Numerical calculation of the Hopf index for three-dimensional magnetic textures

To gain deeper insight into the complex, stable, and robust configurations of magnetic textures, topological characterization has proven essential. In particular, while the skyrmion number is a well-established topological invariant for two-dimensional magnetic textures, the Hopf index serves as a key topological descriptor for three-dimensional magnetic structures. In this paper, we present and compare various methods for numerically calculating the Hopf index, provide implementations, and offer a detailed analysis of their accuracy and computational efficiency. Additionally, we identify and address common pitfalls and challenges associated with the numerical computation of the Hopf index, offering insights for improving the robustness of these techniques.