Fig. 3.

The contour ${\mathcal{A}}_N$ appears in the integral formula (69) for the Kashaev invariant of the ${\mathbf{4}}_1$ knot, and it encircles the N poles (71). By doing the integral along the contour $\mathcal{C}$ and picking the poles in the lower half plane, one obtains a new state-integral with information about the trivial connection