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 Z[A± 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.