Proof of Theorem 2. The semi-infinite bars in a sublevelset filtration of a real-valued function on a space simply recover the homology that space. In our case, the domain of the function V is (S 1 ) n , the n-dimensional torus. Since (S 1 ) n has kdimensional homology of rank n k , it follows that in the sublevelset persistent homology of V , we have n k semi-infinite bars in dimension k, for each 0 ≤ k ≤ n.