Holonomy invariants of links and nonabelian Reidemeister torsion

Abstract

We show that the reduced $\\operatorname{SL}\_2(\\mathbb{C})$-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 $\\operatorname{SL}\_2(\\mathbb{C})$-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.