The complete classification of unital graph C?-algebras: Geometric and strong

We provide a complete classification of the class unital graph $C^*$-algebras - prominently containing full family Cuntz-Krieger algebras showing that Morita equivalence in this case is determined by ordered, filtered $K$-theory. The result geometric sense it establishes any between $C^*(E)$ and $C^*(F)$ can be realized sequence moves leading from $E$ to $F$, way resembling role Reidemeister on knots. As key ingredient, we introduce new such moves, establish they leave invariant, prove after augmentation, list becomes described above. Along way, every reduced $K$-theory isomorphism lifted an stabilized and, as consequence, preserving unit comes $*$-isomorphism themselves. It follows question amongst decidable one. immediate examples applications our results revisit problem for quantum lens spaces verify, case, Abrams-Tomforde conjectures.