Hydrogen bonds in each case were computed using the Luzar-Chandler criterion. For each of these networks, we construct a time-averaged EC value for the subsampled simulations ⟨χ(G ww )⟩, ⟨χ(G cr )⟩, ⟨χ(G wr )⟩. The manifold M for this system is now the entire simulation space (versus the surface of the SAMs described previously), which consists of a 3D cube with a continuous function f : M → R representing the time-averaged density of water in each simulation subsample. We ignore the structure of the corresponding cosolvent topology because it is directly related to the water topology (high density water implies low-density cosolvent). We construct our manifold and function representation by placing a 20 × 20 × 20 grid centered on the reactant molecule. We then accumulate and bin water positions within each grid point (each representing a 0.2 nm 3 volume) over the 200 picoseconds of simulation time. The accumulated data is then averaged over the time frame and represents a continuous density function over the simulation space. The EC curves in Figure look slightly different to those for our previous 2D system. This system, now in 3D, has a 3rd Betti number (β 2 ), which quantifies the number of voids/cavities that appear during filtration. These voids are associated with pockets of high water density (e.g., local maxima) within our 3D manifold and result in a second peak in the EC curve χ = β 0 -β 1 + β 2 . Details for the practical computation of the EC for these 3D manifolds can be found in the supporting information and code. Figure provides another demonstration of the robustness of these topological methods. We illustrate the changes in the water density and hydrogen bonding network ⟨χ(G ww )⟩ during the course of a single MD simulation. Visually, these graphs and manifold functions appear to be distinct, but when quantified and compared through the EC they are almost identical (indicating that they are topologically close). From the ECs of Figure , for instance, it appears that the system quickly reaches a topological steady-state and thus the MD simulation can be terminated early to reduce computational time.