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Knot 10-11

The knot I chose for this project was knot 10-11. That is the eleventh knot that has ten crossings. I picked it for no reason whatsoever. I 3D printed the knot as you can see here.
A knot's crossing number is the smallest number of crossings possible in a 2D representation of the knot. This knot's crossing number is 10. A knot's unknotting number is the smallest number of crossings flipped that will give you the unknot. I found an upper bound on the unknotting number. If you switch the crossings circled below, then the knot will become the unknot and therefore has an unknotting number of at most 3.
The colorability of a knot can help determine whether or not two knots are actually the same. When the number of crossings in a knot is high, it is actually pretty hard to know if two knots are secretly the same and are only represented differently through Reidemeister moves. Colorability can help confirm that two knots are indeed different. Colorability is the ability to have each knot segment designated with a number and to have each crossing have the quality that \((i+j-2k)\mod(n) = 0\) where \(i\) and \(j\) are the values of the segments being separated and \(k\) is the value of the segment on top of the other two. The value of \(n\) depends on what n-colorability is being determined. Colorability helps determine if two knots are the same because a knot's colorability is consistent no matter the representation it has in 2D. So if two knots have the same n-colorabilities, it is possible that they are the same knot, but if they have different n-colorabilities, they are definitely not the same knot. I tested to see if my knot was 3-colorable. When it comes to 3-colorability, you do not have to assign the segments numbers, as long as each crossing has three colors involved or only one color involved, and more than one color is used, the knot is 3-colorable.
The picture above displays my attempt at making my knot 3-colorable. The attempt was going well until the very end. I started by making the large segment on top red. Then, I continued coloring the segments making sure each crossing had three colors until the last white segment. It cannot be red, green, or blue because any one of those colors would cause one of the crossings to have only two colors present. The characteristic of not being 3-colorable means knot 10-11 is not any knot that is 3-colorable, like the trefoil knot.
  A knot is able to have an overall direction. Imagine arrows all pointing the same way as you continue along the knot. The writhe of a knot is the sum of the positive or negative values of its crossings. A crossing is either positive or negative depending on the direction the above and below parts of the crossings have. Assigning each crossing of knot 10-11 a value as pictured below and adding them up gives a writhe of -2.
Every knot has a Q polynomial assigned to it. To find the polynomial, you have to break down the knot by changing each crossing so that the two segments connect and do not cross anymore. There are two ways to connect segments, so you have to keep track of each knot made while breaking down your knot. A knot with 10 crossings has over 1000 knots when broken down, so I only did two steps of finding the knot polynomial. You can see the steps below.
Even though I chose my knot for absolutely no reason, I learned a lot about knot 10-11. Any time someone brings up the mathematically defined knot, I will think of knot 10-11 and its inability to be 3-colorable and the second step of the process to find its Q polynomial.

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