Three-dimensional magnetization textures as quaternionic functions

Thanks to the recent progress in bulk full three-dimensional nanoscale magnetization distribution imaging, there is a growing interest to three-dimensional (3D) magnetization textures, promising new high information density spintronic applications. Compared to 1D domain walls or 2D magnetic vortices/skyrmions, they are a much harder challenge to represent, analyze and reason about. Here we build analytical representation for such a textures (with arbitrary number of singularity-free hopfions and singular Bloch point pairs) as products of simple quaternionic functions. It can serve as a language for expressing theoretical models of 3D magnetization textures and specifying a variety of topologically non-trivial initial conditions for micromagnetic simulations. It also follows from the quaternion algebra properties that three dimensional magnetic states can potentially be useful for implementing topological quantum computation.