Fig. 1: Soft-magnetic annular nanostrips with identical material, geometric parameters (thickness, radius, and width), but differing topology.

a The Möbius strip features a non-orientable surface with constant torsion along the circumference. b The twist-compensated “bent” ring is topologically trivial and has a similar local torsion, but with alternating handedness. In both cases, the equilibrium magnetic state is a macroscopic vortex configuration, with the magnetization closely aligned along the tangential direction. Spin waves propagating along the strips experience antisymmetric boundary conditions in the Möbius strip (a) and ordinary, symmetric boundary conditions in the bent ring (b).