TITLE: hyperbolic structure on Figure–8 knot complement
QUESTION [12 upvotes]: I was trying to understand the proof of the fact that there is a hyperbolic structure on Figure–8 knot complement initially from Thurston's notes and then from some online notes; but unfortunately I could not understand the construction (namely the knot complement can be realized by identifying the boundary of 2 ideal tetrahedrons.)
Can someone please give a good reference for the proof? Thanks!

REPLY [2 votes]: The figure eight knot complement is a fiber bundle over the circle with fiber a once-punctured torus. 
There is a very natural and easy construction of ideal triangulations (and hence hyperbolic structures) for once-punctured torus bundles. It was introduced in Marc Lackenby's paper "The canonical decomposition of one-punctured torus bundles" and you find it nicely explained in Gueritaud's paper http://msp.org/gt/2006/10-3/gt-v10-n3-p01-s.pdf . 
In the case of the figure eight knot complement the monodromy of the fiber bundle is LR, i.e., the product of the Dehn twists at longitude and meridian. You can then easily check that the construction from the above cited papers gives you Thurston's triangulation.