The knot I chose: "8_18".
It is called 8_18 because it has 8 crossings and is the 18th knot on the knot table with 8 crossings.
Picture of my knot (from http://katlas.math.toronto.edu/wiki/8_18)
Crossing number: 8; the smallest number of crossings of any diagram with this knot is eight.
Sequence of crossing switches that gives the unknot (upper bound on the unknotting number):
An unknotting number is the number of changes you can make at intersections by changing which line is over or under at that intersection which at the end of all the changes gives you the unknot, or a simple loop with no knots in it. Basically you can change which line is over and which line is under at each intersection without sacrificing the integrity of the knot by cutting it in any way and the number of crossings you have to alter that give you the unknot becomes the upper bound of the unknotting number because there could be a simpler way to do it that requires less steps. A more formal definition is:
"We can immediately give simple bounds on this. For any n–crossing projection of a knot, the length of any maximum knotted switching sequence will be less than or equal to n. The unknotting number of a knot is the minimum number of crossings that need to be switched to produce the unknot, considered over all projections. So the length of a maximum knotted switching sequence of any projection of a knot must be at least one less than the knot’s unknotting number. Also, if a projection of a knot with n crossings has a knotted switching sequence of length n−1, it has a full sequence. After n−1 switches, exactly one crossing will be unswitched, and as the original knot must have been knotted, its mirror will be as well, so that last crossing may also be switched, creating a full sequence." (http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Milnor.pdf)
My unknotting number: 3 (shown below with notes on notability)
The top loop goes through the bottom loop, untwists, goes over the top where it was and engulfs and untwists the remaining loops giving you just a circle or the unknot.
The actual unknotting number: 2 (from http://katlas.math.toronto.edu/wiki/8_18)
I attempt a coloring. The rules of coloring a knot are 1. At least two colors must be used, and 2. At each crossing, the three incident strands are either all the same color or all different colors.
Due to all colors being the same or different at each crossing this colorability shows that the 8_18 knot is tricolorable. In fact, it has 4 fundamnetally distinct tricolorabilities which is pretty cool and I'm glad I found one of them.
Computing the writhe of my knot. A writhe is simply the sum of all its crossings. How do you assign numbers to crossings in a knot? Well, for every over-under crossing with the over pointing to the top right and the bottom line being pointed to the top left it is assigned a value of positive one. For every crossing where the top line points to the top left and the bottom line points to the top right is is a minus one. The writhe is the sum of all these numbers assigned to each crossing. See below image from (https://en.wikipedia.org/wiki/Writhe) for more details.
When I look at knot 8_18 and compute the +1's and -1's, I get the wraithe to be 0 with 4 negative one crossings and 4 positive one crossings.
The first two steps of the polynomial Q.
The Alxander Polynomial for this knot is \[-t^3+5t^2-10t+13-10t^-1+5t^-2-t^-3\].
It is called 8_18 because it has 8 crossings and is the 18th knot on the knot table with 8 crossings.
Picture of my knot (from http://katlas.math.toronto.edu/wiki/8_18)
Crossing number: 8; the smallest number of crossings of any diagram with this knot is eight.
Sequence of crossing switches that gives the unknot (upper bound on the unknotting number):
An unknotting number is the number of changes you can make at intersections by changing which line is over or under at that intersection which at the end of all the changes gives you the unknot, or a simple loop with no knots in it. Basically you can change which line is over and which line is under at each intersection without sacrificing the integrity of the knot by cutting it in any way and the number of crossings you have to alter that give you the unknot becomes the upper bound of the unknotting number because there could be a simpler way to do it that requires less steps. A more formal definition is:
"We can immediately give simple bounds on this. For any n–crossing projection of a knot, the length of any maximum knotted switching sequence will be less than or equal to n. The unknotting number of a knot is the minimum number of crossings that need to be switched to produce the unknot, considered over all projections. So the length of a maximum knotted switching sequence of any projection of a knot must be at least one less than the knot’s unknotting number. Also, if a projection of a knot with n crossings has a knotted switching sequence of length n−1, it has a full sequence. After n−1 switches, exactly one crossing will be unswitched, and as the original knot must have been knotted, its mirror will be as well, so that last crossing may also be switched, creating a full sequence." (http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Milnor.pdf)
My unknotting number: 3 (shown below with notes on notability)
I attempt a coloring. The rules of coloring a knot are 1. At least two colors must be used, and 2. At each crossing, the three incident strands are either all the same color or all different colors.
Computing the writhe of my knot. A writhe is simply the sum of all its crossings. How do you assign numbers to crossings in a knot? Well, for every over-under crossing with the over pointing to the top right and the bottom line being pointed to the top left it is assigned a value of positive one. For every crossing where the top line points to the top left and the bottom line points to the top right is is a minus one. The writhe is the sum of all these numbers assigned to each crossing. See below image from (https://en.wikipedia.org/wiki/Writhe) for more details.
The first two steps of the polynomial Q.
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