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. There were eight different knots with three crossings, sixteen knots with four crossings, and 32 knots with five crossings. The pattern simply doubled, so I made an educated guess that there would be 64 different knots with six crossings. For some reason though, I found 63, which really gets to me because it just doesn’t make sense. I tried for hours to find the last knot, but I was unsuccessful. After getting use to doing the math work on paper, I made a “number” tree of a knot with seven crossings. Sure enough, I found 128 knots, which is 64 doubled. Again, I went back and

Some topics in this essay:
, Honors Math, knots crossings, crossings pattern, mathematicians studied knots, knots five crossings, five crossings pattern, solid line, knots eight, “family tree, knots five, guidelines class, tree knots, knot theory, five crossings,