A Unified Quantum SO(3) Invariant for Rational Homology 3-Spheres

TL;DR

This paper introduces a unified quantum invariant for rational homology 3-spheres that encompasses all SO(3) Witten-Reshetikhin-Turaev invariants, extending Habiro's invariant to more general cases.

Contribution

It constructs a new unified invariant in a modified Habiro ring for rational homology 3-spheres, generalizing previous invariants and providing a comprehensive framework for quantum invariants.

Findings

The invariant dominates all SO(3) WRT invariants of (M,L).

Recovers Habiro's invariant when b=1 and L is empty.

Introduces new Ohtsuki series for rational homology 3-spheres.

Abstract

Given a rational homology 3-sphere M with the first integral homology of rank b and a link L inside M, colored by odd numbers, we construct a unified invariant I_{M,L} belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev invariants of the pair (M,L). If b=1 and L is empty, I_M coincides with Habiro's invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I_M. One of the applications are the new Ohtsuki series (perturbative expansions of I_M at roots of unity) dominating all quantum SO(3) invariants.