Knots are an object that we come across in our everyday lives. From tying our shoes to tying a mask around the back of our head, tying and creating knots is something that we begin to become exposed to at a very early age. For our final project for the semester, we decided to go more in-depth into knots and see what information we can get out of our individual knots. To begin, I will show you my knot that I chose to use for this project, and then we will go into all of the interesting information we can pull out of it!
My Knot:
My knot is a knot with ten crossings, on the atlas you will find it with the name 10 38, which is the number of times you see the strand go beneath itself. Throughout this post, we will go in-depth into a couple of different things we can find out about this particular knot, this list being: finding a way to make it into the unknot, attempting a coloring of the knot, finding the writhe of the knot, and breaking this knot down into four other knots. If you do not know what any of this means, do not worry! I believe it is easier to see everything laid out at the beginning and then dissect each individual part throughout the whole post! To begin our adventure, lets look at finding a way to turn our knot into this so-called "Unknot".
Unknot:
The Unknot is a knot with the least number of crossings, or you can think of it as the "least knotted knot". While it may be trivial to hear, the Unknot is just a circle that you would make out of string. In order to find the Unknotting number for our knot, we want to attempt to find a series of moves that turns our knot into the Unknot. These series of moves that we will apply to the knot are called "Reidemeister moves", which is a way of increasing the number of crossings of a knot without changing the knot itself. There are three moves that we can use, which will be listed below with pictures. Basically, they show you if you can make a small loop in the knot, or if you can pass one part over another part (in the case of one or two passings).
Moves:
By using a series of these moves, we can find an upper bound for the unknotting number. The unknotting number may be smaller than what we find here, but it is good to know the higher end. Here, I found a series of moves that found an upper bound for the Uknotting number! I found this upper bound to be 4 moves. The actual unknotting number is 2, so try by yourself at the end of this to see if you can get there!
Coloring:
Coloring is sort of what it sounds like, where (in our case) we will see how we can color a knot. In order to do this, we start at some point on the knot, drawing the line with some color, and every time there is a crossing with three parts, we need to follow a couple of rules. For my knot, I will attempt to see if it is "Three Color-able". The rules for tricolor-ability are as follows :
(1.) All colors must be different
(2.) All colors must be the same
(3.) There must be at least two colors used throughout the whole knot.
In order to do this, I will show different drawings of the different attempts at showing this. If we find a way to make it three color-able, then we are done. If not, then we know that that, speific, projection of the knot is not that color-able. Below, you will find a drawing out of my attempt. I found that given my attempt, my projection is not three colorable for the moves I have done. In particular, the part circled in purple is where everything falls apart. The crossing is blue and two green, which does not work given our rules.
Writhe:
Next on our long list of things to do is to find the writhe of a knot. This is not super difficult, as we just need to inspect our projection of our knot. We start by picking any point, and then drawing little arrows in the direction that we wish to travel in. We keep doing it unless we arrive back at our starting points. Next, we look at the different crossings that we have. In particular, we will assign a value of "+1" or "-1" at each of the following crossings (positive crossing goes with "+1", negative crossing goes with "-1":
In order to find the writhe of the knot, we simply add up all of these $\pm 1$'s. Below you will find the projection of my knot with each of these values labeled. Here, I found the writhe to be my knot to be -6. Since we know that the value will switch with a different direction by a factor of -1, then the writhe could be seen as 6, but for this post we will continue using the -6 found. Below, I labeled each crossing and assigned the value found.
The Q Polynomial:
The last thing on our to-do list is to accomplish the first two steps of breaking our knot down into four other knots, using what is called the Q polynomial. Here, we take a series of cuts along the knot, tie different ends to each other. Here are the sort of rules that the Q polynomial follows, where we need to recall the writhe of our knot.
Q Polynomial Information:
Now, we can go on and show all of our work here to get to the Q Polynomial, where we focus on the couple of circles drawn on the knots.
