Rational Surgery Formula for Habiro–le Invariants of Rational Homology 3–spheres

Habiro–Le invariants dominate sl2 Witten–Reshetikhin–Turaev invariants of rational homology 3–spheres at roots of unity of order coprime with the torsion. In this paper we give a formula for the Habiro–Le invariant of a rational homology 3–sphere obtained by rational surgery on a link in S. As an application, we compute this invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that the Habiro–Le invariant separates integral homology Seifert fibered spaces. Introduction In 2001, Habiro [4] announced a construction of an invariant of integral homology 3–spheres with values in the universal ring Ẑ[q] = lim ←−− n Z[q] (q)n where (q)n := (q; q)n = (1− q)(1− q) . . . (1− q). Habiro’s invariant specializes at a root of unity to the value of the sl2 Witten–Reshetikhin–Turaev (WRT) invariant at that root. Recently, Le generalized Habiro’s theory to rational homology 3–spheres [6]. For a rational homology sphere M with |H1(M,Z)| = a, Le constructed an invariant IM which dominates the SO(3) WRT invariants of M at roots of unity of order coprime to a. Habiro’s universal ring was modified by inverting a and cyclotomic polynomials of order not coprime to a. Important applications of the Habiro–Le theory are the new integrality properties of WRT invariants, new results about Ohtsuki series and a better understanding of the relation between LMO invariant, Ohsuki series and WRT invariants. The aim of this paper is to give a rational surgery formula for the Habiro–Le invariant IM . As a main technical ingredient we use the generalized Watson identity for q–hypergeometric series, which allows us to put the result in a smaller ring than considered in [6]. Let us summarize our main results. For a positive integer a let Aa := Z[ 1 a ][q ] and Na the set of positive integers coprime with a. Denote by Φs(t) the s–th cyclotomic polynomial. Let Λa ⊂ Q(q) be the ring obtained from Aa by adding the inverses of each Φs(q ) with s not coprime with a: Λa := Aa [ 1 Φs(q) , s / ∈ Na ] The analog of the Habiro ring constructed in [6] is Λ̂a := lim ←−− n Λa (q)n Let Ua be the set of all complex roots of unity with orders odd and coprime with a. Le showed that the ring Λ̂a plays the role of the Habiro ring, with the set of all roots of 1 replaced by Ua. 1