A Complement of the Figure-Eight Knot

Date started: 2019
Leads: Max Krause and Tashrika Sharma


The figure-eight knot is the simplest hyperbolic knot, meaning that its complement can be given a complete hyperbolic metric. A way to see this is to split the complement into two ideal tetrahedra.

The goal of this project is to show how two such tetrahedra should be glued together in order to obtain the figure-eight complement. The first lead is uses 3D rendering software to create an animation of the gluing, whereas the second lead focuses on creating schematic diagrams to highlight the steps in the gluing process.


We begin to demonstrate gluing the two tetrahedra along all faces (only two faces are highlighted). First we eliminated the vertices (since this is where the figure-eight will be) and then glued the red faces. The last picture shows how we deformed our result to get ready to glue the gray faces.
These stills from the animation show different steps of the gluing process. The face being glued is highlighted in blue, and the green faces eventually will constitute the toroidal boundary of a neighborhood of the figure-eight.


  • William P. Thurston: The Geometry and Topology of Three-Manifolds (2002)