math knots and real knots aren't the same thing and I don't think all possible human tieable knots have been enumerated and classified although I am happy to be pointed to evidence.

Math knots embed circles while real knots are typically made with free ended ropes (although some knots are not). Math knots ignore friction and the width of the rope; real knots can't ignore friction and the width (and other phyhsical properties) matter. See the comment in this article https://en.wikipedia.org/wiki/Overhand_knot

While researching my answer I found out there's a term for what I wanted to do, https://en.wikipedia.org/wiki/Physical_knot_theory

I've had this discussion with mathematicians a few times now and they don't see the difference, so maybe I'm just missing something important.