# hyperbolic structure on Figure–8 knot complement

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!

There's an unconventional way to see the 2-skeleton of the ideal triangulation as a union of two immersed ideal pairs of pants (which are always totally geodesic). The image shows one pair, which has a single line of self intersection.

The figure 8 projection has two bigons, which may be arranged symmetrically (considering the projection onto $$S^2$$), and the other pair of immersed pants has a line of self-intersection along the dotted line in the other bigon. These dotted lines then have 3 branches of the surfaces coming through, and the pants each get divided into two ideal triangles. So the union forms the 2-skeleton of the ideal triangulation, with the two edges having degree 6. Then by the Poincaré polyhedron theorem one concludes that making the two tetrahedra regular ideal tetrahedra with dihedral angles of $$\pi/6$$ gives a complete hyperbolic structure.

Addendum: This is how George Francis describes the decomposition of the figure eight knot complement. Here are the figures from his Topological Picturebook.

This senior thesis looks pretty well written (Alexander Gutierrez, ASU)

A Topological Picturebook by George K. Francis contains a chapter devoted to the figure-8 knot complement and has a very detailed and intuitive illustration.

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.

• I seem to recall that Bill Thurston told me that these ideal triangulations of punctured-torus bundles are due to Troels Jorgensen. The earliest appearance of them in the literature that I am aware of is in a paper by Bill Floyd and myself, "Incompressible surfaces in punctured-torus bundles", Top. and its Appl. 13 (1982), 263-282. Gueritaud's paper seems to have a nice exposition, with nicer pictures than the hand-drawn ones in my paper with Floyd. – Allen Hatcher Dec 1 '15 at 20:37