This work is supported by the National Science Foundation Division of Mathematical Sciences DMS#0074315, DMS#0296098, and DMS#0311010. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
This page provides background information for the ideas of smooth thickness and rope length, and shows example data from my program. This is pretty old stuff and I hope to update it someday.
First, take a piece of rope, tie a few knots in the rope, and splice the ends together to create a closed loop. Next, imagine the radius of the rope becoming infinitely small. The one-dimensional filament closed loop that remains is a topological knot. It is crucial that the infinitely-thin filament never intersects itself, or we have what is called a graph.
Two knots are the same if you can grab one of the knots and move it around in 3-space to get the other knot. You are allowed to stretch, bend, and twist the knot. However, you cannot cut the filament or pass it through itself. For example, a square is equivalent to a unit circle. If a knot is equivalent to a unit circle, it is said to be a "trivial knot" or an "unknot". Any knot which is not an unknot is called "non-trivial". A knot-type consists of all equivalent knots of a given type. So a square has the same knot type as a unit circle.
Can one tie a non-trivial knot with a piece of rope of length twelve inches and radius one inch?
Theory developed in "Thickness of Knots" by R. Litherland, J. Simon, O. Durumeric, and myself, answers no to the preceding question (given that the rope doesn't stretch, no cheating with elastic bands).
Imagine a piece of rope of radius one inch that is flexible enough so that one could make a circle with 2*Pi worth of this rope. Without getting into the nitty-gritty, given one unit length of this rope, the thickness of a knot type is the thickest radius of rope with which one can tie a conformation of that knot-type.
Rope length is similar, except that we set the radius constant at one and
look for the minimal length of rope realizing the knot-type. One can see
that
ropelength(K) = 1 / thickness(K)
and by energy conventions and aesthetics, it is easier to think of
rope length as opposed to thickness.
I have defined a thickness function for polygons. Fix a smooth knot. I have shown that if we take polygonal approximations of smaller and smaller mesh, the polygonal thickness measures of the inscribed polygons converge to the smooth thickness of the original knot. For a given polygonal knot, my program computes its polygonal rope length and uses a random perturbation algorithm to flow the knot to conformations with smaller rope lengths (remember, prettier knots, lower rope lengths).
Below you will find three pages of pictures from the program TOROS. I suggest that you try all of them. If you have a slow connection, this may take a while.
How much rope
does it take to make a trefoil?
What does
the thickness surface look like in equilateral knot space?
Would you
show me some more pretty knots?
become 
?
Evolution
of a trefoil Shows an evolution with a fixed radius and then the same
evolution where the cylinders about the edges are drawn with radius equal
to the polygonal injectivity radius.
The
same evolution as above shown in two columns In the first column, the
trefoil is shown with a fixed radius. In the second column, the trefoil
is drawn with cylinders whose radius is the polygonal injectivity radius.
See the movie
recreation.