Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889–937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmüller space. We explicity compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.