We prove that for the torus with one hole and punctures and the sphere with four holes there is a family of quantum trace functions in the quantum Teichmüller space, analog to the non-quantum trace functions in Teichmüller space, satisfying the properties proposed by Chekhov and Fock.
"Quantum traces in quantum Teichmüller theory." Algebr. Geom. Topol. 10 (3) 1245 - 1283, 2010. https://doi.org/10.2140/agt.2010.10.1245