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, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.

64 pages, 5 figures, diagrams. v3 significantly clarifies the exposition

References in corpus (2)

• Turaev-Viro invariants as an extended TQFT
• Turaev-Viro invariants as an extended TQFT II

Cited by in corpus (1)

• Kashaev--Reshetikhin Invariants of Links