Better pictures of Cannon-Thurston maps

Date started: May 2018
Leads: David Bachman, Saul Schleimer, Henry Segerman

Abstract

The Cannon-Thurston map is a geometric way to produce sphere-filling curves. Roughly speaking, one starts with a surface S immersed in a three-manifold M. The former has a “circle at infinity” S^1_\infty while the later has a “sphere at infinity” S^2_\infty. Thus, sometimes, the immersion i :S \to M leads to a map i_* between the associated objects at infinity. In the good case, when S is a fiber of a fibration of M and when M is hyperbolic, one way to draw the image of S^1_\infty would be to draw a black sphere — after all, the curve is sphere-filling!

Thurston’s classic figure (above right, from this paper) shows an approximation of the sphere-filling curve coming from the inclusion of the Seifert surface S of the figure-eight knot included into its complement M. We seek to understand how this picture was drawn, to draw similar pictures for other pairs (M, S), and to find other techniques for drawing Cannon-Thurston maps.

Media

The unique taut surface inside of the SnapPy manifold s227. This surface is not a fiber. Thus the image (considered as in the universal cover) is a union of Cannon-Thurston maps for its infinitely many lifts.
The Canon-Thurston map for the Seifert surface of the figure-eight knot, as viewed from inside the knot’s complement (m004 in SnapPy notation). A pixel corresponds to a “point near infinity”. The colour of the pixel is computed from the (signed) number of times an interval, from the origin to the point, crosses the surface.
A view of a totally geodesic thrice-punctured sphere inside the SnapPy manifold s776. Since the surface is totally geodesic, its lifts to hyperbolic space are all geodesic planes; their boundaries are round circles.
The SnapPy manifold s000 is a punctured torus bundle (with monodromy RLLLLL). Here we see the Cannon-Thurston map for the fiber.
A screen shot from https://www.shadertoy.com/view/Xd3fzB

References