The moduli spaces of non-isotopic vortex knots are introduced for the ideal fluid flows in invariant domains . 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 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 , where q is the safety factor. For the fluid flow in the first invariant ball , 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 located between the invariant spheres and , while torus knots with are realized only on finitely many invariant tori. The moduli spaces 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 .
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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.