from filtered spaces/complexes with a single filtration parameter $r$:

$$\mathrm{VR}(\mathcal{P},r),f^{-1}((-\infty ,r))$$

It is possible to combine two types of filtration in a single structure having inclusion maps of each type.

Unfortunately there is no “structure theorem” for persistent homology with more than one parameter. This is basically because the structure theorem proof in one parameter involves homology with $\mathbb{R}[x]$ coefficients, while with $n$ parameters it involves homology with $\mathbb{R}[x_1,\dots ,x_n]$ coefficients.

Zig-zag persistent homology

Filtrations can be modified in another way by adding intermediate linking sets. Instead of the usual filtered space like this:

$${\mathcal{K}}_{r_1}\subset {\mathcal{K}}_{r_2}\subset {\mathcal{K}}_{r_3}\subset \cdots$$

We have a “zig-zag” filtered space like this:

$${\mathcal{K}}_1\hookleftarrow {\mathcal{K}}_{1,2}\hookrightarrow {\mathcal{K}}_2\hookrightarrow {\mathcal{K}}_{2,3}\hookleftarrow {\mathcal{K}}_3\hookrightarrow \cdots$$

In general, each arrow is allowed to go either direction. The induced map on homology will follow the same pattern.