Let $A$ be an $N\times N$ irreducible matrix with entries in $\{ 0,1\}$. We present an easy way to find an $(N+3)\times (N+3)$ irreducible matrix $\bar{ A}$ with entries in $\{ 0,1\}$ such that their 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 know that two Cuntz--Krieger algebras ${ \mathcal{ O}}_A$ and ${ \mathcal{ O}}_B$ are isomorphic if and only if the one-sided topological Markov shift $(X_A, \sigma_A)$ is continuously orbit equivalent to $(X_B, \sigma_B)$ or $(X_{ \bar{ B}}, \sigma_{ \bar{ B}}).$
Categories: math.OA