Deformations that are continuous functions and knots

Question

Is typical to illustrate the idea of a continuous transformation by saying something like 'a deformation that don't tear or glue parts of your object'. Seeing those beautifully done videos on the sphere eversion, these transformations seem permissive enough to allow some parts of your manifold to pass through each other (like in this part of Outside In). But if this 'passing-through' is allowed for a continuous function how can we differentiate knots? i.e. taking the trefoil knot for example, isn't what keeps it non-homeomorphic to the circle isn't just the fact that we can't pass some lines through each other (and so changing the number of crossings)?

Answer 1

The sphere transformation in the video is apparently called a regular homotopy of immersions.

The transformations allowed in knot equivalence are called ambient isotopies. These are similar but, as @Qiaochu says, they are stricter. In particular, an isotopy must be an embedding at each moment, which does not allow self-intersections. An immersion does allow the shape being moved to intersect itself, and in the case of the sphere eversion it certainly does.