Hyperbolicity equations for pseudo hyperbolic structures of knot complements

Abstract

Abstract

A knot complement can be decomposed by the Ideal octahedron mod-

ulo two points. In this decomposition, pseudo-developing map and

its holonomy representation show the conditions to construct pseudo-

hyperbolic structure. The conditions are written as hyperbolicity equa-

tion. Therefore, when the shape of each octahedron satisfy the hyper-

bolicity equation, we can give a pseudo-hyperbolic structure to the knot

complement. In this paper, we consider various kinds of variables to rep-

resent and to solve the hyperbolicity equation and especially decide a

general algorithm of obtaining w-variable solutions for this equation.

Keywords: Knot, octahedral decomposition, pseudo-hyperbolic

structure, pseudo-developing.