  We report the computational discovery of complex, topologically charged, and
spectrally stable states in three-dimensional multi-component nonlinear wave
systems of nonlinear Schr{\"o}dinger type. While our computations relate to
two-component atomic Bose-Einstein condensates in parabolic traps, our methods
can be broadly applied to high-dimensional, nonlinear systems of partial
differential equations. The combination of the so-called deflation technique
with a careful selection of initial guesses enables the computation of an
unprecedented breadth of patterns, including ones combining vortex lines,
rings, stars, and ``vortex labyrinths''. Despite their complexity, they may be
dynamically robust and amenable to experimental observation, as confirmed by
Bogolyubov-de Gennes spectral analysis and numerical evolution simulations.

Pattern Formation and Solitons;Numerical Analysis;Dynamical Systems;Numerical Analysis;Computational Physics