A short note on Cuntz splice from a viewpoint of continuous orbit equivalence of topological Markov shifts - Matsumoto, Kengo (MATHEMATICA SCANDINAVICA, 2018)
Let $A$ be an $N× N$ irreducible matrix with entries in $\{0,1\}$. We present an easy way to find an $(N+3)× (N+3)$ irreducible matrix $\bar {A}$ with entries in $\{0,1\}$ such that the associated Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_{\bar {A}}$ are isomorphic and $\det (1 -A) = - \det (1-\bar {A})$. As a consequence, we find that two Cuntz-Krieger algebras ${\mathcal {O}}_A$ and ${\mathcal {O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, σ _A)$ is continuously orbit equivalent to either $(X_B, σ _B)$ or $(X_{\bar {B}}, σ _{\bar {B}})$.