I don’t mean homotopy necessarily in terms of knot theory, I mean it in terms of a smooth (or Ck for some k) map from R x [0,1] to R^3 where f(R,t) is a curve, f(R,0) is bunny ears and f(R,1) is whatever the other method is called.
Yes, in general, this is weaker than what one has in knot theory. What I think you want is actually a slightly stronger claim than even a knot theory homotopy. I think the claim you want is that it satisfies all of that, and one has that has such that the projections onto some plane are strongly equivalent in the sense that one can get from one to the other without any Reidemeister moves.
As to the status issue, I rarely tie my shoes in front of people, or see other people tying their shoes. I guess I could imagine it being a problem in high school gym class?
If this thread is accurate some people can distinguish how the knot was tied by subtle cues. I know that a lot of people claim that there’s all sorts of status junk connected to shoes. I’m skeptical that it exists nearly as much as some people claim but if one does care about such potential environments it might matter.
one has that has such that the projections onto some plane are strongly equivalent in the sense that one can get from one to the other without any Reidemeister moves.
Yes, in general, this is weaker than what one has in knot theory. What I think you want is actually a slightly stronger claim than even a knot theory homotopy. I think the claim you want is that it satisfies all of that, and one has that has such that the projections onto some plane are strongly equivalent in the sense that one can get from one to the other without any Reidemeister moves.
If this thread is accurate some people can distinguish how the knot was tied by subtle cues. I know that a lot of people claim that there’s all sorts of status junk connected to shoes. I’m skeptical that it exists nearly as much as some people claim but if one does care about such potential environments it might matter.
Yes, I think this is what I intended.