Knots are a very interesting topic and a field that has not quite been fully discovered, so mathematicians are still discovering new ideas and invariances about knots even today. While it may seem like knots are a simple skill you learn at camp, they actually have a lot of mathematical properties and in this blog post we are going to look at just a few.
By mathematical definition, a knot is a closed curve in three dimensional space that does not intersect itself. Since we are working with three dimensional space and you are reading this on a two dimensional screen, we need a way to look at knots in two dimensions and that is where knot projections come in. A knot projection is simply a picture of a knot in two dimensions and where a knot crosses itself in the projection is simply a crossing of that projection. The number of crossings of a knot is the smallest number of crossings among all projections of a knot. Since a knot is not necessarily solid, one knot can have many projections when maneuvered properly. I know I just explained a lot of new information quickly, but once we see some pictures and calculate some invariances, it may start to make a little more sense. For my knot example, I chose to use the septoil knot or knot 7_1. An image of the knot projection from wikipedia can be found below.
The seven in the name of the knot stands for the crossing number. The septoil knot has seven crossing and you can count them using the knot projection above.
Next, we are going to look for a sequence of crossing switches that gives the unknot, which is essentially a knot with zero crossings. To do this we will count how many crossings we change and we can make Reidemeister moves (which don't change the knot) in order to end up with the unknot. Below is a visual representation of the crossings I changed and the Reidemeister moves I made.
To explain in words, the pink represents each crossing change that I made, a total of four, and the purple represents the Reidemeister moves I made along the way. I used a combination of Reidemeister one, two, and three moves in order to unknot my knot. Now we know that the unknotting number for the septoil knot is at most four, although it could be less.
Next, we will beattempting a coloring of the septoil knot. If a knot is n-colorable we can confidently say it is not the unknot. Today we will test our knot for 3-colerability. Our knot can be considered 3-colerable if each strand in the knot is given 1 of 3 colors, at each crossing we either see all 3 colors or all the colors are the same, and at least 2 colors are used on the knot. Let's try and see.
Above we see that this knot is in fact not 3-colerable. In the first image, I started with a red strand, so for my next strand I chose a different color (blue), and then I was forced to choose yellow as my next, and then red and so forth. When I got to the end, I ended up with 2 crossings that didn't fit the requirements. In the second image I tried again thinking that I could have a crossing with all three strands the same color, but I got stuck in the same loop and ended without the 3 colors that I needed. Seeing as though these two results would happen every time, I have decided that this knot is not 3-colerable.
Now, we will compute the writhe of the knot, and this is a numerical value associated with the total crossing on the knot. Every crossing is assigned either a +1 or a -1 and the writhe is the sum of them all. Below is an image where the writhe of -7 was calculated. The black arrows represent the orientation of the knot and the red -1s are the numbers assigned to each crossing, totaling a writhe of -7.
One last invariance we want to calculate is the first two steps of the polynomial Q. This polynomial is difficult to compute but a helpful invariance once we have it. Below are the first two steps of how to compute the polynomial Q.
As you can see some of our steps resulted in the unknot which has a P(unknot)= 1, so our polynomial didn't have to get as long! Now I kind of want to go get a rope and tie some knots myself!
By mathematical definition, a knot is a closed curve in three dimensional space that does not intersect itself. Since we are working with three dimensional space and you are reading this on a two dimensional screen, we need a way to look at knots in two dimensions and that is where knot projections come in. A knot projection is simply a picture of a knot in two dimensions and where a knot crosses itself in the projection is simply a crossing of that projection. The number of crossings of a knot is the smallest number of crossings among all projections of a knot. Since a knot is not necessarily solid, one knot can have many projections when maneuvered properly. I know I just explained a lot of new information quickly, but once we see some pictures and calculate some invariances, it may start to make a little more sense. For my knot example, I chose to use the septoil knot or knot 7_1. An image of the knot projection from wikipedia can be found below.
Next, we are going to look for a sequence of crossing switches that gives the unknot, which is essentially a knot with zero crossings. To do this we will count how many crossings we change and we can make Reidemeister moves (which don't change the knot) in order to end up with the unknot. Below is a visual representation of the crossings I changed and the Reidemeister moves I made.
Next, we will beattempting a coloring of the septoil knot. If a knot is n-colorable we can confidently say it is not the unknot. Today we will test our knot for 3-colerability. Our knot can be considered 3-colerable if each strand in the knot is given 1 of 3 colors, at each crossing we either see all 3 colors or all the colors are the same, and at least 2 colors are used on the knot. Let's try and see.
Comments
Post a Comment