Octahedral coordinates from the Wirtinger presentation

TL;DR

This paper provides an explicit algebraic formula linking knot group representations to octahedral decompositions, enabling direct analysis of hyperbolic structures and critical points in the context of knot complements.

Contribution

It introduces a new algebraic formula for geometric parameters of octahedral decompositions derived from knot group representations.

Findings

Explicit algebraic formula for octahedral decomposition parameters

Criterion for representations to be critical points of the potential function

Connection between algebraic data and hyperbolic structures

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.