SL₂ quantum trace in quantum Teichmüller theory via writhe

Quantization of the Teichmüller space of a punctured Riemann surfaceS is an approachto 3–dimensionalquantum gravity, and is a prototypical example of quantization of cluster varieties. Anysimple loop γ in Sgives rise to a natural trace-of-monodromy function II(γ)on the Teichmüller space. For any ideal triangulationΔ ofS, thisfunction II(γ) isa Laurent polynomial in the square-roots of the exponentiated shear coordinates for thearcs of Δ .An important problem was to construct a quantization of this function,II(γ), namelyto replace it by a noncommutative Laurent polynomial in the quantum variables.This problem, which is closely related to the framed protected spin characters inphysics, has been solved by Allegretti and Kim using Bonahon and Wong’sSL ⁡ 2quantum trace for skein algebras, and by Gabella using Gaiotto, Moore and Neitzke’sSeiberg–Witten curves, spectral networks, and writhe of links. We showthat these two solutions to the quantization problem coincide. We enhanceGabella’s solution and show that it is a twist of the Bonahon–Wong quantum trace.