Summary: 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 \(\gamma\) in \(S\) gives rise to a natural trace-of-monodromy function \(\mathbb{I}(\gamma)\) on the Teichmüller space. For any ideal triangulation \(\Delta\) of \(S\), this function \(\mathbb{I}(\gamma)\) is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of \(\Delta\). An important problem was to construct a quantization of this function, \(\mathbb{I}(\gamma)\), 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 D. G. L. Allegretti and H. K. Kim [Adv. Math. 306, 1164–1208 (2017; Zbl 1433.13020)] using F. Bonahon and H. Wong’s [Geom. Topol. 15, No. 3, 1569–1615 (2011; Zbl 1227.57003)] \(\mathrm{SL}_2\) quantum trace for skein algebras, and by M. Gabella [Commun. Math. Phys. 351, No. 2, 563–598 (2017; Zbl 1369.81085)] using D. Gaiotto et al.’s [J. High Energy Phys. 2012, No. 12, Paper No. 82, 169 p. (2012; Zbl 1397.81364); Ann. Henri Poincaré 14, No. 7, 1643–1731 (2013; Zbl 1288.81132); Ann. Henri Poincaré 15, No. 1, 61–141 (2014; Zbl 1301.81262)] 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.