Drawing knots

A problem came to my desk from the hands of Dani (a nice source of problems, btw): drawing knots on a computer! I messed around the web and found a couple of pstricks… but they were not general enough, so I made up my mind to try my own generator. This way xknots was born.

Xknots reads a file in a certain format and renders a postscript file for a 2D view of the knot. What is a 2D view of the knot? OK, here you have one, which goes by the name of trefoil knot:

First of all, mathematical knots are closed curves on R^3. So, no dangling ends. A 2D view is just flattening the knot, and marking at each crossing which thread is above which. My idea was to invent a description rule for each knot. First, we state the number of crossings. So, 3 in our case. In order to understand a crossing, let us look at the next picture:

There are two types of crossings: L and R. All of them have four legs, numbered 1 to 4. So, in order to describe the knot, we give the coordinates of the crossings, along with their type. For example:

(200,100) L

That’s a nice crossing description. We can also, if needed, specify the angles:

(200,100) L A 60 90

The “A 60 90” means that the first leg will point at 60 degrees (counterclockwise) from the X axis, and the angle between the 1 and 2 legs is 90 degrees. Once the crossings are done, we join them with lines, for example:

1-2 3-4

This means: join the 2nd leg of the 1st crossing to the 4th leg of the 3rd crossing. If needed, we can specify how “extended” that line should be. This way,

1-2 3-4 F 2

means that this line wants to go very far away from the shortest path (the default is F 1). The full code for the trefoil is:

3
(120,120) L A 60 120
(180,120) L A 0 120
(150,172) R A 60 60
1-1 3-3
1-2 3-2 F 2
1-3 2-4 F 2
1-4 2-3
2-1 3-1 F 2
2-2 3-4

(all calculated to make a nice equilateral triangle). Here you have a couple of knots more:

The first is the “borromean rings”, the second goes by the funny name of 7_4… So, you can find the source code (C++ for linux, but pretty standard), more ideas and more explanations at the main webpage of the project…

And thanks to Dani & Alberto!