Octahedral developing of knot complement II: Ptolemy coordinates and applications

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 boundary-parabolic [Formula: see text]-representation to boundary-unipotent [Formula: see text]-representation. We also present a diagrammatic algorithm to compute a holonomy representation of the knot group.