  While the existence of stable knotted and linked vortex lines has been
established in many experimental and theoretical systems, their existence in
oscillatory systems and systems with nonlocal coupling has remained elusive.
Here, we present strong numerical evidence that stable knots and links such as
trefoils and Hopf links do exist in simple, complex, and chaotic oscillatory
systems if the coupling between the oscillators is neither too short ranged nor
too long ranged. In this case, effective repulsive forces between vortex lines
in knotted and linked structures stabilize curvature-driven shrinkage observed
for single vortex rings. In contrast to real fluids and excitable media, the
vortex lines correspond to scroll wave chimeras [synchronized scroll waves with
spatially extended (tubelike) unsynchronized filaments], a prime example of
spontaneous synchrony breaking in systems of identical oscillators. In the case
of complex oscillatory systems, this leads to a novel topological
superstructure combining knotted filaments and synchronization defect sheets.
