  Two-dimensional topological solitons, commonly called Skyrmions, are
extensively studied in solid-state magnetic nanostructures and promise many
spintronics applications. However, three-dimensional topological solitons
dubbed hopfions have not been demonstrated as stable spatially localized
structures in solid-state magnetic materials. Here we model the existence of
such static solitons with different Hopf index values in noncentrosymmetric
solid magnetic nanostructures with a perpendicular interfacial magnetic
anisotropy. We show how this surface anisotropy, along with the
Dzyaloshinskii-Moriya interactions and the geometry of nanostructures,
stabilize hopfions. We demonstrate knots in emergent field lines and computer
simulate Lorentz transmission electron microscopy images of such solitonic
configurations to guide their experimental discovery in magnetic solids.
