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*Insert Knot-Based Pun Here*

 A knot is a closed loop which is a transformation of the circle into 3D space! Unlike a physical knot, a mathematical knot has no ends, and thus isn't tied and can't be untied. The fundamental knot is the circle, which is the unknot. 


For a knot more complex than the circle to be formed, the threads of the knot must be crossed a finite number of times. These can be undone through the Reidemeister moves, which move threads past each other. Certain operations do not change the not, and these create invariants of a knot. 


It cannot be trivially proven that a given knot is not actually a clever twist of the circle. Coloring the knot could reveal if a not is not the unknot. However, the failure of coloring a certain number of times does not always mean a knot is the unknot, as different starting points for the coloring may affect the outcome as well as using more colors. The simplest coloring is three-coloring. For this process, three colors are used and the threads of the knots are colored such that at least three colors are used, and that at every crossing either all three colors are present or only one color is present. 

Below I show an attempt at a three coloring of my knot, as well as my uncolored knot. I chose 10-47 which has a crossing number of 10 and is the 47th variation of the 10 crossing knot. 


Uncolored Knot


As circled, my 3-coloring failed! However. this does not mean it is the unknot. This failed because there was a crossing which had 2 colors present. The only options are at a crossing that all 3 colors must be present, or only one can be. 


My knot has a writhe of 4 given the orientation I define below. "Above" crossings are +1 while "negative" crossings have a value of -1, and the sum of the crossings gives the total writhe.







The writhe will change with the number of crossings, and how they are presented in the projection (2D representation, such as the images above). Using the python library, I decided to plot the writhes of the first 150 knots with 10 crossings, and the plot is shown below.


Interestingly, it appears that the writhe is somewhat cyclic! I assigned a value of -1 for knots pyknotid didn't have value for. The maximum writhe was 10 and the minimum was 0. It must attempt to orient the knots such that they have a maximal number of positive crossings. 


Additionally, every knot has an attribute known as its unknotting number. The unknotting number asserts the minimum number of times the knot must be passed through itself to become the unknot. This is not trivial to see, and it is often more intuitive to present the upper bound of this number, since only an especially insightful mathematician could see the moves such that they will produce the minimum number. According to the knot atlas entry for 10-47, it has an unknotting number of 2! I have no Earthly idea of what processes they went through to obtain that, but I applaud their dedication. Below I present what I believe is an upper bound of this process as well as show the steps I took in order to obtain the unknot. 

I suggest that the upper bound of my unknotting number is 11, since it took 11 moves in order to arrive at the unknot. Surely, a more clever mathematician could reduce this much further since the minimal number is 2. Below I show my process










Finally, every knot has an associated polynomial! Below I give an expansion of the polynomial for my knot. 













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