# Complement of figure-8 knot

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?

## 1 Answer

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.

• Thanks so much! I wonder why Prof Thurston does not put this diagram or something of the same effect in his award-winning book. Aug 14, 2013 at 13:16
• @WilsonWong Have you looked at the online notes? library.msri.org/books/gt3m
– Neal
Aug 14, 2013 at 14:01
• yes, I did, but only skim-reading. Aug 14, 2013 at 14:07
• @WilsonWong Some of what I drew, with more explanation, can be found toward the end of chapter 1. There is also an extensive section in chapter 4 on the hyperbolic structure of the figure-8 complement, along with a more complete gluing diagram, if that interests you.
– Neal
Aug 14, 2013 at 14:33