Ideal tetrahedron shape and cusp holonomy

Use the upper half-space model ${\mathbb{H}}^3=\{(x+iy,t):x,y\in \mathbb{R},t>0\},d{s}^2=t^{-2}(d{x}^2+d{y}^2+d{t}^2),$ whose ideal boundary is $\mathbb{C}\cup \{\mathrm{\infty }\}$. An oriented ideal tetrahedron in normalized position has ideal vertices $0,1,\mathrm{\infty },z$, where $\mathrm{Im}z>0$. Its three edge-shape parameters are $z,z^{\mathrm{\prime }}=\frac{1}{1-z},z^{\mathrm{\prime }\mathrm{\prime }}=\frac{z-1}{z},$ Assign $z$ to the edges $0\mathrm{\infty }$ and $1z$, $z^{\mathrm{\prime }}$ to $1\mathrm{\infty }$ and $0z$, and $z^{\mathrm{\prime }\mathrm{\prime }}$ to $z\mathrm{\infty }$ and $01$. The parameter at a corner of a truncated ideal vertex triangle is the parameter of the incident edge. In a triangulated oriented torus link of ideal vertices, use oriented normal paths: paths avoiding link vertices, transverse to link edges, and meeting each triangle in arcs joining distinct sides, each cutting off one corner. Represent a loop by such a path before computing its cusp multiplier. For a chosen normal path $\gamma$, its cusp multiplier $H(\gamma )$ is the product of the corner parameters it cuts off, using the parameter when the corner is on its left and its reciprocal when the corner is on its right.

Jessica S. Purcell, Hyperbolic Knot Theory, §§4.2–4.3

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4