On the integrality of the Witten–Reshetikhin–Turaev 3-manifold invariants

Abstract

We prove that the $\mathrm{SU}(2)$ Witten–Reshetikhin–Turaev invariant of any 3-manifold with any colored link inside at any root of unity is an algebraic integer. As a byproduct, we get a new proof of the integrality of the $\mathrm{SO}(3)$ Witten–Reshetikhin–Turaev invariant for any 3-manifold with any colored link inside at any root of unity of odd order.