Naturality of ${\mathrm{S}\mathrm{L}}_n$ quantum trace maps for surfaces

The ${\mathrm{S}\mathrm{L}}_n$-skein algebra of a punctured surface $\mathfrak{S}$, studied by Sikora, is an algebra generated by isotopy classes of $n$-webs living in the thickened surface $\mathfrak{S}\times (-1,1)$, where an $n$-web is a union of framed links and framed oriented $n$-valent graphs satisfying certain conditions. For each ideal triangulation $\lambda$ of $\mathfrak{S}$, Lê and Yu constructed an algebra homomorphism, called the ${\mathrm{S}\mathrm{L}}_n$-quantum trace, from the ${\mathrm{S}\mathrm{L}}_n$-skein algebra of $\mathfrak{S}$ to a so-called balanced subalgebra of the $n$-root version of Fock and Goncharov’s quantum torus algebra associated to $\lambda$. We show that the ${\mathrm{S}\mathrm{L}}_n$-quantum trace maps for different ideal triangulations are related to each other via a balanced $n$-th root version of the quantum coordinate change isomorphism, which extends Fock and Goncharov’s isomorphism for quantum cluster varieties. We avoid heavy computations in the proof, by using the splitting homomorphisms of Lê and Sikora, and a network dual to the $n$-triangulation of $\lambda$ studied by Schrader and Shapiro.