Octahedral coordinates from the Wirtinger presentation

Abstract

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 $D$. This diagram also determines an ideal triangulation of the complement called the octahedral decomposition. $\rho$ 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 $\rho$ occurs as a critical point of the diagram's Neumann-Zagier--Yokota potential function.