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

Abstract

We introduce a novel computational methodology for indexing the Euler characteristics of N-dimensional objects by overlaying (N+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 (N+1)-dimensional chiral vector fields on N-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.