Itsy bitsy topological field theory

We construct an elementary, combinatorial kind of topological quantum field theory, based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda–Kazez–Matic. This topological field theory stores information in binary format on a surface and has “digital” creation and annihilation operators, giving a toy-model embodiment of “it from bit”.

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.