Chapter 19: Knots

This chapter is about the different types of knots and its components. The general definition for a knot is a closed curve that is embedded in a three dimensional space and it is usually modeled with rope. The study of knots is a mix of different mathematics such as algebra, geometry, group theory, matrix theory, number theory, and more. One of the simplest knots is the trefoil knot which is when the ends of the rope are joined. The reason that it would be the simplest knot is because it can be diagrammed with a minimum of three crossings. The next simplest knot after the trefoil knot is the figure eight with four crossings. There are many ways to divide knots into different classes. One of the ways is alternating and non-alternating. Alternating knots can be diagrammed so that it is possible to go over and under the crossings. Another way of division is prime and composite knots. Prime knots cannot be manipulated to make two or more separated knots. The last basic binary division is called the invertible and the non-invertible knots. Invertible is being able to manipulate the rope so that the structure stays the same but the arrows of direction point the other way. A pretzel knot, on the other hand, has no crossings which proved that all pretzel knots are noninvertible if the crossing numbers are distinct odd integers with absolute values that are greater than one.

In L. Kauffman’s “Knots and Physics” book, she was using a lot of examples to show the different types of knots that there were. She wasn’t using a small variety and I would have wished that Gardner would have included more examples such as a clove hitch and a bowline. In Kauffman’s book she also explained the physics behind the knots not just the mathematics, which is what I wished that Gardner would have included along with all of the different types of mathematics. Friction and tension seem to be some of the main components in making the knot form. While it would have been cool to see all of the different types of knots in the book, I am glad that Gardner didn’t put in a step-by-step for every single structure. Some of the structures are basic and other make you want to think but as I was reading Kauffman’s book I wanted to try to solve the knots myself but it was a bit hard to do so since she included the instructions in both the writing and the images.

In the book “Introduction to Knot Theory” by R.H. Crowell and R.H. Fox, there is a lot of explanation for the mathematics behind the knots. Gardner briefly talks about the mathematics but he does not go into greater detail about how mathematics can be used to form knots or to work with knots in general. As a reader, I would wish for more information regarding how exactly mathematics is involved. I am glad that Gardner didn’t include anything about wild knots because as I was reading about wild knots in Crowell and Fox’s book, it made me very confused about the wild knots and what their purpose is to the talk of knots.

In K. Murasugi’s “Knot Theory and Its Applications,” the author goes in depth on what an actual knot is and the trefoil knot. I wished that Gardner would have explained more about a knot itself rather than give a brief explanation and jump onto different divisions. I would have also wanted for Gardner to talk more about the trefoil knot since it is the simplest knot. The short summary that Gardner provided made me a little confused about the trefoil knot; however, Murasugi’s explanation was a bit longer and more in depth and led me to understand more about the trefoil knot. One more thing that I wished that Gardner had included was a better explanation of mirrored “images” with knots that involve the different mathematics. I’m glad that Gardner omitted parts about the elementary knot moves because that part of Murasugi’s book confused me. There is a lot of mathematics being thrown into that one concept of the elementary knot moves that was hard to follow.

This chapter, overall, was very interesting to read and after reading more into the three author’s works besides Gardner’s, it had made me want to further look into the different types of knots that exist and how some of them are formed. I wanted to try forming them on my own as well. It also had made me want to read more into the more in-depth mathematics and the different formulas used to make these knots.