SL2 quantum trace in quantum Teichmüllertheory via writhe

Quantization of the Teichmüller space of a punctured Riemann surface S is an approach to 3-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties.Any simple loop in S gives rise to a natural trace-ofmonodromy function I. / on the Teichmüller space.For any ideal triangulation of S , this function I. / is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of .An important problem was to construct a quantization of this function, I. /, namely to replace it by a noncommutative Laurent polynomial in the quantum variables.This problem, which is closely related to the framed protected spin characters in physics, has been solved by Allegretti and Kim using Bonahon and Wong's SL 2 quantum trace for skein algebras, and by Gabella using Gaiotto, Moore and Neitzke's Seiberg-Witten curves, spectral networks, and writhe of links.We show that these two solutions to the quantization problem coincide.We enhance Gabella's solution and show that it is a twist of the Bonahon-Wong quantum trace.13F60, 46L85, 53D55, 81R60 1. Introduction 340 2. Quantum Teichmüller theory 346 3. Bonahon-Wong quantum trace 357 4. Gabella's quantum holonomy 378 5. Equality of the two constructions 394