I think there's a close correspondence though. What you naively think of as a knot is a mathematical knot if you join the ends together, and I think any given mathematical knot can give rise to a regular knot.

I know very little about it other than a friend of mine did a PhD in it, but it gets weird fast. Like they often consider it in canonical forms that look nothing like a practice knot, and then of course they start showing correspondences with polynomials &c.

Which is why a math knot isnt the same as a realworld knot. Hitches and bends are also not knots. The word is very overused.

That's where it principally diverges from physics - in reality a knot is about creating friction and wrapping a rope a few times around a tree can for some practical purposes achieve the same thing.

To me the difference is about things like friction, diameter of the rope, and the physical reasons behind why a knot resists binding, tightening, and so on. The fact that mathematical knots don't have ends isn't that important. If you have two ends of a rope and so long as those ends never move into the knot's "territory", then its dynamic the same as a circle knot. Isn't it?

> So, the physical knots you might think of being useful in various situations have only somewhat of a relationship to mathematical knot theory.

No, you can map real world knots to mathematical knots. It is explained in the video, with common examples. You can also map the other way around, but there's way more mathematical knots (prime and non-prime) than people could ever need.