Let $M$ be the complement of a hyperbolic knot in $S^3$.
Through the study of Kashaev's conjecture,
we have found a complex function
which gives the volume and the Chern-Simons invariant of the complete structure of $M$
at the critical point corresponding to the solution to the hyperbolicity equations for $M$.
The purpose of this talk is to construct such functions for the non-complete structures of $M$.