nLab diamond principle

Idea

In zig-zag persistent homology-theory, the diamond principle (Carlsson & de Silva 2010, Sec. 5.2) states that persistence diagrams of pairs of zigzag persistence modules are bijective if the modules coincide away from two consecutive edges where their difference forms an “exact diamond”.

Applied specifically to persistence modules given by the homology groups of zig-zags of topological spaces, the diamond principle is an incarnation of the Mayer-Vietoris sequence in algebraic topology.

In this case it says that the two canonical zigzag persistence modules associated with a filtered topological space (one built from union-zigzags, the other from intersection-zigzags) in fact yield the same persistence diagrams.

Notice how the diamond principle is a result genuinely about persistent invariants, in that the two persistence modules forming an exact diamond do not have the same shape as diagrams and hence are non-comparable when regarded, say, as quiver representations (since they are objects of distinct categories).

Related concepts

• stability of persistence diagrams

References

The diamond principle originates in

• Gunnar Carlsson, Vin de Silva, Section 5.2 of Zigzag Persistence, Found Comput Math 10 (2010) 367–405 $[$arXiv:0812.0197, doi:10.1007/s10208-010-9066-0$]$

Review:

• Sara Kališnik, Sections 1.3-1.4 in: Persistent homology and duality (2013) (pdf, pdf)