Finitely presented isomorphisms of Cuntz-Krieger algebras and continuous orbit equivalence of one-sided topological Markov shifts

We introduce the notion of finitely presented isomorphism between Cuntz–Krieger algebras, and of finitely presented isomorphic Cuntz–Krieger algebras. We prove that there exists a finitely presented isomorphism between Cuntz–Krieger algebras $\mathcal{O}_A$ and $\mathcal{O}_B$ if and only if their one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ are continuously orbit equivalent. Hence the value $\det (I-A)$ is a complete invariant for the existence of a finitely presented isomorphism between isomorphic Cuntz–Krieger algebras, so that there exists a pair of Cuntz–Krieger algebras which are isomorphic but not finitely presented isomorphic.