Some horoball diagrams
of knot complements
The centers and radii of the horoballs were computed by
Jeff Weeks's program
and rendering was accomplished using Larry Gritz's
Blue Moon Rendering Tools .
Click on image for full-size version.
The horoball packing for the figure-eight knot, viewed from
beneath upper-half space. The spheres illustrated are all tangent to the
plane z=0; the spheres which are largest in the Euclidean sense have diameter 1.
There are horoballs of arbitrarily small Euclidean diameter, but here a
"cut-off" diameter of 0.05 has been chosen.
There is one additional horoball (not illustrated), namely the plane z=1.
The knot group acts transitively on the entire collection of horoballs,
but there is a (parabolic) subgroup, isomorphic to Z+Z, which preserves
the "infinite" horoball z=1, and which acts on the remaining horoballs
via Euclidean translations. Thus the 2-dimensional pattern in this picture is
repeated infinitely over the plane z=0. In general, the directions of the translations
corresponding to the meridian and longitude generators of this subgroup aren't
perpendicular, although they must be for amphicheiral knots such as the figure-eight.
We can take a fundamental region for the Z+Z action to be a rectangle whose
vertices are the points of tangency with z=0 of two adjacent full-size
(pink-maroon) spheres on the left edge of the picture, together with the points of
tangency of two adjacent full-size spheres on the right edge of the picture.
Note that pattern of spheres has additional translational and reflectional
symmetries; these correspond to the symmetries of the figure-eight knot.
The horoball packing for the Turk's head knot.
The horoballs included here have Euclidean diameters in the range from 1 down to 0.1.
This 8-crossing alternating knot has symmetry group D8.
The rectangle in the picture indicates a fundamental
region with respect to the action of the group Z+Z of covering
translations which preserve the plane z = 1.
The long horizontal sides correspond to the longitudinal
translation, and the short vertical sides correspond to the meridional
translation. The symmetries of the knot lift to symmetries of the
horoball packing, which are quite easy to spot in this example: for example,
the order 8 symmetry lifts to a glide reflection along a horizontal axis.
A close-up view of the same horoball packing.
A view of the horoball packing for a
15-crossing knot with symmetry group D5. The cut-off diameter
chosen in this picture was 0.1. This time, the fundamental region is not
rectangular, in accordance with the non-amphicheiral nature of this knot.
A close-up view of the previous horoball diagram.