Dehn surgery space

Date started: May 2019
Leads: Saul Schleimer, Henry Segerman


A three-manifold M with torus boundary admits integral Dehn fillings; these are new three-manifolds obtained by gluing on a solid torus onto M. Thurston’s Dehn surgery theorem tells us that if N, the interior of M, is hyperbolic, then most fillings are also hyperbolic manifolds. Furthermore, their hyperbolic structure is closely modelled on that of the compact part of the parent manifold, N.

Thurston proved this by generalising integer surgeries to real surgeries – that is, there is a two-dimensional manifold, called Dehn surgery space that has as its “integer points” the Dehn fillings.  The goal of this project is to draw pictures of Dehn surgery space, using SnapPy.


In the figures below we use the following colour coding: 

  • black – integer point
  • green – all tetrahedra positively oriented
  • blue – contains negatively oriented tetrahedra
  • red – contains flat tetrahedra
  • purple – contains degenerate tetrahedra
  • white – unrecognised solution type
  • grey – no solution found
  • cyan – not attempted

When a positive volume solution is found (green or blue) we shade based on the ratio of its volume to the volume of the unfilled manifold, N.  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 SnapPy seems to do a remarkably consistent job otherwise.

Dehn surgery space for the figure 8 knot, m004
Dehn surgery space for m129
Dehn surgery space for m202



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