Human beings have been inventing knots ever since there was any kind of rope. Mathematicians have studied knots extensively for the last 100 years, but knots have been of interest since ancient times. As mathematicians studied knots, they found that just simple knots could be so rich in profound mathematical connections. Knots appear in illuminated manuscripts, sculptures, paintings, and other art forms all over the world.
The mathematical theory of knots has made major advances in the past decade. One of the most exciting developments has been the discovery of deep connections between knot theory and the branch of physics that studies the fundamental particles and forces that are the building blocks of the universe. It has also been found that DNA is sometimes knotted, and knots may play a role in molecular biology.
Imagine knots as closed loops or paths that you can trace round and round with your finger. It is as though the two free ends of tangled rope have been connected together. I will now explain a simple way to understand how knots work on paper, that is, to an individual who knows close to nothing about knots. The places where the rope crosses itself are shown as a broken line and a solid line. The intent is to show that the part of the rope represented by the broken line is passing under the part represented by the solid line.
I came up with my own family tree of knots. I started by drawing a simple knot with three crossings. A crossing is where two strands of a knot intersect and "cross" each other. Then I switched every crossing, and every combination of crossings to come up with every knot possible from the original knot. As I moved on to more complicating knots, there were more crossings, and more knots to draw. After drawing knots with three, four, and five crossings each, I began to see a pattern that looked similar to that of a family tree. It was a "family tree of knots.