Hopfion

A hopfion is a topological soliton. It is a stable three-dimensional localised configuration of a three-component field formula_1 with a knotted topological structure. They are the three-dimensional counterparts of skyrmions, which exhibit similar topoligical properties in 2D.
The soliton is mobile and stable: i.e. it is protected from a decay by an energy barrier. It can be deformed but always conserves an integer Hopf topological invariant. It is named after the German mathematician, Heinz Hopf.
A model that supports hopfions was proposed as follows
formula_2
The terms of higher-order derivatives are required to stabilize the hopfions.
Stable hopfions were predicted within various physical platforms, including Yang-Mills theory, superconductivity and magnetism.
Experimental observation.
Hopfions have been observed experimentally in Ir/Co/Pt multilayers using X-ray magnetic circular dichroism.