Dehn Surgery Images

Dehn surgery space for the figure 8 knot, m004

Dehn surgery space for m202

Dehn surgery space for m220

Dehn surgery space for m410

At Georgia International Topology Conference 2009, Tracy Hall, Saul Schleimer and I started generating these images of Dehn surgery space. We used SnapPeaPython (now SnapPy) by Marc Culler and Nathan Dunfield, who built on SnapPea by Jeff Weeks.

These images show the result of asking SnapPeaPython to do Dehn surgery to the manifolds (on the 0th cusp), starting from the unfilled manifold, with filling coordinates ranging from -5.0 to 5.0 in both x and y. The colour scheme is as follows:

When a positive volume solution is found (i.e. for green or blue) we shade based on the ratio of this volume to the volume of the unfilled manifold. We colour in ten "bands" moving from lighter to darker green or blue as one moves from the complete structure to the boundary of Dehn surgery space (at which point the volume is zero). There is clearly some junk data points beyond the boundary of Dehn surgery space, but SnapPea seems to do a remarkably consistent job otherwise.

The images are available as zip files of 400x400 pixel png format files:

• SnapPea census of manifolds with fewer than 5 tetrahedra:
  • m003-m412 (15 MB)
• SnapPea census of manifolds with 6 tetrahedra:
  • s000-s961 (47 MB)
• SnapPea census of manifolds with 7 tetrahedra:
  • v0000-v0999 (49 MB)
  • v1000-v1999 (49 MB)
  • v2000-v2999 (48 MB)
  • v3000-v3551 (23 MB)

The images were generated using Pillow with a python script closely related to this script (thanks to Chris Atkinson for updating the script for SnapPy), which generates the image for a single manifold. One can alter the parameters in the script to "zoom in" on interesting features.

Dehn surgery space for m129.

Dehn surgery space for m412.