TITLE: Complement of figure-8 knot
QUESTION [8 upvotes]: I am reading W. Thurston's famous "3-dimensional Geometry and Topology", but I am stuck at the point where it is said that gluing two tetrahedra in an appropriate way give you the complement of the figure-8 knot.
I saw the diagram in which the figure-8 knot is drawn to be like two tetrahedra, but I have no idea how the tetrahedra should be glued to look like that diagram. Can someone explain that?

REPLY [10 votes]: The trick is to remember that the entire knot is going to be pushed off to the vertex of the complex. The $1$-skeleton of the gluing is actually those two little "connecting" lines where the knot twists against itself. The $2$-skeleton will be gluing triangles in an "obvious" way, but the edges of the triangles will be glued to those little connecting lines, not to the knot.
Here's a picture I drew: 
 
(Disclaimer: I ran out of time so I'm not completely sure that what I labeled "inside" is actually the inside of the tetrahedron.)
Also, if you want to work backwards, the explicit face-pairing is given in the online notes in ch. 1 and ch. 4. I don't remember if it's given in the book version.