Q Polynomial Steps:
Conclusion:
I hope you enjoyed seeing some of the interesting things we can do with knots! Overall, this whole topic was very new to me but ties (pun intended) in other branches of mathematics at its core!
My Knot:
My knot is a knot with ten crossings, on the atlas you will find it with the name 10 38, which is the number of times you see the strand go beneath itself. Throughout this post, we will go in-depth into a couple of different things we can find out about this particular knot, this list being: finding a way to make it into the unknot, attempting a coloring of the knot, finding the writhe of the knot, and breaking this knot down into four other knots. If you do not know what any of this means, do not worry! I believe it is easier to see everything laid out at the beginning and then dissect each individual part throughout the whole post! To begin our adventure, lets look at finding a way to turn our knot into this so-called "Unknot".
Unknot:
The Unknot is a knot with the least number of crossings, or you can think of it as the "least knotted knot". While it may be trivial to hear, the Unknot is just a circle that you would make out of string. In order to find the Unknotting number for our knot, we want to attempt to find a series of moves that turns our knot into the Unknot. These series of moves that we will apply to the knot are called "Reidemeister moves", which is a way of increasing the number of crossings of a knot without changing the knot itself. There are three moves that we can use, which will be listed below with pictures. Basically, they show you if you can make a small loop in the knot, or if you can pass one part over another part (in the case of one or two passings).
Moves:
By using a series of these moves, we can find an upper bound for the unknotting number. The unknotting number may be smaller than what we find here, but it is good to know the higher end. Here, I found a series of moves that found an upper bound for the Uknotting number! I found this upper bound to be 4 moves. The actual unknotting number is 2, so try by yourself at the end of this to see if you can get there!
Coloring:
Coloring is sort of what it sounds like, where (in our case) we will see how we can color a knot. In order to do this, we start at some point on the knot, drawing the line with some color, and every time there is a crossing with three parts, we need to follow a couple of rules. For my knot, I will attempt to see if it is "Three Color-able". The rules for tricolor-ability are as follows :
(1.) All colors must be different
(2.) All colors must be the same
(3.) There must be at least two colors used throughout the whole knot.
In order to do this, I will show different drawings of the different attempts at showing this. If we find a way to make it three color-able, then we are done. If not, then we know that that, speific, projection of the knot is not that color-able. Below, you will find a drawing out of my attempt. I found that given my attempt, my projection is not three colorable for the moves I have done. In particular, the part circled in purple is where everything falls apart. The crossing is blue and two green, which does not work given our rules.
Writhe:
Next on our long list of things to do is to find the writhe of a knot. This is not super difficult, as we just need to inspect our projection of our knot. We start by picking any point, and then drawing little arrows in the direction that we wish to travel in. We keep doing it unless we arrive back at our starting points. Next, we look at the different crossings that we have. In particular, we will assign a value of "+1" or "-1" at each of the following crossings (positive crossing goes with "+1", negative crossing goes with "-1":
In order to find the writhe of the knot, we simply add up all of these $\pm 1$'s. Below you will find the projection of my knot with each of these values labeled. Here, I found the writhe to be my knot to be -6. Since we know that the value will switch with a different direction by a factor of -1, then the writhe could be seen as 6, but for this post we will continue using the -6 found. Below, I labeled each crossing and assigned the value found.
The Q Polynomial:
The last thing on our to-do list is to accomplish the first two steps of breaking our knot down into four other knots, using what is called the Q polynomial. Here, we take a series of cuts along the knot, tie different ends to each other. Here are the sort of rules that the Q polynomial follows, where we need to recall the writhe of our knot.
Q Polynomial Information:
Now, we can go on and show all of our work here to get to the Q Polynomial, where we focus on the couple of circles drawn on the knots.
Q Polynomial Steps:
Conclusion:
I hope you enjoyed seeing some of the interesting things we can do with knots! Overall, this whole topic was very new to me but ties (pun intended) in other branches of mathematics at its core!
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