  It has been proposed that localized zero-energy Majorana states can be
realized in a two-dimensional network of quasi-one-dimensional semiconductor
wires that are proximity-coupled to a bulk superconductor. The wires should
have strong spin-orbit coupling with appropriate symmetry, and their electrons
should be partially polarized by a strong Zeeman field. Then, if the Fermi
level is in an appropriate range, the wire can be in a topological
superconducting phase, with Majorana states that occur at wire ends and at $Y$
junctions, where three topological superconductor segments may be joined. Here
we generalize these ideas to consider a three-dimensional network. The
positions of Majorana states can be manipulated, and their non-Abelian
properties made visible, by using external gates to selectively deplete
portions of the network, or by physically connecting and redividing wire
segments. Majorana states can also be manipulated by reorientations of the
Zeeman field on a wire segment, by physically rotating the wire about almost
any axis, or by evolution of the phase of the order parameter in the
proximity-coupled superconductor. We show how to keep track of sign changes in
the zero-energy Hilbert space during adiabatic manipulations by monitoring the
evolution of each Majorana state separately, rather than keeping track of the
braiding of all possible pairs. This has conceptual advantages in the case of a
three-dimensional network, and may be computationally useful even in two
dimensions, if large numbers of Majorana sites are involved.
