Homology of Cube

[alexdkeros]

Dear Javaplex developers,

While trying to compute the persistent homology of a cube, expecting results similar to a 2-sphere, I get the following results:

point_cloud =
0.2500    0.2500    0.2500
0.7500    0.2500    0.2500
0.2500    0.7500    0.2500
0.7500    0.7500    0.2500
0.2500    0.2500    0.7500
0.7500    0.2500    0.7500
0.2500    0.7500    0.7500
0.7500    0.7500    0.7500

intervals =
Dimension: 0
[0.0, 0.5)
[0.0, 0.5)
[0.0, 0.5)
[0.0, 0.5)
[0.0, 0.5)
[0.0, 0.5)
[0.0, 0.5)
[0.0, infinity)
Dimension: 1
[0.5, 0.707)
[0.5, 0.707)
[0.5, 0.707)
[0.5, 0.707)
[0.5, 0.707)
Dimension: 3
[0.707, 0.866)

The issues are the following:

1. There are only 5 generators of dimension 1.
2. There are no generators in dimension 2.
3. There is a generator in dimension 3 that should ideally be in dimension 2.

Could you please shed some light to this issue?

Thank you in advance,
Alexandros

[henryadams]

Hi Alexandros,

Counterintuitively, those are the correct homology groups. At scale parameter r=1, for 1-dimensional homology, the 6-th "hole" is in fact a linear combination of all of the 5 other holes. A related example is that the complete graph on 4 vertices has three 1-dimensional holes, even though it is the 1-skeleton of the tetrahedron (with 4 "faces").

At scale parameter r=2, all edges between vertices of the cube have appeared except for the antipodal pairs (no edge between (1,1,1) and (-1,-1,1), and no edge between (1,-1,1) and (-1,1,-1), for example --- there are two more such missing diagonal edges). It follows that the Vietoris-Rips complex is the boundary of the cross-polytope with 8 vertices (see https://en.wikipedia.org/wiki/Cross-polytope), which is a 3-sphere. That's why there's no 2-dimensional homology but there is 3-dimensional homology when r=2.

So while these homology groups are counterintuitive, they're correct.

Best, Henry