Knot Theory and Applications

Key Points

Many people tie their shoelaces incorrectly, leading to less secure knots.
There are two main ways to tie knots: counterclockwise and clockwise. The clockwise knot (square knot) is superior and more secure.
Exploration of knot tying introduces us to the field of knot theory, an area of mathematics focused on the study of different kinds of knots.

Knot theory is a branch of mathematics that aims to identify, categorize, and understand all possible knots.
There have been 352,152,252 known knots discovered, each with unique properties.
Knot theory is analogous to a periodic table of elements but for knots.

Knot theory is crucial for understanding the structure of proteins and DNA.
It has led to the development of stronger materials and aids in creating life-saving medicines.

Closed Loops: Mathematical knots exist on closed loops, meaning even if the ends are connected, the structure doesn’t fundamentally change.
Unknot: The simplest knot, essentially a circle with no nodes.
Trefoil: The simplest knot after the unknot, one that cannot be turned into a circle without breaking it.

Knot Equivalence Problem: Determining if two knots are the same or different. This problem has been challenging researchers for 150 years.
Famous controversies include Gordian Knot and endless knot recognized across various cultures and used symbolically.

19th Century: Tait, Kirkman, and Little tabulated the first several knots by hand.
Reidemeister Moves: Developed in 1927, comprising three moves (twist, poke, and slide) to transform identical knots into each other.
Algorithms: Created to distinguish between knots, including the computer algorithms developed by Wolfgang Haken in 1961.

Polynomial Invariants: Advanced methods like the Alexander and Jones polynomials help to uniquely identify and differentiate knots.
The discovery of the Jones polynomial earned Vaughan Jones a Fields Medal.
Computer Algorithms: Used to find and categorize millions of knots efficiently. Latest achievements include tabulating knots with up to 19 crossings.

Molecular Knots
- Jean-Pierre Sauvage tied the first synthetic knotted molecule in 1989.
- Increased complexity allows chemists to harness new properties for materials stronger than Kevlar.
DNA and Proteins
- Type II topoisomerase helps to untangle bacterial DNA, which has led to breakthrough antibiotics and chemotherapies.
- 1% of all proteins have knotted structures essential for their function.
Everyday Use
- Improved understanding of knot stability can help in optimizing the way we tie shoelaces, coil wires, etc.

Raymer and Smith Study (2007): Studied knot formation in a controlled environment, leading to practical advice on preventing tangles in cords.

Knot theory's applications have expanded from theoretical pursuits to practical tools that are crucial in technology, medicine, and other scientific domains.
It emphasizes the importance of understanding fundamental concepts and how they can apply to diverse, real-world problems.