  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.
