Drawing Planar Diagrams
My summer student Emily Redelmeier is in the process of writing a program that uses circle packing to draw an arbitrary object given as a PD
as in Planar Diagrams. At the moment her program is still slow, limited and sometimes buggy, but it is already quite useful, as the following lines show:
(For In[1] see Setup)


Thus, for example, here's the torus knot T(4,3):
In[4]:=

Show[DrawPD[TorusKnot[4, 3]]]

Out[4]=

Graphics

One problem we currently have is that crossings come out at nonuniform sizes, hence in the picture below you may need magnifying glasses to decide who's over and who's under:
In[5]:=

MillettUnknot = PD[
X[1,10,2,11], X[9,2,10,3], X[3,7,4,6], X[15,5,16,4],
X[5,17,6,16],X[7,14,8,15], X[8,18,9,17], X[11,18,12,19],
X[19,12,20,13], X[13,20,14,1]
];

In[6]:=

Show[DrawPD[MillettUnknot]]

Out[6]=

Graphics

In such a situation, the option Gap
is sometimes handy:
In[7]:=

Show[DrawPD[MillettUnknot, {Gap > 0.03}]]

Out[7]=

Graphics

How does it work?
DrawPD
uses Andreev's theorem [Andreev1], [Andreev2], which states that every planar graph can be realized, nearly uniquely, as the graph of tangencies of circles drawn within the unit disk. That is, to every vertex of one may associate a disk within the unit disk, so that the interiors of these disks are disjoint and they are tangent iff the corresponding vertices are connected by an edge. The Andreev "circle packing" corresponding to the knot 4_1 is the first picture on the right (circle 13 is the unit disk itself).
But now every ingredient of the original knot (every arc, crossing and face) has a disk in the plane in which it can be cleanly drawn and clashes are guaranteed not to occur. Furthermore, knowing the precise coordinates of all the tangency points allows us to represent each ingredient by some nice smooth arcs that meet smoothly. The result is the second picture on the right. Removing all the circles, what remains is the desired clean planar picture of 4_1.
[Andreev1] ^ A. Andreev, On convex polyhedra in Lobacevskii spaces (in Russian), Math. Sbornik USSR, Nov. Ser. 81 (1970) 445478.
[Andreev2] ^ A. Andreev, On convex polyhedra of finite volume in Lobacevskii spaces (in Russian), Math. Sbornik USSR, Nov. Ser. 83 (1970) 256260.