Introduction
Knots are something we are already familiar with. The difference in the knot we are familiar with, and a mathematical knot is that there are no loose ends in a mathematical knot. These ends would actually come together. A knot is a closed curve in a 3D space that doesn't intersect itself. A knot shaped like a circle would also be considered a knot, it's actually the unknot. The unknot is the simplest knot after that we can have knots with 3 crossings and more.
Crossing Number
The knot that I have chosen has 9 crossings and is knot 9-35 in the knot atlas. In the picture below we can see the 2-dimensional representation of a 3-dimensional knot. This is done by the over and under crossings that we can see.
Colorability
We will be finding if the knot above is colorable. This is where we will be using 3 colors and will be assigning colors to the strands. The knot can be broken up into strands at each under crossing. This means at each under crossing we can switch the color we are using. In addition to this we have a few rules that we need to follow. Each strand that is created must be assigned a color. The second rule is that at each crossing all the strands either have to be the same color or all different colors. The last rule is that there should at least be 2 colors used in the knot. This is so that the entire knot doesn't become a single color. In the picture below we can see an attempt that was made to show colorability. Closely looking at each crossing we are able to see that all three colors are present, so we can say that this knot is 3-colorable.
Unknotting Number
The unknotting number is the next thing we will find for this knot. This is the minimum number of crossings that we would switch to make our knot turn into the unknot. In the picture below you can see how I have switched 3 crossings (highlighted in green) then used the Reidemeister moves to get the unknot. Reidemeister has 3 moves that allow us to move our parts of our knot because they are proved not to change the actual knot. In the image below we can see that the red arrows represent Reidemeister 2 moves, and the highlighted purple represent Reidemeister 1 moves. This doesn't mean that 3 is the unknotting number. This actually is an upper bound for the unknotting number. The upper bound means that this is the lowest unknotting number that I have found so far. However, there could be a possible way to unknot this by switching just 2 or 1 crossings.
Writhe
We have so far assigned colors to our knot. Now we will be assigning numbers to each crossing (+1, -1). In the picture below we can see which type of crossing gets assigned +1 and which crossing gets -1. To get the direction on the crossings we would assign an orientation to the knot, which stays the same for the entire knot. Then to calculate the writhe we would add these numbers up.
In the picture below we can see how I have assigned the orientation by the red arrows on the knot. With the direction of each crossing, I have also assigned a number to each crossing. All of my crossings got a +1 making the writhe to be 9. This know has all the same type of crossings, but this is not the case for all knots.
Knot Polynomial
The last thing we will be looking at is the knot polynomial. This is one a very long thing to calculate for a knot so we will look at how to partly calculate it. In this we would look at a crossing and depending on the over and under crossing we would split and connect it at the top and bottom. To explain this better lets look at the image below showing our knot.
In the top row we see that the green highlighted crossing shows where we split the crossing. This gave us two different knots which were then further split and can be seen on the second row. This split caused us to have 4 outcomes. This would typically be done until there are no crossings left, which you can imagine would take a very long time. The third row shows what our equation is in a simplified form. We can calculate the polynomial Q for up to these steps. For this we would also need the writhe that we calculated above. In the image below we can see the Q-polynomial.
Conclusion
We have done many things with our knot that were all interesting and different to do. Each thing we did is considered as an invariant of our knot.
Knots are something we are already familiar with. The difference in the knot we are familiar with, and a mathematical knot is that there are no loose ends in a mathematical knot. These ends would actually come together. A knot is a closed curve in a 3D space that doesn't intersect itself. A knot shaped like a circle would also be considered a knot, it's actually the unknot. The unknot is the simplest knot after that we can have knots with 3 crossings and more.
Crossing Number
The knot that I have chosen has 9 crossings and is knot 9-35 in the knot atlas. In the picture below we can see the 2-dimensional representation of a 3-dimensional knot. This is done by the over and under crossings that we can see.
Colorability
We will be finding if the knot above is colorable. This is where we will be using 3 colors and will be assigning colors to the strands. The knot can be broken up into strands at each under crossing. This means at each under crossing we can switch the color we are using. In addition to this we have a few rules that we need to follow. Each strand that is created must be assigned a color. The second rule is that at each crossing all the strands either have to be the same color or all different colors. The last rule is that there should at least be 2 colors used in the knot. This is so that the entire knot doesn't become a single color. In the picture below we can see an attempt that was made to show colorability. Closely looking at each crossing we are able to see that all three colors are present, so we can say that this knot is 3-colorable.
Unknotting Number
The unknotting number is the next thing we will find for this knot. This is the minimum number of crossings that we would switch to make our knot turn into the unknot. In the picture below you can see how I have switched 3 crossings (highlighted in green) then used the Reidemeister moves to get the unknot. Reidemeister has 3 moves that allow us to move our parts of our knot because they are proved not to change the actual knot. In the image below we can see that the red arrows represent Reidemeister 2 moves, and the highlighted purple represent Reidemeister 1 moves. This doesn't mean that 3 is the unknotting number. This actually is an upper bound for the unknotting number. The upper bound means that this is the lowest unknotting number that I have found so far. However, there could be a possible way to unknot this by switching just 2 or 1 crossings.
Writhe
We have so far assigned colors to our knot. Now we will be assigning numbers to each crossing (+1, -1). In the picture below we can see which type of crossing gets assigned +1 and which crossing gets -1. To get the direction on the crossings we would assign an orientation to the knot, which stays the same for the entire knot. Then to calculate the writhe we would add these numbers up.
The last thing we will be looking at is the knot polynomial. This is one a very long thing to calculate for a knot so we will look at how to partly calculate it. In this we would look at a crossing and depending on the over and under crossing we would split and connect it at the top and bottom. To explain this better lets look at the image below showing our knot.
We have done many things with our knot that were all interesting and different to do. Each thing we did is considered as an invariant of our knot.
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