The moduli spaces S ( D ) of non-isotopic vortex knots are introduced for the ideal fluid flows in invariant domains D . The analogous moduli spaces of the magnetic fields B knots are defined. We derive and investigate new exact fluid flows (and analogous plasma equilibria) satisfying the Beltrami equation which have nested invariant balls B k 3 with radii Rk ≈ (k + 1) π, k⟶∞. The first flow is z-axisymmetric; the other ones do not possess any rotational symmetries. The axisymmetric flow has an invariant plane z = 0. Due to an involutive symmetry of the flow, its vortex knots in the invariant half-spaces z > 0 and z < 0 are equivalent. It is demonstrated that the moduli space 𝒮(ℝ3) for the derived fluid flow in ℝ3 is naturally isomorphic to the set of all rational numbers p/q in the interval J 1 : 0 . 25 < q < M ̃ 1 0 . 5847 , where q is the safety factor. For the fluid flow in the first invariant ball B 1 3 , it is shown that all values of the safety factor q belong to a small interval of length ℓ ≈ 0.1261. It is established that only torus knots Kp,q with 0.25 < p/q < 0.5847 are realized as vortex knots for the constructed flow in ℝ3. Each torus knot Kp,q with 0.25 < p/q < 0.5 is realized on countably many invariant tori T k 2 located between the invariant spheres S k 2 and S k + 1 2 , while torus knots with 0 . 5 < p / q < M ̃ 1 are realized only on finitely many invariant tori. The moduli spaces S m ( B a 3 ) ( m = 1 , 2 , ) of vortex knots are constructed for some axisymmetric steady fluid flows that are solutions to the boundary eigenvalue problem for the curl  operator on a ball B a 3 .

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The safety factor q(P) is connected to the pitch p(P) of the helical magnetic force lines on the torus P by the relation p(P) = 2πq(P). For vortex helical lines, we use both of these notions.

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