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 of 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 aspects of the knot diagram.

We introduce a vanishing property of adjoint Reidemeister torsions of a cusped hyperbolic 3-manifold derived from the physics of wrapped M5-branes on the manifold. To support our physical observation, we present a rigorous proof for the figure-eight knot complement with respect to all slopes. We also present numerical verification for several knots.

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Given a braid presentation D of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by D. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic PSL(2, C)-representation of the knot group. They conjecture the existence of a solution corresponding to the geometric representation. In this paper, we show that a boundary-parabolic representation ρ arises from a solution if and only if the length of D modulo 2 equals the obstruction to lifting ρ to a boundary-parabolic SL(2, C)-representation (as an element in Z 2 ). In particular, the Hikami-Inoue conjecture holds if and only if the length of D is odd. This can always be achieved by adding a kink to the braid if necessary. We also explicitly construct the solution corresponding to a boundary-parabolic representation given in the Wirtinger presentation of the knot group.