Starting with a persistence module -a functor M : P → Vec k for some finite poset P -we seek to assign to M an invariant capturing meaningful information about the peristence module. This is often accomplished via applying a Möbius 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.
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