TL;DR
This paper constructs quantum trace functions in quantum Teichmüller spaces for specific surfaces, extending classical trace functions to the quantum setting and verifying their properties as proposed by Chekhov and Fock.
Contribution
It introduces a family of quantum trace functions for the torus with one hole and the sphere with four holes, aligning with Chekhov and Fock's proposed properties.
Findings
Quantum trace functions are constructed for the specified surfaces.
The functions satisfy properties analogous to classical trace functions.
The work extends classical Teichmüller theory into the quantum domain.
Abstract
We prove that for the torus with one hole and p greater than or equal to 1 punctures and the sphere with four holes there is a family of quantum trace functions in the quantum Teichm\"uller space, analog to the non-quantum trace functions in Teichm\"uller space, satisfying the properties proposed by Chekhov and Fock in their paper "Observables in 3D Gravity and Geodesic Algebras."
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