Knot Theory

Veritasium published an interesting video touching on Knot Theory.

Turns out this theory is quite complicated.

Some learnings I got out of it include:

To study knots, they are made into closed loops so that they don’t untangle.
The only way to make one knot into another is to break the loop, make changes and rejoin it again.
There’s a periodic table for knots! In that table, only prime knots are listed. This is because you can take any two knots and join them together to form componud knots.
A knot can be simplified into its reduced form.
Two knots are the same if you can translate one to the other using only Reidemeister moves. There are three such moves:
- twising
- poking
- sliding
Upper bounds of Reidemeister moves tell whether two knots can be translated into one another (assuming that they are equivalent)
It’s very hard to tell if two knots are the same.
Knots have invariances
- Crossing number (of the reduced form) is one
- There’s a whole set of knot polynomial algebra that can be used to categorize knots
  - A new polynomial that was discovered led to the author gaining a Fields medal
- There’s also p-colourability to distinguish knots from each other

Applications

Knots are useful in many real-world applications from