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Knot 9-31

Knot 9-31
A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into.
The Crossing Number
For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings!
The Unknotting Number
The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still want to perform the correct moves to untie the knot. From the Knot Atlas, knot 9-31 has an unknotting number of 2. Below are the steps/moves that I took to "untie" my knot.
From the image, both the moves are marked with a circle. By switching these crossings, the knot can easily be untwisted, resulting in the unknot. This is better understood with the 3D model. The model almost already wants to unknot itself, but these two crossings restrict that. This confirms that the minimum number of times the knot must be passed through itself to untie it is 2!
Colorability
Next, we will find the Colorability of my knot. For a simple definition, tricolorability, is the ability of a knot to be colored with three colors subject to certain rules. These rules include: Each strand in the knot is given 1 of 3 colors, at least two colors are used on the knot, and at each crossing there is either all 3 colors or all the colors are the same. Now, this is just for a knot that is three colorable. A knot can be N colorable where N is any number that is less than or equal to the crossing number. Below is my attempt at seeing if my knot is three colorable!
From my image, it is clear that my attempt to see if my knot was three-colorable failed. This does not mean that my knot is not entirely three-colorable, but it means that the route I chose or the decisions that I made were not the correct ones. I started in the top right corner of the knot. I ended up having two crossings that resulted in repetition of colors.
Writhe
In simple terms, the writhe is the total number of positive (+1) crossings summed with the total number of negative (-1). The way that we determine if a crossing is positive or negative is using a variation of the right-hand rule. This is shown here:
Now, I am going to calculate the writhe for my knot! The image below shows my process.
From my computation, I found the writhe of my knot to be: -1
Knot Polynomial
The knot polynomial is one of the hardest things to compute for a knot. The knot polynomial is a knot invariant in the form of a polynomial, in our case Q, in which the coefficients encode some of the properties of a given knot. Below is my computation for the first 2 steps of my knot.
This computation was very challenging, but I am confident with how both my image came out from a computation perspective and an artistic perspective. I had to hand draw all of my knots from this blog, so I am pretty satisfied with how they all came out. In conclusion, knots are very unique topics of mathematics and are far one of the most interesting things I have learned. I never knew the number of properties knots had and how intriguing they are to study.

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