Rotating magnetized compact objects and their accretion discs can generate strong toroidal magnetic fields driving highly magnetized plasmas into relativistic jets. Of significant concern, however, has been that a strong toroidal field in the jet should be highly unstable to the non-axisymmetric helical kink (screw) $m=1$ mode leading to rapid disruption. In addition, a recent concern has been that the jet formation process itself may be unstable due to the accretion of non-dipolar magnetic fields. We describe large-scale fully three-dimensional global general relativistic magnetohydrodynamic simulations of rapidly rotating, accreting black holes producing jets. We study both the stability of the jet as it propagates and the stability of the jet formation process during accretion of dipolar and quadrupolar fields. For our dipolar model, despite strong non-axisymmetric disc turbulence, the jet reaches Lorentz factors of $Gammasim 10$ with opening half-angle $theta_jsim 5^circ$ at $10^3$ gravitational radii without significant disruption or dissipation with only mild substructure dominated by the $m=1$ mode. On the contrary, our quadrupolar model does not produce a steady relativistic ($Gammagtrsim 3$) jet due to mass-loading of the polar regions caused by unstable polar fields. Thus, if produced, relativistic jets are roughly stable structures and may reach up to an external shock with strong magnetic fields. We discuss the astrophysical implications of the accreted magnetic geometry playing such a significant role in relativistic jet formation, and we outline avenues for future work.
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