Holonomy invariants of links and nonabelian Reidemeister torsion

Abstract

We show that the reduced ${\mathrm{SL}}_2(\mathbb{C})$-twisted Burau representation can be obtained from the quantum group ${\mathcal{U}}_q({\mathfrak{sl}}_2)$ for $q=i$ a fourth root of unity and that representations of ${\mathcal{U}}_q({\mathfrak{sl}}_2)$ 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–Reshetikhin, and we interpret their invariant as a twisted Conway potential.