so knots do exist

so we start off with a knot, a single continuous closed loop in 3 dimensinoal euclidean space.
two knots and the same is one can be transformed into the other without breaking the loop or doing anything dodgy. generally in knot theory we work with pictures of knots (knot diagrams) rather than the knots in 3d because it's easier and it's basically the same thing since we can draw a picture of any knot so that we could make it again in real life. with pictures, we get crossings (the point on the picture which correspond to teo points on the knot) whereas in 3d we don't. crossing are very important in knot theory i think because thats basically what makes them different. With knots diagrams we know there are 4 things you can do to a knot diagram to change the diagram in a way so that it still represents the same knot. these are called the reidemeister moves, r0 is basically bending and stretching the knot without changing any crossings, r1 is adding or removing a single twist on a single isolated strand on the knot. r2 is letting one strand poke over or under another and the reverse and r3 is letting a strand slide across a crossing. these 4 moves are enough for any transformation. we can prove two knots are the same simply by changing one into the other but to prove they are different requires some cleverness. this is where the reidemeister moves come in. if we can find(create? discover?) properties of the knot diagram which only depend on the actual knot the diagram is of, no matter how the diagram is drawn then we can use these properties to show that the knot are actually different, these are called invariants, we know that the only things you can do to a knot diagram are those 4 moves and so to check if something is and invariant we simply need to check that those 4 moves don't change the invariant and thats it. obviously there are trivial invariants such as let x=0 for all knots. thats quite useless. but there are useful ones. ones such as tricolourability, if you can colour a knot with 3 colours where at each crossing you use either 1 colour or all 3 colours then the knot is tricolourable. the unknot ie the knot which isn't knotted, ie just a circle is not tricolourable because it can't be coloured with all 3 colours no matter how you draw it, whereas the trefoil (a knot with 3 crossings) can be tricoloured any way you draw it. so this proves that the trefoil is actually different to the unknot, and so not every knot is an unknot and hence knots exist!