Twisty itsy bitsy topological field theory

We extend the topological field theory (“itsy bitsy topological field theory”‘) of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on those surfaces respecting signs (sutures). It has information-theoretic (“itsy”) and quantum-field-theoretic (“bitsy”) aspects. In the process we extend some results of sutured Floer homology, consider associated ribbon graph structures, and construct explicit admissible Heegaard decompositions.