TITLE: How to see isometries of figure 8 knot complement
QUESTION [16 upvotes]: The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this involution is an isometry. Is there some way of visualizing this isometry?
I know how to produce one relatively easy to see isometry of $M$. The fundamental group of $M$ is a two generator group. Lift geodesic representatives of a pair of generators to $\mathbb{H}^3$. These geodesics have a mutual perpendicular, and $180^{\circ}$ rotation about that geodesic descends to an involution of $M$. However, this map has fixed points, and I'd like to "see" one that doesn't.

REPLY [3 votes]: In terms of 'seeing' the isometry $\rho$ needed to realize the Gieseking manifold as a quotient of the figure 8 knot complement $S^3-K$, we require a slight of hand, namely thinking of another name for the manifold. The reason for this is simple, the Gieseking manifold is non-orientable so $\rho$ must reverse orientation. However, there are no non-orientable manifolds covered by $S^3$, so we will not be able to 'see' $\rho$ acting on $S^3$.
Instead, we can think of $S^3-K$ by another name as follows. The figure 8 knot complement is fibered. In fact, it can be thought of as a once-punctured torus bundle with monodromy given by $\pmatrix{2 & 1\\ 1 & 1}$. It turns out $\rho$ acts on once-punctured torus bundle by a glide reflection (loosely translate 'half-way' up and reflect).