Points of quantum ${\mathrm{SL}}_n$ coming from quantum snakes

Abstract

We show that the quantized Fock-Goncharov monodromy matrices satisfy the relations of the quantum special linear group ${\mathrm{SL}}_n^q$. The proof employs a quantum version of the technology invented by Fock-Goncharov called snakes. This relationship between higher Teichmüller theory and quantum group theory is integral to the construction of a ${\mathrm{SL}}_n$-quantum trace map for knots in thickened surfaces, partially developed in a companion paper (arXiv:2101.06817).

47 pages (double-spaced), 21 figures. Version 3: Final version after publication