Numerical simulation of knotted solutions for Maxwell equations

In this work, we use the finite differences in time domain (FDTD) numerical method to compute and assess the validity of Hopf solutions, or hopfions, for the electromagnetic field equations. In these solutions, field lines form closed loops characterized by different knot topologies which are preserved during their time evolution. Hopfions have been studied extensively in the past from an analytical perspective but never, to the best of our knowledge, from a numerical approach. The implementation and validation of this technique eases the study of more complex cases of this phenomena; e.g., how these fields could interact with materials (e.g., anisotropic or nonlinear), their coupling with other physical systems (e.g., plasmas), and also opens the path on their artificial generation by different means (e.g., antenna arrays or lasers).