A Unified Witten-reshetikhin-turaev Invariant for Integral Homology Spheres

We construct an invariant JM of integral homology spheres M with values in a completion Ẑ[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten-ReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhin-Turaev invariants of M . As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) as a function on ζ behaves like an “analytic function” defined on the set of roots of unity. That is, the τζ(M) for all roots of unity are determined by a “Taylor expansion” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.