Derived invariants for surface cut algebras of global dimension 2 II: the punctured case

For algebras of global dimension 2 arising from a cut of the quiver with potential associated with a triangulation of an unpunctured surface, Amiot-Grimeland defined integer-valued functions on the first homology groups of the surfaces. Derived equivalences translate to the existence of automorphisms of surfaces preserving these functions. We generalize this to punctured surfaces. Moreover, we show that all algebras of global dimension 2 arising from an arbitrarily punctured polygon are derived equivalent. We also combinatorially characterize the QPs of triangulations that do not admit cuts and those cuts that yield algebras of global dimension greater than 2.