There is of course an enormous body of work on knot invariants,
the 3-manifold topology of knot complements,
connections between knot theory and statistical mechanics, etc.
I am instead interested here primarily in geometric questions
arising from knot embeddings.
Atlas of oriented knots and links, Corinne Cerf extends previous lists of all small knots and links,
to allow each component of the link to be marked by an orientation.
Borromean rings don't exist.
Geoff Mess relates a proof that
the Borromean ring configuration
(in which three loops are tangled together but no pair is linked)
can not be formed out of circles.
Dan Asimov discusses some related higher dimensional questions.
Matthew
Cook conjectures the converse.
Are
Borromean links so rare?
S. Javan relates the history of the links and describes
various generalizations with more than three rings.
For more history and symbolism of the Borromean rings,
see Peter Cromwell's
web site.
A
Brunnian link. Cutting any one of five links allows the remaining
four to be disconnected from each other, so this is in some sense a
generalization of the Borromean rings. However since each pair of links
crosses four times, it can't be drawn with circles.
Curvature of knots.
Steve Fenner proves
the Fary-Milnor
theorem that any smooth, simple, closed curve in 3-space must have
total curvature at least 4 pi.
Cut-the-knot logo.
With a proof of the origami-folklore that this folded-flat overhand
knot forms a regular pentagon.
Geometry and the Imagination in Minneapolis.
Notes from a workshop led by Conway, Doyle, Gilman, and Thurston.
Includes several sections on polyhedra, knots, and symmetry groups.
Hyperbolic Knot.
From Eric Weisstein's treasure trove of mathematics.
Maille Weaves.
Different repetitive patterns formed by linked circles along a plane in space,
as used for making chain mail. Along with some linear patterns for
jewelry chains.
Mathematically
correct breakfast. George Hart describes how to cut a single bagel
into two linked Möbius strips. As a bonus, you get more surface
area for your cream cheese than a standard sliced bagel.
Meru Foundation appears to be
another sacred geometry site, with animated gifs of torus knots
and other geometric visualizations and articles.
Modularity in art.
Slavik Jablan explores connections between art, tiling, knotwork, and
other mathematical topics.
Mutations and knots.
Connections between knot theory and dissection of hyperbolic polyhedra.
Orthogonal discrete knots.
Hew Wolff asks questions about the minimum total length, or the minimum volume of a rectangular box, needed to form different knots as three-dimensional polygons using only integer-length axis-parallel edges.
The
Pretzel Page. Eric Sedgwick uses animated movies of twisting pretzel knots
to visualize a theorem about Heegard splittings
(ways of dividing a complex topological space into two simple pieces).
Programming for 3d
modeling, T. Longtin. Tensegrity structures, twisted torus space frames,
Moebius band gear assemblies, jigsaw puzzle polyhedra, Hilbert fractal helices,
herds of turtles, and more.
In search of the ideal knot.
Piotr Pieranski applies an iterative shrinking heuristic to find the
minimum length unit-diameter rope that can be used to tie a given knot.
SnapPea, powerful software for computing geometric properties of
knot complements and other 3-manifolds.
Trefoil
knot stairs. Java animation of an Escher-like infinite stair construction,
intended as a Montreal metro station sculpture,
by Guillaume LaBelle.
Triangulating 3-dimensional polygons.
This is always possible (with exponentially many Steiner points)
if the polygon is unknotted, but NP-complete if no Steiner points are allowed.
The proof uses gadgets in which quadrilaterals are
stacked like Pringles to form wires.
