Hopfions

Abstract

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.