I'm not sure what you mean here. This is discussing a 1-dimensional structure embeded in 4-dimensional space. If you're not sure it works for something else,

I'm not sure what you mean here. This is discussing a 1-dimensional structure embeded in 4-dimensional space. If you're not sure it works for something else, well, that isn't what's under discussion. If you just mean you're just unclear on the first step, of laying the knot out in 2D with crossings marked over/under, that's always possible after just some ordinary 3D adjustments. Although, yeah, if you asked me to prove it, I dunno that I could give one, I'm not a topologist... (and I guess now that I think about it the "finitely many" crossings part is actually wrong if we're allowing wild knots, but that's not really the issue)