We show that the reduced -twisted Burau representation can be obtained from the quantum group for a fourth root of unity and that representations of satisfy a type of Schur–Weyl duality with the Burau representation. As a consequence, the -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–Reshetikhin, and we interpret their invariant as a twisted Conway potential.
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Calvin McPhail-Snyder, Holonomy invariants of links and nonabelian Reidemeister torsion. Quantum Topol. 13 (2022), no. 1, pp. 55–135DOI 10.4171/QT/160