The Problem:

Topologists care about surfaces (2-dimensional objects) embedded in 4-dimensional space. Our goal is to embed surfaces with boundary into the 4-ball. The boundary of a surface embedded in the boundary of a 4-ball is a knot embedded in a 3-sphere, so we can start here. To create these interesting spaces we wish to triangulate a knot embedded in a 3-sphere. We need the knot to be in the 1-skeleton (set of 1-simplices) of a simplicial complex homeomorphic to the 3-sphere.

Fun slides (with cool pictures!)

Paper!

To be updated:

Try out our software!

Here is linked our github branch of Regina containing the script for embedding knots. We used the language Python, but a C++ update is on the way.

Software Used

We are working with the software Regina, created by Ben Burton, which produces triangulations for smooth manifolds. You can check out the Github Here.