# Knot Theory’s Impact Across Disciplines

This article is based on the video titled “How The Most Useless Branch of Math Could Save Your Life” by Veritasium. The video gets into the complexities and applications of knot theory, a mathematical discipline that has evolved from a failed 19th-century theory of atoms to a versatile tool in modern science and technology.

## The Genesis of Knot Theory

Knot theory originated from Lord Kelvin’s Vortex theory, which proposed that atoms were knotted vortices in the ether. Peter Guthrie Tait, a mathematician, took up the task of tabulating knots to support this theory. Despite the theory’s eventual failure, Tait’s work laid the foundation for modern knot theory (which is quite ironic given that the theory was initially considered a failure).

## Classifying The Basics

Knots are classified based on their “crossing numbers,” the number of times the string crosses over itself. The simplest knot is the “unknot,” a loop with zero crossings. Tait began his tabulation with the “trefoil,” a three-crossing knot, and continued to classify knots with increasing complexity. He also introduced the concept of “prime knots,” which are knots that cannot be decomposed into simpler knots (much like prime numbers in arithmetic).

## The Challenge of Knot Equivalence

One of the central problems in knot theory is determining if two knots are equivalent. German mathematician Kurt Reidemeister proposed that any two identical knots can be transformed into each other using three types of moves: the twist, the poke, and the slide. While this was a significant advancement, it still left the question of how to prove that two knots are different.

## Invariants: The Identifiers of Knots

Invariants are properties of a knot that remain unchanged regardless of how the knot is twisted or tangled. One such invariant is “tri-colorability,” which involves coloring a knot diagram with three colors while following specific rules. Another powerful invariant is “P-colorability,” where P can be any prime number. These invariants help in distinguishing knots that appear similar but are fundamentally different.

The Alexander polynomial and the Jones polynomial are mathematical expressions that serve as invariants for knots. These polynomials are highly specific and can distinguish between a wide variety of knots. The discovery of these polynomials has been a significant milestone in knot theory, leading to further advancements like the HOMFLY polynomial.

## Knot Theory in the Real World

Knot theory has found applications in various fields, from creating synthetic knotted molecules to understanding DNA replication in bacteria. Inhibiting an enzyme called “type 2 topoisomerase” in bacteria, which is responsible for unknotting DNA, serves as the basis for some of the most common antibiotics. Similarly, knot theory is used in chemotherapy treatments to inhibit human topoisomerases, thereby killing rapidly dividing cancer cells.

## Everyday Knots and Practical Applications

Research has shown that the way we tie our shoelaces can be analyzed using knot theory. For example, a square knot is more secure than a granny knot. The study of how knots form in everyday objects like headphone wires has even led to patents for stiffer earbud wires.

## From Obscurity to Ubiquity

What began as a speculative theory about the nature of atoms has evolved into a mathematical discipline with applications in diverse fields. From its humble beginnings, knot theory has grown to become an indispensable tool in modern science, proving that even the most abstract mathematical concepts can have practical applications in the real world.

Knot theory, once a mathematical curiosity with no apparent practical use, has evolved into a versatile tool that impacts multiple disciplines ranging from chemistry to medicine. This article has traced the journey of knot theory from its origins in 19th-century atomic theory to its modern applications in creating new materials and life-saving drugs. The field has demonstrated its value in solving real-world problems, making it a compelling subject not just for mathematicians but for anyone interested in the interconnectedness of science and everyday life (like how the way you tie your shoelaces can be a subject of scientific inquiry).