While knots are something that I have used in everyday life since I learned how to tie my shoes, I had never thought of them as a mathematical phenomenon until now. Topologists describe knots as “closed curves embedded in three-dimensional space”. Chapter 19 starts off by talking about the simplest of knots, the overhand knot as shown below
Imagine this, you take one end of the string and loop it through the knot and pull both ends. What will happen? Will the knot stay or will it dissolve? The knot actually dissolves, to make another knot you would need to loop it through three times instead of two.
Knots are divisible into many different categories there are alternating and non-alternating, prime and composite, amphicheiral and non-amphiceiral, and invertible and non-invertible. Alternating knots can be diagrammed so that if you follow the curve in either direction you alternate going over and under. Alternating knots possess many properties that are not possessed by non-alternating knots. Prime knots cannot be manipulated to make two or more separate knots. Amphicheiral knots are ones that can be manipulated into making their mirror image. To explain invertible knots you have to use a bit of imagination. Imagine and arrow is pointing in the direction that follows the curve of the rope. If it is possible to manipulate the rope so that the structure of the knot remains the same but the arrows change direction then the knot is invertible. It wasn’t proven until 1963 that non-invertible knots existed. It wasn’t proven until Hale F. Trotter wrote a paper called “Non-invertible Knots Exist”. Trotter described a family of infinite pretzel knots that will not invert.
Knots can also pose as puzzles, one of these is a string with two overhand knots, as shown above, and a ring is placed on one of the knots. Without untying the knots, you are to move the ring from one overhand knot to the other. This can only be done by manipulating the string and creating new knots and moving the ring by doing so.
Another type of knot that can be used as a “party trick” is a knot invented by Roger Penrose, a British mathematician and physicist. This knot, as shown above, is based what crocheting, sewing and embroidery call a chain stitch. If you have someone pinch any one knot on the chain stitch, then pull both ends all of the knots will dissolve except there will be a tight not where ever said person was pinching.
The best was to understand the Jones polynomial, a knot polynomial discovered by Vaughn Jones in 1984, is through statistical mechanics and quantum theory. Sir Michael Atiyah was the first to see the connections between these, then Edward Witten did the work in developing these connections. Since these discoveries have been made knot theory has applications to superstrings, a theory that explains basic particles by treating them as tiny loops, and to quantum field theory. In today’s society physicists and topologists work closely together. Discoveries in either field lead to new discoveries in the other.
This chapter KNOT only focuses on the games and puzzle aspects of knots, but also the scientific side as well. It was very interesting sitting down and actually creating the knots that were described in this chapter and taking the time to understand how and why each one works the way it does. Never would I have imagined that something as simple as tying your shoes could be used to understand quantum theory.
Upon reading this blog post, I was surprised by several things, the first being the fact that knots can be categorized as “prime” and “composite”. Here I was, thinking that those properties could only be used to describe numbers. Another surprising concept to me is the “amphicheiral knot” which can be manipulated in a way which mirrors itself. I was always lead to believe that knots were just tangles of rope or string – sure, I knew there was some basic method to the madness – but I had never considered there was a science to them. Nowadays, knot theory is one of the most impelling fields in the realm of mathematics. Admittedly, it was a bit confusing to me to learn how knots can be related to polynomials, but it seems I’m not alone. Even though there have been significant advances in simplifying these polynomials, there is yet to discover a way to distinguish all knots using an algebraic sequence. I was intrigued by the story of Roger Penrose, and thoroughly impressed by his invention of the mysterious appearance of a knot, all while only being in grade school. Lastly, it’s amazing how knots have aided in the broadening of our understanding of DNA molecules – how something physical and recognizable in form can play a part in something so complex and, in a way, so abstract in our everyday lives.
ReplyDeleteWhen I first read the title of this chapter I did not really see how knots were related what so ever to mathematics. Everybody knows what a knot is from either to tie your shoes or make fishing knots, but there was never anything math related to it which was why this chapter was intriguing to read. Maybe if you were in boy scouts you would categorize a knot for maybe when to use it or why its used but this chapter focuses on the math side of categorizing knots. I never knew the different names for the knots that they give like " Alternating and Amphicheiral." Also how you are able to change a knot to a puzzle was neat. This would make more since to me how knots would be related to mathematics if I hadn't read about the other things before hand. One thing that I thought about that raised a question to me was would it be possible for a knot not to end. What I mean by this is we all know of double knotting shoe strings so why not just keep making knots would that change anything. Also if you kept interlacing the string through and through how would that be categorized as? This chapter was a fun read just to see the different kinds of knots and how they were related to the realm of mathematics.
ReplyDeleteI found this chapter to be a bit weird. Just like Alex pointed out, I never thought about knots in a mathematical way. I also never knew you could become a “knot theorists.” Compared to other chapters I have read so far, this chapter was easier to understand because it used simple words. The only complicated part to this chapter was not knowing what some of the specific type of knots were. For example I still have no idea what a granny knot is. The part which talked about explaining knots with polynomial equations interested me the most out of this chapter. I like that math is all the same language but has many different ways to express it. For instance, an unknotted knot can have the polynomial equation of x2-x+1. There are two separate ways to write a knot with no crossings, and they both are acceptable. This chapter made me think about how we classify math. I personally never considered knots to have mathematical merit, but mathematicians are still advancing on ways to prove and make more polynomials for knots. It is radical that knot theory is a thing,
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