Triangulating knot complements in 4D
Another picture from our wonderful blackboard artist Ryan Budney in Victoria, Canada. He’s taken a 4-dimensional sphere and removed a knotted 2-dimensional sphere (i.e. the surface of a beach ball), and tried to triangulate the resulting space. Triangulation means breaking up a complicated space into simpler pieces – triangles! – and then explaining how these pieces fit together. The triangulation here has precisely one vertex (a 0-dimensional triangle), one edge (a 1-dimensional triangle), 4 triangles (i.e. your usual garden-variety triangles), 5 tetrahedra (3D triangles) and the two pentachora (4-dimensional triangles). The bottom picture shows how the pieces are glued together. The top picture is completely unrelated.
The knotted 2-dimensional sphere is called a Cappell-Shaneson knot because it is sitting inside a ‘homotopy 4-sphere’. That is, the space the knot is sitting in looks like a normal sphere from a distance but might turn out not to be upon closer inspection. If it isn’t, then it would provide a counterexample to the Poincaré Conjecture, which is still open in 4 dimensions (but was famously solved for 3 dimensions by Grigori Perelman in 2003).