Project

Knot Theory Project

Megan Stine



More Applets and Resources

Title Source Description
Knot Plot Knot Plot Program to design 3-D knots. Also has many example pages of knots already made. Great open exploratory tool.
Knot Zoo Knot Plot My favorite page on Knot Plot that has many examples of knots to chose from. Click on one to be able to rotate it. Includes knots involving one, two, or three rings.
Sort and Search Knot Info A page where you select the attributes and see all resulting knots with that invariant. Can sort by any property. Also can search for a knot by name. No visual representations.

Tricolorability Applet



Knot Theory

From large knots that hold down a floating barge to small ones that piece together microscopic material, knots are integral in society. They are hidden everywhere, yet many hold important jobs. Some we build ourselves, such as to climb a cliff or pitch a tent, while many knots appear naturally, even down to the blueprint of our own bodies (Zhang). Scouts learn of many knots with many names and uses. But how can we match any knot to its name? Can we even name them all? How do we know if one mess of string is truly the same or different from another?

History & Settup

Using the powers of mathematics is an intelligent way to approach these questions, which is what some great scholars, such as Frederick Gauss, did (Richeson). However, he did not start going into as great a depth as Guthrie Tait, who is credited for establishing the subject of Knot Theory. Looking at smoke rings floating in the air, with his fellow scientist, Sir William Thomson, they pondered whether the fundamental structure of the universe, atoms, were made of swirling vortexes (Richeson). Tait then took it upon himself to attempt to categorize and document all of the various twisted rings he could come up with (Richeson). Notice how knots took the form of closed rings in this new study, as mathematicians needed to be able to open up each knot and compare them. If there were loose ends, the knot would fall apart and fundamentally change its shape. Though objects with multiple rings or which exist in higher dimensions are studied in Knot Theory, the focus of this paper will be on simpler objects using the following definition: “Mathematical knots are closed loops in three-dimensional space” (Prevos).

To start with the basics, Tait began to relate knots to math by attempting to reduce them to their simplest form and create a table of knots (Richeson). Specifically, a knot is reduced when it crosses itself the fewest number of times it can. By definition, the crossing number of knot K is the “minimum number of crossings in any diagram of K” (Gordon). In addition, knots can be prime or composite, depending whether it is made of other knots. For example, “Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the knot sum…” (Wikipedia). Just like prime factors form composite numbers in number theory, primeness is much more efficient to study because the rest of the system is composed of prime objects. As time passed, Tait managed to find all of the prime knots with crossing numbers up to seven by himself (Richeson). With the help of two more colleagues, Rev. Thomas Kirkman and Charles Little, they figured out all prime knots with 11 crossings with only one duplicated error (Richeson). Keep in mind that the more crossings there are, the more possible knots and the more difficult it is to tell them apart visually.

Mathematics

Because of this obstacle, scholars needed a better way of proving whether two knots were foundationally equivalent besides making educated guesses and then trying to check their work. Starting this process off, Kert Reidemeister proved that to change one knot to any other equivalent knot, there are only a few types of moves needed to do so. These moves (see Figure 1 below) were dubbed Reidemeister moves and were founded on the mathematics behind combinatorics and isomorphisms (Kauffman). However, as there are too many combinations of ways to perform these four moves anywhere on a knot, this is not a sufficient way to prove two knots are not equivalent. From this point on, mathematicians through the years have developed many proofs, theorems, conjectures, and new processes for understanding various attributes of knots. Though they started out highly complicated, unusable, and even impossible to calculate in a lifetime, over the years proofs for Knot Theory became more manageable.

One connecting motive of these proofs is to find a “property, quantity or algebraic entity that is associated with the knot and can often be computed simply”, such as the crossing number for example (Richeson). If such a property is always shared by equivalent knots, this is called an invariant (Zhang). All equivalent knots have matching invariants, though trouble is that they can still share characteristics with other knots which are not equivalent. An example of this concept, outside of Knot Theory, is categorizing polygons by the number of sides they have (an invariant). All squares have four sides, but other four-sided polygons exist too, such as rectangles, trapezoids, and parallelograms. Because of this, one cannot show that two shapes are both squares just because they both have four sides. However, one can easily see that a triangle and square are not the same shape because they have a different number of sides. Therefore, it is still useful to label polygons with this attribute. Going back to knots, a shared quality does not prove that two knots are equivalent. Only if a property is different between two knots can it be concluded that those knots must not be equivalent (Zhang).

