  We investigate the dynamical propagation of the South-East jet from the Crab
pulsar interacting with supernova ejecta by means of three-dimensional
relativistic MHD numerical simulations with the PLUTO code.
  The initial jet structure is set up from the inner regions of the Crab
Nebula.
  We study the evolution of hot, relativistic hollow outflows initially
carrying a purely azimuthal magnetic field.
  Our jet models are characterized by different choices of the outflow
magnetization ($\sigma$ parameter) and the bulk Lorentz factor ($\gamma_{j}$).
  We show that the jet is heavily affected by the growth of current-driven kink
instabilities causing considerable deflection throughout its propagation
length.
  This behavior is partially stabilized by the combined action of larger flow
velocities and/or reduced magnetic field strengths.
  We find that our best jet models are characterized by relatively large values
of $\sigma$ ($\gtrsim 1$) and small values of $\gamma_{j}\simeq 2$.
  Our results are in good agreement with the recent X-ray (\textit{Chandra})
data of the Crab Nebula South-East jet indicating that the jet changes
direction of propagation on a time scale of the order of few years.
  The 3D models presented here may have important implications in the
investigation of particle acceleration in relativistic outflows.
