(29 pages) – on the arXiv – published in International Journal of Mathematics.

**Abstract:** We use the theory of sutured TQFT to classify contact elements in the sutured Floer homology, with \(\mathbb{Z}\) 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 \(\mathbb{Z}\) 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.

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Sutured TQFT, torsion, and tori