Holonomy invariants of links and nonabelian Reidemeister torsion

We show that the reduced ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group ${\mathcal{U}}_q({\mathfrak{s}\mathfrak{l}}_2)$ for $q=i$ a fourth root of unity and that representations of ${\mathcal{U}}_q({\mathfrak{s}\mathfrak{l}}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the ${\mathrm{SL}}_2(\mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.