  We prove the existence of knotted and linked thin vortex tubes for steady
solutions to the incompressible Euler equation in R^3. More precisely, given a
finite collection of (possibly linked and knotted) disjoint thin tubes in R^3,
we show that they can be transformed with a C^m-small diffeomorphism into a set
of vortex tubes of a Beltrami field that tends to zero at infinity. The
structure of the vortex lines in the tubes is extremely rich, presenting a
positive-measure set of invariant tori and infinitely many periodic vortex
lines. The problem of the existence of steady knotted vortex tubes can be
traced back to Lord Kelvin.
