Just for fun, I doodled some knots and made a video.
So here is what I MEANT to say in the video before I wandered off into swirly knot land. The first knot I drew, called an "unknot," is a 1 by 1 square knot with 1 component (strand). The second one is 2 units wide by 2 units high and it has 2 components. The third is 3 by 3 with 3 components, and the fourth is 4 by 4 with 4 components. So, you might have noticed a pattern here. If you have a square knot of this sort, and it is p units wide by p units high, it will have p components.
The next one I drew was rectangular, 2 by 5, and it has 1 component. There's a theorem that says
that the number of components in a p by q knotwork panel is given by the greatest common divisor gcd(p, q). So applying this to the rectangular knot, gcd (2, 5) is 1, and the theorem works. If we apply this to square knots, when the panel is square, we get p = q and the gcd(p, p) = p.
Then I got bored with my lesson so I started doodling. Sorry, but I don't know any good math to go with the rest of the doodles.
I made this video with Doceri and lots of coffee. Too much coffee.