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Contact topology and hydrodynamics. III: Knotted orbits. (English) Zbl 0960.76020
Summary: We employ the relationship between contact structures and Beltrami fields derived in part I of this series [see the authors, Nonlinearity 13, No. 2, 441-458 (2000; Zbl 0982.76021)] to construct a steady nonsingular solution to the Euler equations on a Riemannian \(S^3\) whose flowlines trace out closed curves of all possible knot and link types. Using careful contact-topological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on \(S^3\). Sufficient review of concepts is included to make this paper independent of the previous works in this series.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
57M25 Knots and links in the \(3\)-sphere (MSC2010)
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37J55 Contact systems
37C27 Periodic orbits of vector fields and flows
53D10 Contact manifolds, general
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References:
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