Chapter 2: Two Knots
Knot theory is the study of patterns that can be made with string. Two of the simplest such patterns are the trefoil knot, also known as the overhand knot, and the figure eight knot.
The Trefoil
The Figure Eight
Here is a technique for tying a trefoil knot. After crossing as is shown in the second photo, the end is brought up the front and put through the opening at the hand.
And here is a similar technique for tying a figure eight knot. After the end in the lower hand is wrapped all the way around the other strand, it is brought up the front and passed through the opening at the upper hand.
The trefoil knot is chiral, meaning that there is a right hand version and a left hand one, and they are different, like your shoes.
We start with a right twist and a left twist.
Ends are brought up through to complete the knots.
Here is a picture of a left trefoil and a right trefoil tied in the same length of rope.
The figure eight is achiral, meaning there is no left/right distinction, like a tennis ball. Here we tie two Figure Eights, one beginning with a left hand twist, the other beginning with a right hand twist. We then flip over the knot on the right twice, first horizontally then vertically. It doesn’t matter which flip we do first, the result will be the same. And the result is, as we can see, is that the two knots are the same.
A left and a right trefoil cannot be made to look the same no matter what you do.
We start with a left hand twist and a right hand twist.
Ends passed through to complete Figure Eights.
The knot on the right has been flipped horizontally.
The knot on the right was then flipped vertically.
In a practical sense, the trefoil and the figure eight can have very different properties. The trefoil can made to walk along the rope fairly easily — the rope can often be pulled through it. This makes it unsafe to use in many applications. It is especially bad as a stopper knot — when one hopes the knot will prevent the rope from passing through an aperture. One of the first things beginning rock climbers are taught is to never use the trefoil (overhand) knot on a climbing rope. The figure eight knot is much more resistant to walking — letting the rope slide through. We can see why this is by looking at directed diagrams of the knots. In a directed diagram we follow the rope through the knot, always moving in the same direction. We can denote this direction with arrows along the rope.
For the knot to walk along the rope, it moves forward following the arrows. If we look at some of the places where the rope needs to slide by itself, we can see that in the trefoil, the strands are moving in roughly the same direction. So the motion doesn’t have to fight much friction. But in the figure eight, at some of those places the arrows are opposing, so any walking has to overcome friction.
A directed trefoil.
A directed figure eight.