Figure 8 knot incomplete hyperbolic structure

Question

The incomplete hyperbolic structure of the figure-8 knot $M$ is nicely reviewed in the notes by J.Purcell. The incomplete hyperbolic structure can be described by the holonomy representation of $\pi_1(M)$, under which the meridian $m$ and longitude $l$ of the boundary torus are mapped to upper triangular matrices $$ \rho(m) = \begin{pmatrix} e^u & 1 \\ 0 & e^{-u}\end{pmatrix}, \quad \rho(l) = \begin{pmatrix} e^v & * \\ 0 & e^{-v}\end{pmatrix} $$ for some $u,v \in \mathbb{C}$, up to conjugation by $PSL(2,\mathbb{C})$. The complete hyperbolic structure corresponds to $u=v=0$.

It is stated in the above reference (p.110) that the isometries $\rho(m)$ and $\rho(l)$ fix a unique geodesic in the upper half space $\mathbb{H}^3$. Can I find this geodesic explicitly for the figure-8 knot?

Answer 1

The fixed points of $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, acting on $\mathbb{CP}^1$, are found by solving, for $z$ and $w$, the equation $azw + bw^2 = cz^2 + dzw$. Much of the time you can pretend $w = 1$ and solve $\frac{az + b}{cz + d} = z$ instead.

When there are two fix-points then they are the ideal points (that is, the points at infinity) of the desired geodesic in hyperbolic space.