use Wehrheim and Woodward’s theory of quilts, we will define this invariant directly in the Floer field theory framework, and then prove in Proposition \[calculscindement\] that its definition agrees with the one we just outlined. This will have two advantages: first, from this definition, the proof of Theorem \[sommecnx\] is a simple application of Künneth formula, although it is unclear (at least for the author) how to prove it from the Heegaard splitting definition. Second, when proving the surgery exact sequence, it will be convenient to work directly inside the moduli space of the punctured torus, and deal with generalized Dehn twists rather than fibered ones as in [@WWtriangle].

The proof of Theorem \[trianglechir\] will follow from a generalization of Seidel’s long exact sequence, and from the fact that a Dehn twist on a punctured torus induces a generalized Dehn twist on its extended moduli space.

I would like to warmly thank my advisors Paolo Ghiggini and Michel Boileau for suggesting me this problem and for their constant support. I also would like to thank Chris Woodward for suggesting the definition of the twisted version, Christian Blanchet for suggesting the application to alternating links, and Jean-François Barraud, Frédéric Bourgeois, Laurent 