Quantum hyperbolic invariants for diffeomorphisms of small surfaces

TL;DR

This paper explicitly computes quantum hyperbolic invariants for specific surface diffeomorphisms, advancing understanding of quantum Teichmuller space representations in low-puncture cases.

Contribution

It provides explicit calculations of quantum hyperbolic invariants for the 1-puncture torus and 4-puncture sphere, building on previous theoretical frameworks.

Findings

Explicit formulas for invariants of the 1-puncture torus

Explicit formulas for invariants of the 4-puncture sphere

Enhanced understanding of quantum Teichmuller representations

Abstract

An earlier article with Francis Bonahon introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmuller space. We explicity compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.