Extended Data Fig. 1: Field evolution of the magnetic state with double hopfion rings and three skyrmions.

This is an extended version of Fig. 2 (top row), showing over-focus Lorentz TEM images recorded at a specimen temperature of 95 K. In a, the skyrmion string is attached to the outer ring. In b, the skyrmion moves to the left edge of the sample but becomes more visible in c with increasing field. In d, the skyrmion moves to the top left-top corner. Between e and f, the texture rotates slightly. In g and h, one can observe compression of the rings. The outer hopfion ring collapses at 301 mT to either a chiral bobber or a dipole string, as marked by a dashed circle in i. In j and k, the single ring continues to contract with increasing field. The entire system collapses at 343 mT to three bobbers or dipole strings (see l).