  Three manifold topology is used to analyze the effect of anyonic
interferometers in which the probe anyons' path along an arm crosses itself,
leading to a "twisted" or braided space-time trajectory for the probe anyons.
In the case of Ising non-Abelian anyons, twisted interferometry is shown to be
able to generate a topologically protected $\pi/8$-phase gate, which cannot be
generated from quasiparticle braiding.
