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Knot Theory

Today in my Calculus III class we had a professor come down from the University of Ottawa to give us a speech on Knot Theory. My first impression was that it sounded a lot like the String Theory heard of in most of my pervious physics classes, but I was proven wrong.

Knot Theory is a mathematical branch of topology. If you were to take a piece of string and join the ends (or tips) together you would be forming a knot. A variety of knots can be equivalent to one another using simple transformations that do not involve the string being cut or passing through itself. The most common way to represent knots is by the use of planar diagrams.

You can prove mathematically that if the trefoil knot was to be flipped, it wouldn't represent the same knot, due to the Jones polynomial not be equal to one another. Here is an example of a trefoil knot:

In the past few decades Knot Theory has been used to provide answers as to why certain viruses can "change" the DNA in a nucleus by simply twisting an arc in the DNA. The twist in a certain arc is enough to change the description of that DNA, which would than lead to the change in function of that DNA. This is proven by simply looking at the example of the flipped trefoil that isn't equivalent to that of the upright trefoil. The flip has caused a change in the Jones polynomial leading then to become different from one another.

Here are the three ways you can get more than one knot being equivalent to one another.
  1. Twist and untwist in either direction.
  2. Move one strand completely over another.
  3. Move a strand completely over or under a crossing.

Visual aid:

By trying to understand how viruses affect DNA using Knot Theory we can learn more about how to fix these infections. I haven't even touched on the basics of Knot theory, but I hope this has helped you understand a little more about how viruses can effect your DNA by using the Knot Theory.

To give a better representation of how complicated using the Knot Theory to understand DNA. Think of a basketball as the nucleus of a cell and take 200 km of fishing wire and put it inside the basketball and there you have it an exploded version of a nucleus.

Now try to find which arc the virus has infected to cause the DNA to change its functions.


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