Does anyone know something about the space of all topological knots (injective continuous maps from $S^1$) in $\mathbb R^3$ (or in some manifold)?

In addition, what is known about wild knots?

I found only one wonderful fact:

"Montesinos also proved that there exists a universal wild knot, i.e. every closed orientable 3-manifold is a 3- fold branched covering of $S^3$ with branched set a wild knot. This shows how rich the wild knot theory can be." (http://arxiv.org/pdf/math/0509124v1)

Added: I mean the situation same as finite-type invariant theory for tame knots - homotopy type of injective part of this space (I found nothing about it) or homotopy type of topological knots with self-intersections. So, any general-topological properties of singular knot (knot with self-intersection in this context) neighborhood is interesting too.

Added2: Ryan, thanks for clarification! Theo is right, question is "what is the correct topology on the space of topological knots for which knot theory is interesting". Usually only piecewise-smooth knots are studied.

4more comments