Sutured TQFT, torsion, and tori

We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with Z coefficients, of certain sutured manifolds of the form \( (\Sigma \times S^1, F \times S^1) \) where \( \Sigma \) is an annulus or punctured torus. Using this classification, we give a new proof that the contact invariant in sutured Floer homology with Z coefficients of a contact structure with Giroux torsion vanishes. We also give a new proof of Massot’s theorem that the contact invariant vanishes for a contact structure on \( (\Sigma \times S^1, F \times S^1) \) described by an isolating dividing set.