Each graph on the plane codes a knot. For example, the triangle
codes the trefoil knot:
Graphs are very easy to produce. A large class of them is even called
"dessins d'enfants" (children's drawings) by mathematicians.Here is a graph:
The process to
draw the knot it codes is in four steps:
It is important to draw the crossings neatly so that there is no doubt about the direction a bit of thread points to. In particular, don't simply tick the middle of the edge.
You have to connect these bits of threads to one another in order for them to flow nicely without beginning and end. To do that, think of your graph as being a maze where each edge is a wall with an open door in the middle, where the crossing occurs. Choose a bit of thread, it points in a given direction. Follow the wall in that direction, turn the corner, follow the wall until you reach another crossing. One of its threads points towards you, it's the one to connect to.
At this stage,
you need to straighten things a bit so that each thread flows nicely. But
don't do it too quickly, don't try to guess too fast as mayhem will occur.
So don't be too smart and apply the rules.
Third, at every crossing, you have to decide which thread is
over and which is under. You could set the first one arbitrarily and then
follow it, under then over, then under, changing its status at each crossing
along the way. That may work for small designs but it is unmanagable for
larger knots. You end up with inconstistencies between parts of your drawing.
Fortunately, there is an easier way: Draw this little guide on a corner
of a page and rip it off. Then, for each edge of your graph, place your
guide alongside with both edges aligned. It will decide for you which of
the two threads is over.
Then inflate your knot. Think of the thread you just draw as the yellow line at the middle of a road, you need now to draw the left and the right side of the road. Be careful that at each crossing, there is a bridge, a road goes under the bridge and the other goes over. You should start by underlining the bridges so that you know where to start and where to stop.
Now you are ready for some exercices.