Diagrams of Persistence Modules Over Finite Posets

Starting with a persistence module - a functor M from a finite poset to the category of finite dimensional vector spaces - we seek to assign to M an invariant capturing meaningful information about the persistence module. This is often accomplished via applying a Mobius inversion to the rank function or birth-death function. In this paper we establish the relationship between the rank function and birth-death function by introducing a new invariant: the kernel function. The peristence diagram produced by the kernel function is equal to the diagram produced by the birth-death function off the diagonal and we prove a formula for converting between the persistence diagrams of the rank function and kernel function.Comment: Added funding acknowledgemen