Unravelling Knot Theory

Have you ever wondered about the intricate world of knots beyond tying shoelaces or earning Boy Scout badges? Knot theory is a branch of mathematics that tries to understand the fascinating complexities of knots.

What is a knot?

Mathematicians define a knot as the continuous embedding of a circle into three-dimensional space. Think of it as a piece of string tied into an intricate, tangled shape. What makes knot theory truly intriguing is that you can think of these knots just like real knots—meaning you can twist and turn them, but you can't cut, glue, or untangle them without physically passing one part through another.

Two of the simplest (and most well-known) knots are the ‘unknot’ and the ‘trefoil’ knot. The unknot is exactly what it sounds like—an untangled loop resembling a circle. The trefoil knot is the next simplest type of knot, looking like a set of three overlapping rings.

At this point, you may already be having a hard time visualizing these knots mentally, and they quickly become far more complex. To study knots more systematically, mathematicians use a tool called a "knot diagram". These diagrams are 2D representations of knots, projected from their 3D forms. While simplifying the knots into diagrams, they maintain crucial information about which part of the rope is on top during a crossing. In the images below, both the unknot (left) and trefoil (right) are represented as knot diagrams.

Knot Equivalence

One of the central challenges in knot theory is the "equivalence problem". Mathematicians use three precise operations called Reidemeister moves to determine whether two different knot diagrams represent the same knot or different ones.

Reidemeister Type I (R1): If there is an isolated twist in the knot, we can untwist it to simplify the knot.

Reidemeister Type II (R2): When two loops in the diagram are close together, we can place one loop over the other, introducing two new knot crossings. This transforms the diagram, but it preserves the knot's essential nature.

Reidemeister Type III (R3): When three threads cross over, with one on top of (or below) the other two, we can slide the top thread to the other side.

Preserving knot equivalence is a crucial aspect of Reidemeister moves. In other words, if you start with two different knot diagrams representing the same knot, you can connect them through a sequence of Reidemeister moves. The question then becomes, how many moves do we need to perform to transform one diagram into another?

Mathematicians Coward and Lackenby made a groundbreaking discovery in 2011, proving that the maximum length of a sequence of Reidemeister moves required to verify knot equivalence can be represented by an exponent tower of 10¹⁰⁰⁰⁰⁰⁰⁰ 2s. That is to say 2 to the 2 to the 2 to the 2 ... and so on 10¹⁰⁰⁰⁰⁰⁰⁰ times. Despite its staggering size, this number is computable, meaning that, theoretically, we can check, in a finite amount of time, whether two knots are equivalent. It's a testament to the power of mathematics and human ingenuity.

Applications of Knot Theory

While knot theory has its roots in algebraic topology, it finds applications in various fields, including molecular biology, where it can be used to study the structure of DNA and proteins, and in physics, where it's relevant to understanding the behavior of certain physical systems. Knot theory invites us to explore the beauty and intrigue of the intricate world of knots, reminding us that even in the most tangled of problems, there is order waiting to be discovered.

Further Reading:

Untangling the beautiful math of KNOTS : https://www.youtube.com/watch?v=pSgf9mJXn9o

An Upper Bound on Reidemeister Moves:

https://johncarlosbaez.wordpress.com/2018/03/09/an-upper-bound-on-reidemeister-moves/