For most people, knots are a common thing that they encounter on a regular basis; but in mathematics, an entire section of topology is devoted to the study of different types of knots. In this article, we will take a closer look at the basis of knot theory and also learn about its practical applications.
What is a knot and link?
While typical knots can be made by simply tying a rope, mathematical knots require a condition where the two ends of the knot must be joined, so it cannot be undone. The simplest type of knot in topology is just a rope with connected ends that forms a ring, called an unknot. More complicated knots consist of different types of crossings, for example trefoil and figure-of-eight knots (figure 1).
Figure 1
(In knot notations of the form Yₓ, Y indicates the crossing number, and X is an arbitrary index assigned to a specific knot with the same crossing number.)
2 or more knots that are intertwined are called links. The simplest non-trivial link is the Hopf Link, in which each of two components loops once around the other. If two knots are separated, they form the unlink (figure 2).
Figure 2
Reidemeister moves
Some knots that looks different at first glance are actually the same (figure 3):
Figure 3
The knots in figure 3 are the same, because they can be transformed into each other.
Such knots are formed using just three movements, called Reidemeister moves, these are (figure 4):
Figure 4
I. Twist (R1) move: allows either of the following replacements of a fragment of a link diagram to be made.
II. Poke (R2) move: allows either of the following replacements of a fragment of a link diagram to be made.
III. Slide (R3) move: allows the following replacement of a fragment of a link diagram to be made.
Tangled question about tangled knots
Since the beginnings of knot theory in the 19th century, more than 350 000 000 knots and links have been tabulated. However, scientists still can’t be sure if there are no repeating knots there. Nowadays, there are several ways to distinguish knots in some specific cases. One of these methods is tricolorability.
The essence of tricolorability is that we can color a knot using three colors, subject to certain rules:
1. At least two colors must be used, and
2. At each crossing, the three incident strands are either all the same color or all different colors.
Tricolorability can be used to distinguish between two different knots. For example, we can color a trefoil knot with three colors using these rules, but we cannot do this with a figure-of-eight knot (figure 5). It follows that these two knots are not equal:
Figure 5
However, this method is not suitable for all cases; mathematicians have not yet decided on a reliable universal way to distinguish nodes from each other. The algorithms that make this possible require enormous computations—so large that even supercomputers would take millennia to solve this problem.
Why do we need knot theory?
For a long time, knot theory was just a pastime for mathematicians, the so-called “knowledge for the sake of knowledge.” However, after a while, scientists found practical applications for it in various fields.
Biology: knots are used to examine the ability of topoisomerase enzymes to add or remove tangles from DNA.
Chemistry: knots allow us to describe the structure of topological stereoisomers, or molecules with the same atoms but different configurations.
Physics: graphs used in knot theory can be used to create Ising models for examining the way in which particles interact.
Works Cited
Wikipedia.org (2023). Knot theory [online] Available at: https://en.wikipedia.org/wiki/Knot_theory
Joy Lim and Eve Martin (2019). The Application of Knot Theory to Models in Biology and Physics [online]. Available at: https://math.mit.edu/research/highschool/primes/circle/documents/2019/Lim_Martin_2019.pdf
Scientific Russia. (2023). Tangled theory of tangled knots: knots and why they are needed [online]. Available at:
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