Generalized Navier-Stokes (GNS) equations describing three-dimensional (3D) active fluids with flow-dependent spectral forcing have been shown to possess numerical solutions that can sustain significant energy transfer to larger scales by realising chiral Beltramitype chaotic flows. To rationalise these findings, we study here the triad truncations of polynomial and Gaussian GNS models focusing on modes lying in the energy injection range. Identifying a previously unknown cubic invariant, we show that the asymptotic triad dynamics reduces to that of a forced rigid body coupled to a particle moving in a magnetic field. This analogy allows us to classify triadic interactions by their asymptotic stability: unstable triads correspond to rigid-body forcing along the largest and smallest principal axes, whereas stable triads arise from forcing along the middle axis. Analysis of the polynomial GNS model reveals that unstable triads induce exponential growth of energy and helicity, whereas stable triads develop a limit cycle of bounded energy and helicity. This suggests that the unstable triads dominate the initial relaxation stage of the full hydrodynamic equations, whereas the stable triads determine the statistically stationary state. To test this hypothesis, we introduce and investigate the Gaussian active turbulence model, which develops a Kolmogorov-type −5/3 energy spectrum at large wavelengths. Similar to the polynomial case, the steady-state chaotic flows spontaneously accumulate non-zero mean helicity while exhibiting Beltrami statistics and upward energy transport. Our results suggest that self-sustained Beltrami-type flows and an inverse energy cascade may be generic features of 3D active turbulence models with flow-dependent spectral forcing.

We show that mixed bipartite CC and CQ states are geometrically and topologically distinguished in the space of states. They are characterized by non-vanishing Euler-Poincaré characteristics on the topological side and by the existence of symplectic structures on the geometric side.

Abstract. Polyak proved that the set {Ω1a, Ω1b, Ω2a, Ω3a} is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining 32 different types of moves, which we call directed oriented Reidemeister moves. In this article we prove that the set of 8 directed Polyak moves {Ω1a ↑ , Ω1a ↓ , Ω1b ↑ , Ω1b ↓ , Ω2a ↑ , Ω2a ↓ , Ω3a ↑ , Ω3a ↓ } is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a L-generating set for a link L. The same set is proven to be a minimal L-generating set for any link L with at least 2 components. Finally, we discuss knot diagram invariants arising in the study of K-generating sets for an arbitrary knot K, emphasizing the distinction between ascending and descending moves of type Ω3.