Space of all topological knots (tame and wild) 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.
 A: A "long topological knot" in $\mathbb R^n$ is a topological embedding $f : \mathbb R \to \mathbb R^n$ such that $f(x) = (x,0)$ for all $x \in \mathbb R \setminus (-1,1)$.  
Let $K_n$ be the space of all long topological knots in $\mathbb R^n$ with the compact-open topology.  Then $K_n$ is contractible.  The contraction is given by
$F : [0,1] \times K_n \to K_n$ defined by
$F(t,f)(x) = (1-t)f(\frac{x}{1-t})$ provided $t \in [0,1)$ and $F(1,f)(x) = (x,0)$. 
This map, $F$, is sometimes called "The Alexander Trick".  See the Wikipedia Alexander Trick page for context. 

In response to your edit, perhaps an interesting topology on $K_n$ could be given this way.  Given $f \in K_n$ and $\epsilon > 0$ we'll say an $\epsilon$-ball about $f$ consists of all knots $\phi \circ f$ where $\phi : \mathbb R^n \to \mathbb R^n$ is a homeomorphism which agrees with the identity map outside of $D^n$, and such that $|\phi(x)-x|<\epsilon$ for all $x \in \mathbb R^n$.  The topology on $K_n$ could be the topology generated by all $\epsilon$-balls about all $f \in K_n$.   Presumably this kind of topology has a name?
A: This comes 10 years late, but maybe useful to put it out there. A natural way to define the space of all topological knots, which is also equivalent (up to, say, Borel reducibility) with many other ways of defining it, is to look at knots as compact subsets of the 3-sphere and define the Hausdorff metric on them. The space of knots construed in this way is a Borel subset of $K(S^3)$, the compact space of all compact subsets of the 3-sphere. See this (my) paper for reference (page 4 of the PDF):
https://www.ams.org/journals/tran/2017-369-08/S0002-9947-2017-06960-9/
