SnapPy views

Date started: September 2019
Leads: Matthias Goerner, Saul Schleimer, Henry Segerman


Our goal is to add a viewer to the popular three-manifold software SnapPy [1]. It will show what a hyperbolic three-manifold looks like “from the inside”, in the spirit of Thurston’s article [2] (see especially Sections 2 and 3). This will be accomplished by hyperbolic raytracing. The technique has already been explored in other projects such as [3] and [4]; by integrating it into SnapPy, we are able to see, interactively, how Dehn surgery affects the inside view (see [5]). Other applications include visualising how cusp neighborhoods become “Margulis tubes” as the geometry is deformed away from the complete structure.

A further, more ambitious, goal is to relate the inside view of a hyperbolic link complement to the “outside view”: that of the link diagram. This will require finding an explicit homeomorphism from the exterior of the link diagram to the hyperbolic triangulation. A user could then, for example, fly through the three-manifold in the inside view and at the same time see her or his trajectory in the exterior of the link diagram. Another application would be to show how the hyperbolic systole (or other parts of the length spectrum) is topologically related to the original link.


Looking along a geodesic in s431. The “balls” are horoballs and the “beams” are equidistant neighbourhoods of edges. The green and red stripes on the horoballs are elevations of the geometric meridian and longitude.
The complete (left) and an incomplete (right) geometric structure of the figure eight knot complement, equipped with its usual triangulation. Note how, on the right hand side, the horoballs have elongated to become Margulis tubes about the incomplete locus.
A sequence of incomplete structures on the figure eight knot complement. These approach a point on the boundary of “Dehn surgery space” where both tetrahedra are flat (see [5] for a closely related video).