by Knot Nerds
What is knot theory?
Have you ever tied your shoes before? I mean, I hope you have! So, what in the world does tying your shoes have to do with math? Here comes knot theory! Knot theory is a sub-topic of topology that studies mathematical knots, links, and their permutations. Knots can be represented by polynomials and can thus be compared to one another.
- Invariable: not changing; when looking at transformations of knots, two configurations are invariable if they are the same knot before and after being manipulated
- Equivalent: same; equal
- Unknot: a circle; the simplest knot
- Prime Knot: a singular knot
- Composite Knot: two separate knots that have each been cut open and then connected to each other
So, what is a knot?
You might not even know it, but you use knot theory in your day to day lives, like making a friendship bracelet. Using knot theory, we can untwist or retwist DNA strands. Take a look at this parasite DNA strand below, this is a figure-eight knot. So, what even is a knot? A knot is a closed path in a 3-dimensional space that does not intersect itself. This can be easily represented through a knot diagram, a 2-dimensional representation of a knot’s configuration. There are many types of knots, here are some examples: trefoils, square knots, and figure-eight knots. Although there is a special case, it is called the unknot. This is a circle, but is not twisted or looped.
And a link?
Is knot theory only about knots? No! Knot theory also concerns links. A link is several knots which don’t intersect each other, forming a link that cannot be pulled apart. A common link is the Hopf Link, which is two unknots put together.
There are three important rules in manipulating a knot: the Reidemister (RHY-duh-my-ster) rules. (Kurt Werner Friedrich Reidemeister was a German topologist.) These specific moves are used to change the shape of knots while keeping them invariable, and are the only three that can conserve this property.
So how can we really use math to define knots? Be ready to welcome in bracket polynomial rules! There are three of these rules as you can see below.
However, note that the way the knot is arranged matters. For example, in the first bracket polynomial rule, the a and b parts of the equation show the separation of the knot. But here’s the catch; the equation only looks like this when the top knot is going from the bottom left to the bottom right. Otherwise, it looks like this – reversed.
We use bracket polynomials to determine whether knots are equivalent and to prove invariance in our Reidemeister Moves. This may all seem overwhelming and way too difficult but don’t be discouraged! This was the hardest part for our group to get past, but once we did, everything flowed smoothly.
Alright gang, this was your knot theory group of Girls Talk Math at WPI: Irene, Abby, Jennifer, Arushi, Ila, and Timory!