Robust PCA Study of TEM Images of Gold Nanospheres: Stable and Dynamic Closed Knot-Like Phase-Space Loops
In this study, we apply Robust PCA to transmission electron microscopy (TEM) images of gold nanoparticles (GNPs) to decompose their geometrical, topological, and dynamical organization into static and dynamic knot-like phase-space loops. Dynamic loops show higher crossing numbers and longer rope lengths, while static loops show higher mean curvature. Although most Alexander determinants were D(t)=1 with isolated values of D(t)=3 at smaller nanoparticle diameters, implying that geometrical complexity does not require topological complexity. The different writhe signs indicate opposite handedness and distinct phase-space geometries in static and dynamic cases. For both static and dynamic knot-like loops, the graphs show an inverse relationship between mean curvature and mean torsion. Flatter trajectories occur at high curvature, and more torsional twisting occurs at low curvature. The small changes in Moffatt-Ricca helicities (~10^-14<<0) are due to circulation-dependent scaling, yet Moffatt-Ricca helicities still follow the same self-linking trend in both static and dynamic configurations. As nanoparticle diameter increases, the dynamic knot-like loops show stronger Kelvin-like spectral localization and smaller minimum self-approach distances, indicating more concentrated spatial modes and tighter geometric folding. At the same time, the decreasing CTEI suggests reduced global phase-space organization. These contrasting trends indicate that the geometrical organization tends to be less global and coherent, and more localized and mode-specific. These results show that the PCA-reconstructed phase space could serve as a mathematical analog to Kelvin's aether, because all disturbances exhibit similar topological configurations with varying degrees of geometric difference.