Octahedral coordinates from the Wirtinger presentation

Let \(\rho\) be a representation of a knot group (or more generally, the fundamental group of a tangle complement) into \(\operatorname{SL}_2(\mathbb{C})\) expressed in terms of the Wirtinger generators of a diagram \(D\). In this note we give a direct algebraic formula for the geometric parameters of the octahedral decomposition of the knot complement associated to \(D\). Our formula gives a new, explicit criterion for whether \(\rho\) occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function.