About the Background
The knots in the backgrounds were found by Kofman,
Patterson and me, and 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.
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
Callahan, Dean, and Weeks.
The simplest hyperbolic knots, J. Knot Theory Ramifications, 8
(1999), no. 3, 279--297.
(Hyperbolic knots with 6 or fewer tetrahedra)
Champanerkar, Kofman, and Patterson.
The next simplest hyperbolic knots,
J. Knot Theory Ramifications 13 (2004), no. 7, 965-987.
(Hyperbolic knots with 7 tetrahedra)
Champanerkar, Kofman, and Mullen.
The 500 simplest hyperbolic knots,
J. Knot Theory Ramifications 23 (2014), no. 12,
1450055[1-34]. (Hyperbolic knots with 8 tetrahedra)
The beautiful pictures for these knots were
made by Rob
. More pictures:
Hyperbolic Knots 1
Hyperbolic Knots 2
The hyperbolic knot census is included
. The tetrahedral census name (when available) is included in
Table of Knot Invariants