On torsion in the Kauffman bracket skein module of 3-manifolds

We study Kirby problems 1. 92 (E) - (G), which, roughly speaking, ask for which compact oriented 3-manifold M the Kauffman bracket skein module S (M) has torsion as a ZA^ 1-module. We give new criteria for the presence of torsion in terms of how large the SL₂ (C) -character variety of M is. This gives many counterexamples to question 1. 92 (G) - (i) in Kirby's list. For manifolds with incompressible tori, we give new effective criteria for the presence of torsion, revisiting the work of Przytycki and Veve. We also show that S (R P³# L (p, 1) ) has torsion when p is even. Finally, we show that for M an oriented Seifert manifold, closed or with boundary, S (M) has torsion if and only if M admits a 2-sided non-boundary parallel essential surface.