In part because once you’ve connected the ends of your string the knot you’ve got is the knot you’ve got, and that invariance is very attractive to math folk. It turns out that if you have an ordinary knot, you can use it to create a higher dimensional knot. That’s not a knot (knot at all). And yes: you can keep going into higher and higher dimensions using the same idea. The spun knot picture was lifted remorselessly from the second paper mentioned earlier.