So why on earth would we want to study knots?! Well, if you dont mind me offloading some work from here go watch that video I mentioned earlier. However I can highlight some of the motivation I had to read about knots and their theories.

Let's start with trying to draw some knots, but lets think about it for a second. Our first attempt of a knot on a page might be some tight, scrambled looking mess with a start and end of our rope, far from our knot.

Lets try to expand them to highlight some detail in their turns and folds.

Now we have something more appealing, with some symmetry perhaps, though not very symmetric to any other part, but smooth in shape. Still our rope looks kind of bad. We think of a rope in 3 dimensions after all so lets add some "dimensionality" to our rope by removing bits of line as our rope goes under and over itself. You may think of the solid line as a bit of rope passing over another, while the slighlty erased line may imply the rope passing below the other.

If we would like to keep moving about little sections of this rope, we could move the start and end of our rope or string and untie the knot, rather sad, because we want to keep these knots in our structure we created. So we join start and end together to kind of "seal" this knot within our rope.

And so we end up with what is known as a "knot diagram". Pretty cool eh? Reviewing this paragraph I realised I didn't provide much motivation for where you may start to think about knots in the first place, but in the end I find the continiuity and symmetry of these knots rather interesting and fun to look at, trying to decipher what kind of knot it is or what it may look like. Then again classification of knots is THE problem in knot theory so, the more you know.

The following is a little collection of prime knots.

First some simple ones

Slighlty more complex ones with 9 crossings. I put these in here as an example of how for knots with a higher amount of crossings you need to get a bit more creative one how to show the bloody knot. So many twists and turns.

We are mathematicians here after all so, what rules do we have for moving these mathematical knots around? There is a set of operations called the Reidermeister Moves, which represent the ways you can arrange real knots. These allow us to call different knot diagrams I illustrate them below. Try convincing yourself that you could preform these on a real physical knot.

**(1)** The first one can be thought of as twisting the strand about an axis.

**(2)** In this one, think one of the two strands being ontop the other and you pull both apart.

**(3)** Finally, here two strands crossing, and the third horizontal one is in this case below both strands.

Thats cool and all but what are we going to do with this? With these knot bending rules established we can now try to show what knots are quivalent under these relations, with the aim of reducing each knot to its simlpest form, a prime knot. For the non mathematically minded here, I use the word 'prime' to encourage thinking of these knots as the most fundamental form of knot, or like building blocks of knots, just like we have prime numbers and each number having a prime number decompostition.

Here I'll show an example of, what we can call, unkotting a knot using the reidermeister moves. Here I use the first knot we seen on this page which was drawn arbitrarily and happens to be an unknot!