Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion

TL;DR

This paper constructs a unified quantum invariant for rational homology 3-spheres with 2-torsion, linking SO(3) and SU(2) invariants at various roots of unity, and explores their refinements and applications.

Contribution

It introduces a new unified invariant that encompasses different quantum invariants and their refinements for 3-spheres with 2-torsion, advancing understanding of their structure and integrality.

Findings

Unified invariant interpolates between SO(3) and SU(2) invariants.

Splits into refined invariants for spin and cohomological structures.

Applications include new results on Ohtsuki series and integrality of quantum invariants.

Abstract

For every rational homology 3-sphere with 2-torsion only we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invariant splits into a sum of the refined unified invariants dominating spin and cohomological refinements of quantum SU(2) invariants. New results on the Ohtsuki series and the integrality of quantum invariants are the main applications of our construction.