As outlined in Section IV B, the Künneth formula explains how to find each k-dimensional bar for the sublevelset persistence of V n+1 , in terms of the sublevelset persistence bars of V n and g n+1 . Since the building block energy function g n+1 : S 1 → R has sublevelset persistent homology only in dimensions 0 and 1, we can obtain k-dimensional bars of V n+1 only by the following three types of pairings: For (i), note that by induction there are n k + (3 n -1) n-1 k bars of dimension k in V n , and since the building block energy function g n+1 has three 0-dimensional bars, the number of bars of type (i) is 3 times that amount. For (ii), by induction there are and k-1 + (3 n -1) n-1 k-1 ways to combine a (k -1)dimensional bar from V n with the lone 1-dimensional semiinfinite bar from the building block energy function g n+1 . Finally, we count the number of bars given by mathematical torsion in (iii). Of the n k-1 + (3 n -1) n-1 k-1 bars of dimension k -1 in V n , the number of semi-infinite bars is n k-1 because the domain of V n is the n-dimensional torus (S 1 ) n , which has (k -1)-dimensional homology of rank n k-1 . Therefore there are (3 n -1) n-1 k-1 finite length (k -1)-dimensional bars for V n . For (iii), we count the number of ways for a finite length (k -1)-dimensional bar of V n to combine with a finite length 0-dimensional bar from the building block energy function g n+1 , of which there are two, meaning, there are 2