[altrego99]

Take X = all continuous well behaved 1-1 mapping f from [0,1] to R^3, with f(0)=f(1).

A knot is an equivalence class of such functions, equivalence defined as given two functions f and g, you can "morph" one into other continuously - i.e. if you can find a parametrized well-behaved (i.e. continuous, differentiable etc.) function h(x;t) s.t. h(x;0)=f(x) and h(x;1)=g(x) and h(x;t_ is in X for all t.

It's pretty much the definition you'd come up with too.