  We elaborate on the construction of the Evans chain complex for higher-rank
graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of
the differential maps. These block matrices are then used to identify a wide
family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally,
in the specialized case where the higher-rank graph consists of one vertex, we
are able to use the K\"unneth theorem to explicitly compute the homology groups
of the Evans chain complex.
