Octahedral developing of knot complement II: Ptolemy coordinates and applications

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