Seminar

This talk will take place on campus, in Evans Hall room 736.

Many links $L$ in $S^3$ have a complement admitting a hyperbolic structure. Given a combinatorial description of the link diagram, SnapPy has an algorithm to find this hyperbolic structure (due to Jeffrey Weeks; based on work of William Thurston, Terry Lawson, and William Menasco). That is, it finds a triangulation that is homeomorphic to the complement and that admits a geometric structure. But it does not give an explicit description of the homeomorphism and thus cannot show hyperbolic objects such as closed geodesics or the hyperbolic tetrahedra in the link diagram. Our goal is to fill this gap.

To do this, Henry Segerman and I developed a new approach to triangulating a link complement which gives us an explicit homeomorphism from the hyperbolic triangulation to $S^3 \setminus L$. In particular, we implemented that a user can see their trajectory around the link in $S^3$ as they fly through the hyperbolic 3-manifold in SnapPy's inside view. This was inspired by the Geometry Center's "Not Knot" video and our wish to make an interactive version of it.