Persistent homology of the sum metric

Given finite metric spaces $(X, d_X)$ and $(Y, d_Y)$, we investigate the persistent homology $PH_*(X \times Y)$ of the Cartesian product $X \times Y$ equipped with the sum metric $d_X + d_Y$. Interpreting persistent homology as a module over a polynomial ring, one might expect the usual K\"unneth short exact sequence to hold. We prove that it holds for $PH_0$ and $PH_1$, and we illustrate with the Hamming cube $\{0,1\}^k$ that it fails for $PH_n,\,\, n \geq 2$. For $n = 2$, the prediction for $PH_2(X \times Y)$ from the expected K\"unneth short exact sequence has a natural surjection onto $PH_2(X \times Y)$. We compute the nontrivial kernel of this surjection for the splitting of Hamming cubes $\{0,1\}^k = \{0,1\}^{k-1} \times \{0,1\}$. For all $n \geq 0$, the interleaving distance between the prediction for $PH_n(X \times Y)$ and the true persistent homology is bounded above by the minimum of the diameters of $X$ and $Y$. As preliminary results of independent interest, we establish an algebraic K\"unneth formula for simplicial modules over the ring $\kappa[\mathbb{R}_+]$ of polynomials with coefficients in a field $\kappa$ and exponents in $\mathbb{R}_+ = [0,\infty)$, as well as a K\"unneth formula for the persistent homology of $\mathbb{R}_+$-filtered simplicial sets -- both of these K\"unneth formulas hold in all homological dimensions $n \geq 0$.