A dual graph construction for higher-rank graphs, and $K$-theory for finite 2-graphs

Given a $k$-graph $\mathrm{\Lambda }$ and an element $p$ of ${\text{NN}}^k$, we define the dual $k$-graph, $p\mathrm{\Lambda }$. We show that when $\mathrm{\Lambda }$ is row-finite and has no sources, the $C^{\ast }$-algebras $C^{\ast }(\mathrm{\Lambda })$ and $C^{\ast }(p\mathrm{\Lambda })$ coincide. We use this isomorphism to apply Robertson and Steger’s results to calculate the $K$-theory of $C^{\ast }(\mathrm{\Lambda })$ when $\mathrm{\Lambda }$ is finite and strongly connected and satisfies the aperiodicity condition.