I am reading Low Dimensional Geometry by Bonahon, but I am stuck at a certain point in the proof of theorem 11.10 concerning the figure-8 knot complement.
On page 308, he modifies the polyhedra $X^\pm$ obtained from the knot complement by collapsing the digon faces so that the edges of the resulting tetrahedra $Y^\pm$ can correspond to geodesics in $\mathbb H^3$. I am unclear as to how the resulting quotient space obtained from gluing the $Y^\pm$'s is homeomorphic to the quotient spaces from gluing the original $X^\pm$'s, so I would appreciate if someone could help explain.