Octahedral coordinates from the Wirtinger presentation

Let ρ be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into SL2(C) expressed in terms of the Wirtinger generators of a diagram D. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. ρ induces a hyperbolic structure on the complement of D, and in this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition induced by this structure. Our formula gives a new, explicit criterion for whether ρ occurs as a critical point of the diagram’s Neumann–Zagier–Yokota potential function.