About the Background

The knots in the backgrounds appear in the census of simplest hyperbolic knots . Below are the various knots which appear as backgrounds on my homepage. The census knot number as well as the SnapPea manifold number is indicated.

398 1300 1690 1921 2759 2930
k7_13 k7_48 k7_59 k7_69 k7_101 k7_111
v398 v1300 v1690 v1921 v2759 v2930

Simplest hyperbolic knots

The complement of a hyperbolic knot can be triangulated using hyperbolic ideal tetrahedra. The minimum number of ideal tetrahedra needed to triangulate a hyperbolic knot complement gives a natural measure of its geometric complexity. The census of knots using this geometric complexity gives a different view of the space of knots than the view using the crossing number of knots. For example, many of the geometrically simple knots have very high crossing number. This different view is interesting and intriguing from the point of view of knot invariants, in particular the Jones polynomial which appears to be unusually simple for the simplest hyperbolic knots. The numbering scheme is based first on the number of required tetrahedra, and then on volumes and geodesic lengths. The simplest hyperbolic knot census is enumerated in the following papers.
The beautiful pictures for these knots were made by Rob Scharein using KnotPlot . More pictures: Hyperbolic Knots 1 Hyperbolic Knots 2

The hyperbolic knot census is included in SnapPy. The tetrahedral census name (when available) is included in Table of Knot Invariants.