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?


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: http://i.imgur.com/MpVm11q.jpg (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.

  • $\begingroup$ Thanks so much! I wonder why Prof Thurston does not put this diagram or something of the same effect in his award-winning book. $\endgroup$ – wilsonw Aug 14 '13 at 13:16
  • $\begingroup$ @WilsonWong Have you looked at the online notes? library.msri.org/books/gt3m $\endgroup$ – Neal Aug 14 '13 at 14:01
  • $\begingroup$ yes, I did, but only skim-reading. $\endgroup$ – wilsonw Aug 14 '13 at 14:07
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    $\begingroup$ @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. $\endgroup$ – Neal Aug 14 '13 at 14:33

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