Knot Your Average Blog Post

There exists an entire branch of mathematics dedicated to the concept of knots. This method of tying ropes taught to every boy scout in America poses many interesting mathematical questions that many desire to answer. In the math world, a knot is defined as a closed curve. This can be as simple as an ordinary rubber band, or infintely complex with many crossings. To investigate all of the fun operations and characterstics of knots, this blog will focus on what speficic knot that will be referred to as 9_23. 9_23 looks like the following:

Credit to katlas.org for the picture. This knot contains 9 crossings, and it is importatnt to note that for each crossing there is great significance for which strand is the over and which strand is the under.

Reidemeister Moves and Simplifying the Knot

One set of useful operations that can be performed on a knot is the Reidermeister moves. These moves allow for a knot's strands to be moved around without changing any of the defining characteristics of the knot. There are 3 of these moves and are simply called Reidermeister moves 1,2 and 3 respectively. The moves are best explained by the following chart:

This chart depicts the three moves that can be performed without changing the properties of the knot. These moves are invertible, meaning they work both ways. These moves are very helpful for simplifying a knot presentation that is twisted. This will be important for analyzing the unknotting number later.

Colorability

An interesting property of knots is their n-colorability. Without getting super technical, 2-colorability states that if one were to color each strand of a knot with 2 distinct colors, then the knot is 2-colorable if at each crossing, the strands are either all the same color, or are n distinct colors. This is easiest to see when n is 3. If the knot can meet the criteria with 3 unqiue colors, then the knot is 3 colorable. The following image shows a coloring of the knot:

Looking at the image, some of the crossings follow the requirements listed above. However, there are multiple crossings in which there are 3 strands with 2 distinct colors. This indicates that the knot is not 3 colorable. The knot may be colorable for other n, but more than 3 colors are needed.

Unknotting Numbers

For any knot, there exists a minimum number of crossing switches that can be performed such that the knot is transformed into the unknot. My claim is that the the upper bound for this number is 2. Consider the image shown here:

If the two circled crossings are changed so that the under and over strands are swapped, then the result is the unknot. This can be shown using Reidermeister moves 1 and 2, with some R2 moves skipped over for brevity. The following image shows the process done quickly over multiple steps:

Wish I had a tablet to do this on, but paint has served its purpose! The fact that switching 2 crossings resulted in something that was equivalent to the unknot, 2 is the maximum value of the unknotting number for 9_23.

The Writhe

Moving on to another propery of knots: the writhe. In my opinion this one sounds the coolest. Start by picking a direction to move on the knot, and move keep track of that direction all the way through. If an under strand is moving to the right, the value assigned to that crossing is -1. If the understrand is moving towards the left, this value is positive 1. For this to work however, the crossing needs to be rotated such that both orientations are moving upwards. The writhe is the sum of the values at all crossings. The following image shows an oriented version of 9_23:

Looking at the knot, there are 8 instances where the crossing has a value of -1, and 1 where there is a value of 1. This means the writhe of the knot is -7.

Knot Polynomial

Last but not least: the knot polynomial. The knot polynomial is a way of writing the knot as the sum of two other knots. These addend knots are created by changing the crossings of the sum knot. The following image shows 2 steps of this process:

The goal is to reduce so that the only thing that remains within the paranthesis are unknots.