  Streamlines, vortex lines and magnetic flux tubes in turbulent fluids and
plasmas display a great amount of coiling, twisting and linking, raising the
question as to whether their topological complexity (continually created and
destroyed by reconnections) can be quantified. In superfluid helium, the
discrete (quantized) nature of vorticity can be exploited to associate to each
vortex loop a knot invariant called the Alexander polynomial whose degree
characterizes the topology of that vortex loop. By numerically simulating the
dynamics of a tangle of quantum vortex lines, we find that this quantum
turbulence always contains vortex knots of very large degree which keep
forming, vanishing and reforming, creating a distribution of topologies which
we quantify in terms of a knot spectrum and its scaling law. We also find
results analogous to those in the wider literature, demonstrating that the
knotting probability of the vortex tangle grows with the vortex length, as for
macromolecules, and saturates above a characteristic length, as found for
tumbled strings.
