  It is known that a knot complement can be decomposed into ideal octahedra
along a knot diagram. A solution to the gluing equations applied to this
decomposition gives a pseudo-developing map of the knot complement, which will
be called a pseudo-hyperbolic structure. In this paper, we study these in terms
of segment and region variables which are motivated by the volume conjecture so
that we can compute complex volumes of all the boundary parabolic
representations explicitly. We investigate the octahedral developing and
holonomy representation carefully, and obtain a concrete formula of Wirtinger
generators for the representation and also cusp shape. We demonstrate explicit
solutions for $T(2,N)$ torus knots, $J(N,M)$ knots and also for other
interesting knots as examples. Using these solutions we can observe the
asymptotic behavior of complex volumes and cusp shapes of these knots. We note
that this construction works for any knot or link, and reflects systematically
both geometric properties of the knot complement and combinatorial aspect of
the knot diagram.
