A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

Abstract

We construct an invariant J M of integral homology spheres M with values in a completion \(\widehat{\mathbb{Z}[q]}\) of the polynomial ring ℤ[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten–Reshetikhin–Turaev invariant τζ(M) of M at ζ. Thus J M unifies all the SU(2) Witten–Reshetikhin–Turaev invariants of M. It also follows that τζ(M) as a function on ζ behaves like an “analytic function” defined on the set of roots of unity.