# What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?

Let $$K$$ be a hyperbolic knot in $$S^3$$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $$\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$$, the holonomy representation.

This is hard to compute in practice. A way that is easier is to decompose the knot complement into ideal tetrahedra, whose geometry is described by shape parameters. It is then possible to write down equations for these shape parameters. Solving them gives a description of the hyperbolic structure, and can be used to compute things like the volume. [Warning: I know less about this paragraph, so I may be wrong.]

Is there a source that works out the details of how to go between these two descriptions? I would be especially interested in explicit examples of how to obtain the shape parameters from the matrices of the holonomy and vice-versa.

• I think what you're looking for is in chapter 4 of Thurston's notes. – Neal Aug 3 at 23:59