Topological transition from superfluid vortex rings to isolated knots and links

Knots and links are fundamental topological objects that play a key role in both classical and quantum fluids. In this paper, we propose a scheme to generate torus vortex knots and links through the reconnections of vortex rings perturbed by Kelvin waves in trapped Bose-Einstein condensates, which makes it possible to generate topologically complex structures from topologically trivial objects controllably. The transfer of helicity between knots, links, and coils occur with different pathways which can be controlled through designing specific initial states. The generation of a knot or link can be achieved by setting the parity of the Kelvin wave number. The stability of knots and links can be greatly improved with tunable parameters, including the ideal relative angle and the minimal distance between the initial vortex rings.