Let's consider the space of long knots in $\mathbb R^n, n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I think the last result is about convergency in $E^1$ term (http://palmer.wellesley.edu/~ivolic/pdf/Papers/VassilievCollapseFinal-G%26T.pdf).

But my question is about what these cohomologies are precise. So, is it true that $H^1(long\ knots\ in\ \mathbb R^4)=0$ ?

Does there exist some table with $H^i(space\ of\ long\ knots\ in\ \mathbb R^j)$ at least for small $i,j$ ?

allembeddings, as a function space. I think you might be interpreting the space as the knot complement. – Ryan Budney Nov 14 '11 at 17:51