More than 40 years ago, Faddeev proposed the existence of three-dimensional topological solitons classified by the integer-valued Hopf invariant. These solitons are now known as hopfions and have been investigated in a range of systems, including the original model suggested by Faddeev, where a variety of stable knot and link solutions have been computed numerically. Very recently, numerical computations have predicted the existence of nanoscale hopfions in frustrated magnets and experiments have realized micrometer-sized hopfions in chiral ferromagnetic fluids. All these examples of hopfions will be described and their similarities and differences discussed.
Sutcliffe, P. (2018). Hopfions. In M. Ge, A. Niemi, K. Phua, & L. Takhtajan (Eds.), Ludwig Faddeev memorial volume : a life in mathematical physics (539-547). World Scientific Publishing. https://doi.org/10.1142/9789813233867_0025