Topological invariants play a central role in structured light, providing robust, discrete descriptors of complex optical fields. In an effort to systematically expand such invariants, we detail and generalize two complementary mechanisms for extracting lower-dimensional components of a higher-dimensional field: one associated with phase vortices, in which a gauge structure is relevant, and one based on polarization, for which it is not. Together, these mechanisms form part of a broader framework for defining topological invariants in high-dimensional fields, enabling high-density and robust information encoding through topological field engineering.