  Over the last decade, substantial progress has been made in understanding the
topology of quasi-2D non-equilibrium fluid flows driven by ATP-powered
microtubules and microorganisms. By contrast, the topology of 3D active fluid
flows still poses interesting open questions. Here, we study the topology of a
spherically confined active flow using 3D direct numerical simulations of
generalized Navier-Stokes (GNS) equations at the scale of typical microfluidic
experiments. Consistent with earlier results for unbounded periodic domains,
our simulations confirm the formation of Beltrami-like bulk flows with
spontaneously broken chiral symmetry in this model. Furthermore, by leveraging
fast methods to compute linking numbers, we explicitly connect this chiral
symmetry breaking to the entanglement statistics of vortex lines. We observe
that the mean of linking number distribution converges to the global helicity,
consistent with the asymptotic result by Arnold. Additionally, we characterize
the rate of convergence of this measure with respect to the number and length
of observed vortex lines, and examine higher moments of the distribution. We
find that the full distribution is well described by a k-Gamma distribution, in
agreement with an entropic argument. Beyond active suspensions, the tools for
the topological characterization of 3D vector fields developed here are
applicable to any solenoidal field whose curl is tangent to or cancels at the
boundaries in a simply connected domain.
