what i mean is, from a mathematical perspective, knots are interesting because of the ways you can “untie” them. you formalize knots as “ways of putting a circle (S^1) into 3d space (R^3)” which is really the same as a continuous, injective function f:S^1 -> R^3. this also sets you up to define “untying” as an “ambient isotopy” between such functions. Once you pin all these things down, you can notice that the notion of ambient isotopy of such functions is nontrivial for all f:S^n -> R^n+2, ie when you are looking at the sphere (meaning the surface of a sphere) in 4 dimensions, or the higher dimensional analogs of the sphere in the space 2 dimensions up (using the topological notion of dimensions where the so called 3d sphere is 2d, because it locally looks like R^2). the slogan i was taught in my topology class was “knots are a co-dimension 2 phenomenon “
this is the extent of what i know about knots, and i dont know any physics. So there may be something special about 3d that has to do with knots (or their role in physics), but the existence of knots does not distinguish R^3 from the other R^n on its own!