The issue with the first few invariants discovered, such as the crossing number, is that they relied on a knot being in its reduced form. This projection is quite hard to find and prove for many cases. How could one know that a knot is the least tangled form that it could be? In addition, invariants are most useful in larger amounts (Zhang). If there are many, then there are just as many attributes to label per knot, which can be enough to narrow down its possible equivalences on Tait’s table of knots. Looking at a polygon's number of sides, angle measurements, and side lengths all at once is more useful for proving similarity.

Arguably, the most ground-breaking invariant began with the discovery of tricolorability, which does not rely on such conditions (Zhang). Start with a 2-D drawing of a knot with gaps to signify which curve is crossing over another. All one needs to do is color each of the sections of the knot using at least two colors (Zhang). A knot is considered to be tricolorable if it can be colored in such a way that at each crossing the three parts are either all the same or all different colors. (see Figure 2).

This method is usually easy to perform and prove whether a knot has or lacks this attribute. Though this only splits all knots into two distinct categories (tricolorable or not), this idea was extended further to p-colorability, where p is some prime number and there is a new, generalized set of rules (Zhang). From this, many more invariants developed, which further narrowed the effort needed to prove two knots are not alike.

Knot Theory’s progression does not end there, such as with the development of high-degree polynomials, which stood for some time until better methods and invariants were later discovered by Vaughan Jones (Zhang). As mathematicians continued to add onto his theories, they now have enough features of a knot to search for to determine a knot’s uniqueness relatively quickly (Zhang). With the aid of computer algorithms classifying and comparing knots’ invariants, humanity has now discovered more than 350 million prime knots, which include all knots with crossing numbers up to 19 (Zhang). This does not stop mathematicians from continuing to study this topic, delving further into other types of knots, such as the published collection of proofs Knots and Links, which incorporate many more rings (Rolfsen).

Application & Conclusion

These added rings help apply Knot Theory to objects in the broader world, moving beyond the pure mathematics behind it. DNA is an interesting example because it is composed of a tangled mess which not only gets duplicated by enzymes, but also is physically separated into the two copies with little error (Strogatz). The enzymes essentially perform an illegal move in Knot Theory, which temporarily breaks one strand, passing it around another strand in the way, then reconnecting on the other side. Figuring out how to prevent these enzymes from duplicating DNA and therefore creating more cells is key to chemotherapy and antibiotics (Strogatz). Similarly, all materials form bonds similar to knots at the atomic level which can be studied to make stronger material from new knots which are less able to be untangled (Strogatz). Consider how the addition of Artificial Intelligence will allow for further application of these mathematical discoveries!

Tying it all up, Knot Theory is a thorough example of how something is studied and proven in mathematics. It demonstrates how mathematicians work together and build off of one another’s ideas. As more theorems are unlocked and connections made to other branches of mathematics and technology, the subject’s effectiveness rises increasingly. Similarly, its usefulness to the world is just beginning to open as scientists apply these ideas to real life solutions. What will Scouts be learning about knots one hundred years from now?

Sources

Gordon, Cameron McA. “KNOTS.” University of Texas, 29 Oct. 2019. Kauffman, Louis H., and Sofia Lambropoulou. “Hard Unknots and collapsing tangles.” Introductory Lectures on Knot Theory, 2011, pp. 187–247, https://doi.org/10.1142/9789814313001_0009. “Knot Theory.” Wikipedia, Wikimedia Foundation, 4 Sept. 2023, en.wikipedia.org/wiki/Knot_theory. Prevos, Peter. “Unknot Diagrams: The Art and Magic of the Trivial Knot.” The Horizon of Reason, The Horizon of Reason, 28 Sept. 2021, horizonofreason.com/science/unknot-diagrams-trivial-knot-collection/. Richeson, David S., and substantive Quanta Magazine moderates comments to facilitate an informed. “Why Mathematicians Study Knots.” Quanta Magazine, Simons Foundation, 31 Oct. 2022, www.quantamagazine.org/why-mathematicians-study-knots-20221031/. Rolfsen, Dale. Knots and Links. American Mathematical Society, 2004. Strogatz, Steven, and substantive Quanta Magazine moderates comments to facilitate an informed. “Why Knots Matter in Math and Science.” Quanta Magazine, Simons Foundation, 7 Apr. 2022, www.quantamagazine.org/why-knots-matter-in-math-and-science-20220406/. Zhang, Emily, director. How The Most Useless Branch of Math Could Save Your Life. YouTube, 3 Sept. 2023, https://youtu.be/8DBhTXM_Br4. Accessed 29 Oct. 2023.


2023