Inside a piece of magnetic material, electron spins, tiny magnetic arrows, usually arrange into simple patterns. Sometimes they swirl into vortices, like water down a drain, and these vortices can tie themselves into knots. Previously, such knots, called hopfions, could only be made in samples with special shapes, cut with jeweler-like precision.
Now researchers have found a simpler way. They pass an electric current through an ordinary magnetic material, and the current helps the vortices twist into three-dimensional knots. The shape and size of the sample no longer matter — the current itself organizes the magnetic arrows into the desired pattern. The resulting knots do not fall apart even when an external magnetic field is applied against them.
Such stability is rare for magnetic structures. Usually, vortices are easily destroyed, but these hold. The authors also built a mathematical classification that distinguishes hopfions from their flat relatives — skyrmions and merons. It's like sorting all possible knots onto shelves: which ones are truly three-dimensional, and which are merely projections.
The hopfion is named after mathematician Heinz Hopf, who in 1931 described a topological invariant — a number that does not change under continuous deformations. This invariant is precisely what distinguishes a true knot from a simple ring.