# Octahedral developing of knot complement II: Ptolemy coordinates and applications

## TL;DR

This paper explores the Ptolemy coordinates for octahedral decompositions of knot complements, providing explicit formulas and algorithms to analyze representations and their obstructions, advancing understanding of knot invariants and geometric structures.

## Contribution

It introduces explicit Ptolemy coordinate formulas and a diagrammatic algorithm for holonomy representations in the context of octahedral knot decompositions, linking to Thurston's gluing equations.

## Key findings

- Explicit Ptolemy coordinates in terms of segment and region variables
- A formula for the obstruction to lifting representations
- A diagrammatic algorithm for computing holonomy representations

## Abstract

It is known that a knot complement (minus two points) decomposes into ideal octahedra with respect to a given knot diagram. In this paper, we study the Ptolemy variety for such an octahedral decomposition in perspective of Thurston's gluing equation variety. More precisely, we compute explicit Ptolemy coordinates in terms of segment and region variables, the coordinates of the gluing equation variety motivated from the volume conjecture. As a consequence, we present an explicit formula for computing the obstruction to lifting a $(\mathrm{PSL}(2,\mathbb{C}),P)$-representation of the knot group to a $(\mathrm{SL}(2,\mathbb{C}),P)$-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.