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

Abstract

Abstract. We construct an invariant J M of integral homology spheres M with values in a completion Z[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 τ ζ (M ) of M at ζ. Thus J M unifies all the SU (2) WittenReshetikhin-Turaev invariants of M . As a consequence, τ ζ (M ) is an algebraic integer. Moreover, it follows that τ ζ (M ) as a function on ζ behaves like an "analytic function" defined on the set of roots of unity. That is, the τ ζ (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, τ ζ (M ) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.

A unified quantum SO(3) invariant for rational homology 3-spheres

“…When L ′ = ∅, this is Theorem 8.2 in [7]. This generalization, essentially also due to Habiro, can be proved similarly as in [7].…”

“…The Ohtsuki series [26,16], originally defined through some arithmetic congruence property of the SO(3) invariant, can be identified with the Taylor expansion of I M at q = 1 [7,15]. We will also investigate the Taylor expansions of I M at roots of unity and show that these Taylor expansions satisfy congruence relations similar to the original definition of the Ohtsuki series, see Section 4.…”

“…The positive answer to this question was given first for integral homology spheres by Habiro [7], and then for arbitrary 3-manifolds by the first and third author [3], in connection with the study of "strong integrality".…”

“…We also construct a Frobenius type isomorphism to get rid of the formal fractional power of q that appeared in [15], [3]. As a byproduct, we generalize the deep integrality result of Habiro (Theorem 8.2 in [7]), underlying the construction of the unified invariants, to a union of an algebraically split link with any odd colored one.…”

“…Two elements f, g ∈ Z[q] are the same if and only if their Taylor series at a root of unity coincide. In addition, each function f (q) ∈ Z[q] is totally determined by its values at, say, infinitely many roots of order 3 n , n ∈ N. Due to these properties the Habiro ring is also called a ring of "analytic functions at roots of unity" [7]. Thus belonging to Z[q] means that the collection of the SO(3) WRT invariants is far from a random collection of algebraic integers; together they form a nice function.…”