  Advanced vector imaging techniques provide us with 3D maps of magnetization
fields in which topological concepts can be directly applied to describe
real-space experimental textures in non-ideal geometries. Here, the 3D
magnetization of a low symmetry permalloy microstructure is obtained by X-ray
vector magnetic tomography and analysed in detail in terms of topological
charges and emergent fields. A central asymmetric domain wall with a complex 3D
structure is observed in which magnetization chirality transitions are mediated
by Bloch points arranged in several dipoles and a triplet. The ideal spherical
symmetry of the emergent field of an isolated monopole is severely modified due
to shape effects of the permalloy microstructure. Emergent field lines
aggregate into bundles that either connect adjacent Bloch points within a
topological dipole or tend towards the surface. These bundles may present
different textures such as helical vortices or half merons, depending on
asymmetries and confinement, but are constrained by topological charge
conservation in a given sample volume. This precise description of the
singularities in realistic systems, enabled by the quantitative experimental
information on magnetization vector fields, can significantly improve our
understanding of topological constraints in 3D magnetic systems and provide
advancements in the design of magnetic devices.
