  We study the entanglement properties of a three dimensional generalization of
the Kitaev honeycomb model proposed by Ryu [Phys. Rev. B 79, 075124, (2009)].
The entanglement entropy in this model separates into a contribution from a
$Z_2$ gauge field and that of a system of hopping Majorana fermions, similar to
what occurs in the Kitaev model. This separation enables the systematic study
of the entanglement of this 3D interacting bosonic model by using the tools of
non-interacting fermions. In this way, we find that the topological
entanglement entropy comes exclusively from the $Z_2$ gauge field, and that it
is the same for all of the phases of the system. There are differences,
however, in the entanglement spectrum of the Majorana fermions that distinguish
between the topologically distinct phases of the model. We further point out
that the effect of introducing vortex lines in the $Z_2$ gauge field will only
change the entanglement contribution of the Majorana fermions. We evaluate this
contribution to the entanglement which arises due to gapless Majorana modes
that are trapped by the vortex lines.
