Think of a knot as a twisted up loop of string. You can draw this as a picture with a line representing the string and the line breaking when one part of the string goes underneath another. Knot theory is the study of the topological properties of knots, and has been around since the late 1700s. In the 1860s Lord Kelvin conjectured that atoms were knots of aether and this led Peter Guthrie Tait to start drawing classification tables for knots. In fact Peter Guthrie Tait’s original research into the theory of these knot diagrams is said to have led to the birth of topology, now one of the main fields of pure mathematics. Here are some knot drawings from Tait’s classification of knots:
you can find some more here.
I was recently on a choir trip in Princeton and I decided to go to some math(s) classes. I went to a class by Peter Ozsváth titled ‘Algebraic Topology’ in which he taught the snake lemma and 5-lemma extremely well (lots of coloured chalk and enthusiastic diagram chasing). He then applied them to something the class had been learning: grid homology of knots. I wont try to explain this all in one post but I’ll show you how a grid diagram relates to a knot.
A grid diagram is a square grid, with boxes either filled in with an X, an O, or left blank such that:
- every column has exactly one X and one O,
- every row has exactly one X and one O.
The number of boxes in each row/column is called the grid number of the grid diagram. When all the Xs and Os are in place we join the X and O in the same row and the X and O in the same column with lines, letting the horizontal lines cross under the vertical lines. We will do an example to show how this represents a knot!
Example : a grid diagram with grid number 7.
Start with a grid (7 by 7) with Xs and Os as specified:
we see that really it looks like a very simple sudoku.
Join the Xs and Os with lines in each row/column, letting the horizontal lines cross under the vertical lines:
Forget the grid (!):
Forget the Xs and Os (!):
Morph to make it look a bit more ‘knotty’:
So there we have an example which illustrates the general transition from knot diagram to knot! You can trace this backwards to get a knot diagram for every knot but this was is a bit trickier as you have to get the Xs and Os to be one per row/column. In fact many different looking knot diagrams represent the same knot. Next time I’ll explain the grid homology you can define with these diagrams.
Below are some references for those interested: