Computing Euler characteristic of Ν-dimensional objects via a Skyrmion-inspired overlaying (Ν+1)-dimensional chiral field.

We introduce a novel computational methodology for indexing the Euler characteristics of [Formula: see text]-dimensional objects by overlaying ([Formula: see text]+1)-dimensional chiral vector fields. Analogous to how the skyrmion number characterizes a two-dimensional magnetic skyrmion through the integration of the solid angle of its spin field, we generalize this principle to arbitrary dimensions. By iteratively applying a simple numerical process, we generate ([Formula: see text]+1)-dimensional chiral vector fields on [Formula: see text]-dimensional objects. The Euler characteristics of these objects are calculated by aggregating the local solid angles subtended by neighboring chiral vectors. In this study, we focus on verifying our method in two and three dimensions. For dimensions higher than three, we conduct preliminary experiments on simple objects to explore potential applicability. Although our method shows promising potential in higher dimensions, further investigation is required to fully understand its applicability beyond three dimensions.