Introduction to Knot Theory

By Jia Hao Liang

Have you ever wondered why some knots you tie seem to untangle so easily? That may be because the knot you tied is actually an unknot—a knot that is simply a plain loop. By pulling and adjusting, you’ve effectively transformed your knot back into this trivial loop. This naturally raises a question: how can we find a way to distinguish knots from the unknot? Further, how can we identify and categorize every distinct knot? These challenges form the heart of the knot equivalence problem, the central focus of knot theory, a branch of topology (the study of shapes under continuous change) devoted to the mathematical study of knots.

Figure 1: Different knots (Schaufelberger, 2020)

To understand the complexity of the knot equivalence problem, we first need to make sense of the concept of knot equivalence: Two knots are considered equivalent if one can be transformed into the other through a sequence of Reidemeister moves, which are simple modifications that do not change the main structure of the knot. There are three types of these moves: a twist, which adds or removes a loop in a strand; a poke, which moves one strand over or under another; and a slide, which shifts a strand across a crossing of two others. Surprisingly, even knots that appear extremely complicated—such as the famous Thistlethwaite unknot—are found to be equivalent to the trivial unknot once simplified through these moves.

Figure 2: Thistlethwaite Unknot (Blair et al., 2017)

To approach the larger problem of classification, mathematicians have developed various ways to assign each knot a kind of identification. Just as people can be categorized by features like height, birthday, or blood type, knots can be classified using specific properties. Two of the most important of these categorization methods are colorability and polynomial invariants.

Colorability is an elegant and intuitive concept in knot theory. Imagine trying to distinguish between two black-and-white subway maps with no labels—just lines. The differences become much more obvious once each segment is colored: different maps may require a different number of colors to be distinguished. Knot colorability, as the name implies, works in a similar manner. If you can assign each strand of the knot one of “p” colors, following two rules—at every crossing, either all three strands have the same color, or the sum of two strands is congruent modulo p to the third (the difference of the sums are divisible by p)—then the knot is p-colorable. Using this colorization, knots can be distinguished. For example, the unknot is not colorable, because there are no intersections, hence you cannot color a loop with more than 1 color, while the famous trefoil knot is 3-colorable. That being said, there are also limitations to this method, as both the left and right trefoil are 3-colorable, but they are not the same knot.

Figure 3: The left and right trefoils (Distinguishing the left-hand, 2014)

Polynomials are a more comprehensive, albeit more complicated, way to distinguish knots. Each knot is assigned a polynomial (for example, the Jones knot polynomial for the knot on the cover will be 1-t-1+3t-2-3t-3+3t4-4t-5+3t-6-2t-7+t-8, computed by analyzing its crossings. Since polynomials allow for far greater variability than colorability, they provide a much more precise method of classification. The first polynomial invariant—the Alexander polynomial—was introduced by James Waddell Alexander II in 1923. In 1984, Vaughan Jones introduced an even more powerful invariant, the Jones polynomial, which earned him the Fields Medal, the highest honor in mathematics. His work was also influenced by Columbia professor Joan Birman, a leading topologist. Later, the HOMFLY-PT polynomial, which involves two variables, was developed independently by eight mathematicians—its name formed from the initials of their surnames. Although these polynomial invariants significantly refine the classification of knots, there still exist distinct knots that share the same polynomial “ID.”

Figure 4: The knot on the cover of this article (2025)

Currently, the methods used to identify different knots are either not specific enough, like colorability, or require algorithms that take astronomical amounts of time to compute, as with polynomial invariants. This leaves us in a situation similar to being able to distinguish people by names or birthdays separately, but lacking a tool that can capture every property of a person at once. Regardless, we should not feel discouraged: in 1954, Alan Turing, the father of computer science, described the knot equivalence problem as an “undecidable problem.” Since then, mathematicians have established an upper bound for determining knot equivalence—though the bound remains extraordinarily large (to be precise, it is exp(cn)(n), c=101,000,000  reidemeister moves). Therefore, the crucial leap from zero to one has already been made, and the field now strives to push beyond one toward more efficient and comprehensive solutions.

Knot theory might seem abstract, but it has powerful applications across many fields, ranging from the study of molecular structures in chemistry to advances in material science, and even to understanding DNA and protein folding. For example, in 1989, chemists synthesized the first molecular knot—a trefoil knot made from long organic molecules coordinated around chloride ions as a template. This breakthrough showed that knot theory is not just theoretical, but can directly guide the design of new chemical compounds whose unique properties arise from their knotted molecular structures. Therefore, developing more efficient and comprehensive invariants remains a central goal in the field, as knot theory continues to reveal both the beauty of pure mathematics and its greatness when applied to real-world problems.

References:
Alexander, J. W. (1928). Topological Invariants of Knots and Links. Transactions of the American Mathematical Society, 30(2), 275–306. https://doi.org/10.2307/1989123

Blair, Ryan & Kjuchukova, Alexandra & Ozawa, Makoto. (2017). The incompatibility of crossing number and bridge number for knot diagrams. Discrete Mathematics. 342. 10.1016/j.disc.2019.03.013.

Dietrich-Buchecker, C., & Sauvage, J.-P. (1989). A Synthetic Molecular Trefoil Knot. Angewandte Chemie, 28(2), 189–192. https://doi.org/10.1002/anie.198901891

Distinguishing the left-hand trefoil from the right-hand trefoil by colouring. (2014, January 24). Low Dimensional Topology. https://ldtopology.wordpress.com/2014/01/24/distinguishing-the-left-hand-trefoil-from-the-right-hand-trefoil-by-colouring-them/

Jones, V. F. R. (1985). A polynomial invariant for knots via von Neumann algebras. Bulletin of the American Mathematical Society, 12(1), 103–111. https://doi.org/10.1090/s0273-0979-1985-15304-2

Lackenby, M. (2015). A polynomial upper bound on Reidemeister moves. Annals of Mathematics, 491–564. https://doi.org/10.4007/annals.2015.182.2.3

Millett, K. C., & Panagiotou, E. (2023). HOMFLY-PT polynomials of open links. Journal of Knot Theory and Its Ramifications. https://doi.org/10.1142/s0218216523400175

Schaufelberger, F. (2020). Open questions in functional molecular topology. Communications Chemistry, 3(1). https://doi.org/10.1038/s42004-020-00433-7

Turing, A. (1954). Solvable and Unsolvable Problems (1954). https://www.ivanociardelli.altervista.org/wp-content/uploads/2018/04/Solvable-and-unsolvable-problems.pdf

​(2025). Knotfol.io. https://knotfol.io