SL2 quantum trace in quantum Teichmüller theory via writhe

Quantization of the Teichm\"uller space a punctured Riemann surface $S$ is an approach to $3$-dimensional quantum gravity, and prototypical example quantization cluster varieties. Any simple loop $\gamma$ in gives rise natural trace-of-monodromy function $\mathbb{I}(\gamma)$ on space. For any ideal triangulation $\Delta$ $S$, this Laurent polynomial square-roots exponentiated shear coordinates for arcs $\Delta$. An important problem was construct $\mathbb{I}(\gamma)$, namely replace it by noncommutative variables. This problem, which closely related framed protected spin characters physics, has been solved Allegretti Kim using Bonahon Wong's ${\rm SL}_2$ trace skein algebras, Gabella Gaiotto, Moore Neitzke's Seiberg-Witten curves, spectral networks, writhe links. We show that these two solutions coincide. enhance Gabella's solution twist Bonahon-Wong trace.