  Heterostructures of spin-orbit coupled materials with s-wave superconductors
are thought to be capable of supporting zero-energy Majorana bound states. Such
excitations are known to obey non-Abelian statistics in two dimensions, and are
thus relevant to topological quantum computation (TQC). In a one-dimensional
system, Majorana states are localized to phase boundaries. In order to bypass
the constraints of one-dimension, a wire network may be created, allowing the
exchange of Majoranas by way of junctions in the network. Alicea et al. have
proposed such a network as a platform for TQC, showing that the Majorana bound
states obey non-Abelian exchange statistics even in quasi-one-dimensional
systems. Here we show that the particular realization of non- Abelian
statistics produced in a Majorana wire network is highly dependent on the local
properties of individual wire junctions. For a simply connected network, the
possible realizations can be characterized by the chirality of individual
junctions. There is in general no requirement for junction chiralities to
remain consistent across a wire network. We show how the chiralities of
different junctions may be compared experimentally and discuss the implications
for TQC in Majorana wire networks.
