Geometry and topology of vortices in random quantum eigenfunctions

Disordered complex 3D scalar wave fields typically contain a dense tangle of nodal lines (quantized vortices), which are important in diverse physical wave systems including turbulent superfluids, optical volume speckle, the quantum eigenfunctions of chaotic 3D cavities, and liquid crystal phases. Based on extensive numerical simulations these nodal tangles are known to have fractal properties on large scales, although more subtle topological quantities such as the probability of knotted or linked vortices are sensitive to the details of the model. We numerically generate many examples of wave chaos in three random systems at fixed energy (3D cube with periodic boundary conditions, 3-sphere and 3D harmonic oscillator), analysing aspects of their statistical geometry and identifying the knot types of the vortex curves which appear. Knots tend to occur with high probability even at comparatively low energies, and the statistics of knot complexity vary significantly amongst the three systems. Furthermore, the different symmetries and boundary conditions of these systems strongly affect the knotted conformations that can occur, and we discuss how this relates to the statistics of knotting with mode count in different systems.