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.

Sutured TQFT, torsion, and tori