The following theorem outlines a recipe, using the Künneth formula , for calculating the persistence of the topological features of V based on the persistence of the building block functions. The bars we obtain come from combining bars in two different ways; standard combinations as in line (4) of Theorem 1, or combining via mathematical torsion as in line (5) of Theorem 1. Here X and Y are persistence barcodes, and X ⊗ Y is the tensor product barcode. We let bcd n (X) denote the set of n-dimensional bars in the persistence diagram for X, and similarly for bcd n (Y ) and bcd n (X ⊗ Y ) Additionally, denotes the birth time of a bar and r denotes the death of a bar.